4.2: Quadratic Functions
- Recognize characteristics of parabolas.
- Understand how the graph of a parabola is related to its quadratic function.
- Determine a quadratic function’s minimum or maximum value.
- Solve problems involving a quadratic function’s minimum or maximum value.
Try these questions prior to beginning this section to help determine if you are set up for success:
-
Simplify:
- \(\quad \sqrt{-9}\)
- \(\quad \sqrt{-24}\)
-
Solve:
- \(\quad 7h^2+15=26h\)
- \(\quad 3y^2-10=0\)
- \(\quad 4x^2+7=5x\)
- Answer
-
-
- \(\quad 3i\)
- \(\quad 2i\sqrt{6}\)
If you missed this problem or feel you could use more practice, review [ 2.23: Complex Numbers ]
-
- \(\quad h=3, \dfrac{5}{7}\)
- \(\quad y=\pm\dfrac{\sqrt{30}}{3}\)
- \(\quad x=\dfrac{5\pm i\sqrt{87}}{8}\)
If you missed this problem or feel you could use more practice, review [ 2.8: Solving Polynomial Equations , 2.24: Solving Quadratic Equations Using the Square Root Property , 2.25: Solving Quadratic Equations by Completing the Square , and 2.26: Solving Quadratic Equations Using the Quadratic Formula ]
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Curved antennas, such as the ones shown in Figure \(\PageIndex{1}\), are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The shape used in many antenna and satellite designs are called paraboloids which are studied in multivariable calculus. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function.
Figure \(\PageIndex{1}\): An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr)
In this section, we will investigate quadratic functions, which frequently model problems involving area and projectile motion. Working with quadratic functions can be less complex than working with higher degree functions, so they provide a good opportunity for a detailed study of function behavior.
Characteristics of Parabolas
The graph of a quadratic function is a U-shaped curve called a parabola. One important feature of the graph is that it has a local extrema point, called the vertex . If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the axis of symmetry . The y-intercept is the point at which the parabola crosses the \(y\)-axis. The x-intercepts are the points at which the parabola crosses the \(x\)-axis. These features are illustrated in Figure \(\PageIndex{2}\).
Determine the vertex, axis of symmetry, x-intercepts, and y-intercept of the parabola shown in Figure \(\PageIndex{3}\).
Solution
The vertex is the lowest point of the graph. We can see that the vertex is at \((3,1)\). The axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is \(x=3\). This parabola does not cross the x-axis, so it has no x-intercepts. It crosses the \(y\)-axis at \((0,7)\) so this is the y-intercept.
We have discussed transformations of functions. Given the function \(f(x)=3(x+2)^{2} -13\), we know that the parabola open upward and that the vertex is at \((-2,-13)\). With quadratics we sometimes call this transformation form of the function vertex form .
Now, consider what happens when we expand this function out:
\[\begin{array}{l} {f(x)=3(x+2)^{2} -13} \\ {f(x)=3(x+2)(x+2)-13} \\ {f(x)=3(x^{2} +4x+4)-13} \\ {f(x)=3x^{2} +12x+12)-13} \\{f(x)= 3x^{2} +12x-1}
\end{array} \nonumber \]
This expanded form, \(f(x)= 3x^{2} +12x-1 \), is commonly called general form of a quadratic function. (Note that this terminology is not universal but we will refer to this as general form throughout this text).
Also, notice that the leading coefficient of 3 was the same value in the vertex and general form of the function. This value tells us whether the graph has a vertical stretch or compression as well as an x-axis reflection.
The general form of a quadratic function is \(f(x)=ax^2+bx+c\) where \(a\), \(b\), and \(c\) are real numbers and \(a{\neq}0\).
- If \(a>0\), then the parabola opens upward.
- If \(a<0\), then the parabola opens downward.
Next, how do we determine what the vertex of a quadratic function is when the function is given in general form? To do so, we rewrite the function in vertex form by completing the square. Let's look at an example of this:
Given the function \(f(x)=x^2+2x+4\), begin by grouping the variable terms together, \[f(x)=(x^2+2x)+4 \nonumber \] Then complete the square by taking half of the linear term coefficient and the squaring it, \((\frac{1}{2}(2))^2=1\). Next, add and subtract this value to the function, \[f(x)=(x^2+2x+1)-1+4 \nonumber \]
Now factor and simplify to get the vertex form, \[f(x)=(x+1)^2+3\nonumber \]
Completing the square is a very useful skill for writing quadratic functions in vertex form. It is also useful for writing many other equations in more useful forms in calculus. Let's look at another example using this important skill.
Rewrite \(f(x)=2x^{2} -12x+14\) into vertex form by completing the square and identify the vertex.
Solution
We start by factoring the leading coefficient from the quadratic and linear terms.
\[2(x^{2} -6x)+14 \nonumber \]
Take half of the linear coefficient, -6, which is -3, then square to get (-3)\({}^{2}\) = 9. \[(x^{2} -6x+9)=(x-3)^{2} \nonumber \]
Now, we can’t just add 9 into the expression – that would change the value of the expression. In fact, adding 9 inside the parentheses actually adds 18 to the expression, since the 2 outside the parentheses will distribute. To keep the expression balanced, we can subtract 18.
\[2(x^{2} -6x+9)+14-18 \nonumber \]
Simplifying, we are left with vertex form.
\[2(x-3)^{2} -4 \nonumber \]
The vertex is \((3,-4)\).
The vertex form of a quadratic function is
\[f(x)=a(x-h)^{2} +k \nonumber \]
The vertex of the quadratic function is located at (\(h\), \(k\)), where \(h\) and \(k\) are the numbers in the transformation form of the function.
In the previous example, we saw that it is possible to rewrite a quadratic function in vertex form by completing the square. It would be useful to reverse this process since the vertex form reveals the vertex.
Expanding out the vertex form of a quadratic gives:
\[\begin{array}{l} {f(x)=a(x-h)^{2} +k} \\{f(x)=a(x-h)(x-h)+k} \\ {f(x)=a(x^{2} -2xh+h^{2} +k} \\{f(x)=ax^{2} -2ahx+ah^{2} +k} \end{array}\ \nonumber \]
This is equivalent to the general form of the quadratic:
\[ax^{2} -2ahx+ah^{2} +k=ax^{2} +bx+c\ \nonumber \]
The second degree terms are already equal and have the same labels. For the linear terms to be equal, the coefficients must be equal. So,
\[-2ah=b \nonumber \]
then solving for h we get: \[h=-\dfrac{b}{2a}\ \nonumber \]
Likewise, set the constant terms equal,\[ah^{2} +k=c \nonumber \]
Solving for k we get: \[k=c-ah^{2} =c-a\left(-\dfrac{b}{2a} \right)^{2} =c-a\dfrac{b^{2} }{4a^{2} } =c-\dfrac{b^{2} }{4a}\ \nonumber\]
In practice, though, it is usually easier to remember that \(k\) is the output value of the function when the input is \(h\), so \(k=f(h)\).
Given a quadratic function in general form, \(f(x)=ax^2+bx+c\), the vertex is \((h,k)\) where
\(h=-\dfrac{b}{2a}\ \) and \(k=f(h)\)
Find the vertex of the quadratic \(f(x)=2x^{2} -6x+7\).
Solution
The x-coordinate of the vertex will be at
\[h=-\dfrac{b}{2a} =-\dfrac{-6}{2(2)} =\dfrac{6}{4} =\dfrac{3}{2} \ \nonumber \]
The y-coordinate of the vertex will be at
\[f(\dfrac{3}{2} )=2(\dfrac{3}{2})^{2} -6(\dfrac{3}{2} )+7=\dfrac{5}{2} \ \nonumber \]
The vertex is \(\left(\dfrac{3}{2}, \dfrac{5}{2}\right)\)
Intercept Behavior
As with any function, we can find the y-intercepts of a quadratic by evaluating the function at an input of zero. All quadratics functions have one y-intercept since the domain is all real numbers. We can find the x-intercepts by solving for when the output will be zero. Notice that depending upon the location of the graph, we might have zero, one, or two horizontal intercepts.
Find intercepts of the quadratic \(f(x)=3x^{2} +5x-2\)
Solution
We can find the y-intercept by evaluating the function at an input of zero:
\[f(0)=3(0)^{2} +5(0)-2=-2\ \nonumber \] Vertical intercept at (0,-2)
For the x-intercepts, we solve for when the output will be zero
\[0=3x^{2} +5x-2\ \nonumber \]
In this case, the quadratic can be factored easily, providing the simplest method for solution
\[0=(3x-1)(x+2)\ \nonumber \]
\[\begin{array}{l} {0=3x-1} \\[4pt] {x=\dfrac{1}{3} } \end{array} \qquad\text{ or }\qquad\begin{array}{l} {0=x+2} \\[4pt] {x=-2} \end{array}\ \nonumber \]
x-intercepts at \(\left(\dfrac{1}{3} ,0\right)\) and (-2,0)
Notice that in the standard form of a quadratic, the constant term c reveals the vertical intercept of the graph.
Find the x-intercepts of the quadratic \(f(x)=2x^{2} +4x-4\)
Solution
Again we will solve for when the output will be zero
\[0=2x^{2} +4x-4\ \nonumber \]
Since the quadratic is not easily factorable in this case, we solve for the intercepts by first rewriting the quadratic into transformation form.
\[h=-\dfrac{b}{2a} =-\dfrac{4}{2(2)} =-1\ \ \ k=f(-1)=2(-1)^{2} +4(-1)-4=-6\ \nonumber \]
\[f(x)=2(x+1)^{2} -6\ \nonumber \]
Now we can solve for when the output will be zero
\[\begin{array}{l} {0=2(x+1)^{2} -6} \\ {6=2(x+1)^{2} } \\ {3=(x+1)^{2} } \\ {x+1=\pm \sqrt{3} } \\ {x=-1\pm \sqrt{3} } \end{array}\ \nonumber \]
The graph has x-intercepts at \((-1-\sqrt{3} ,0)\)and \((-1+\sqrt{3} ,0)\)
For the function \(g(x)=13+x^{2} -6x\), find the y and x-intercepts.
- Answer
-
The y-intercept at (0, 13), There are no x-intercepts since the vertex is above the \(x\)-axis and the graph opens upwards.
Note that if there are no x-intercepts, it is important to explain how you determined that.
The process for finding the x-intercepts is done commonly enough that sometimes people find it easier to solve the problem once in general and remember the formula for the result, rather than repeating the process each time. Based on our previous work we showed that any quadratic in general form can be written into vertex form as:
\[f(x)=a \left(x+\dfrac{b}{2a} \right)^{2} +c-\dfrac{b^{2} }{4a}\nonumber \]
Solving for the horizontal intercepts using this general equation gives:
\[0=a \left(x+\dfrac{b}{2a} \right)^{2} +c-\dfrac{b^{2} }{4a}\ \nonumber \] start to solve for \(x\) by moving the constants to the other side
\[\dfrac{b^{2} }{4a} -c=a \left(x+\dfrac{b}{2a} \right)^{2}\ \nonumber \] divide both sides by \(a\)
\[\dfrac{b^{2} }{4a^{2} } -\dfrac{c}{a} = \left(x+\dfrac{b}{2a} \right)^{2}\ \nonumber \] find a common denominator to combine fractions
\[\dfrac{b^{2} }{4a^{2} } -\dfrac{4ac}{4a^{2} } =\left(x+\dfrac{b}{2a} \right)^{2}\ \nonumber \] combine the fractions on the left side of the equation
\[\dfrac{b^{2} -4ac}{4a^{2} } =\left(x+\dfrac{b}{2a} \right)^{2}\ \nonumber \] take the square root of both sides
\[\pm \sqrt{\dfrac{b^{2} -4ac}{4a^{2} } } =x+\dfrac{b}{2a}\ \nonumber \] subtract \(b/2a\) from both sides
\[-\dfrac{b}{2a} \pm \dfrac{\sqrt{b^{2} -4ac} }{2a} =x\ \nonumber \] combining the fractions
\[x=\dfrac{-b\pm \sqrt{b^{2} -4ac} }{2a}\ \nonumber \] Notice that this can yield two different answers for \(x\)
For a quadratic function given in standard form \(f(x)=ax^{2} +bx+c\), the quadratic formula gives the horizontal intercepts of the graph of this function.
\[x=\dfrac{-b\pm \sqrt{b^{2} -4ac} }{2a} \nonumber \]
Let's practice a few more examples of applying all the concepts we've learned about quadratics.
Find the domain and range of \(f(x)=−5x^2+9x−1\).
Solution
As with any quadratic function, the domain is all real numbers.
Because \(a\) is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding the x-value of the vertex.
\[\begin{align*} h&=−\dfrac{b}{2a} \\[5pt] &=−\dfrac{9}{2(-5)} \\[5pt] &=\dfrac{9}{10} \end{align*}\nonumber \]
The maximum value is given by \(f(h)\).
\[\begin{align*} f \left(\dfrac{9}{10}\right)&=5\left(\dfrac{9}{10}\right)^2+9\left(\dfrac{9}{10}\right)-1 \\[5pt] &= \dfrac{61}{20}\end{align*}\nonumber \]
Since we know the parabola opens down, we know the range is \(f(x){\leq}\dfrac{61}{20}\), or \(\left(−\infty,\dfrac{61}{20}\right]\).
A backyard farmer wants to enclose a rectangular space for a new garden within her fenced backyard. She has purchased 80 feet of wire fencing to enclose three sides, and she will use a section of the backyard fence as the fourth side.
- Find a formula for the area enclosed by the fence if the sides of fencing perpendicular to the existing fence have length \(L\).
- What dimensions should she make her garden to maximize the enclosed area?
Solution
Let’s use a diagram such as Figure \(\PageIndex{4}\) to record the given information. It is also helpful to introduce a temporary variable, \(W\), to represent the width of the garden and the length of the fence section parallel to the backyard fence.
a. We know we have only 80 feet of fence available, and \(L+W+L=80\), or more simply, \(2L+W=80\). This allows us to represent the width, \(W\), in terms of \(L\).
\[W=80−2L\nonumber \]
Now we are ready to write an equation for the area the fence encloses. We know the area of a rectangle is length multiplied by width, so
\[\begin{align*} A&=LW=L(80−2L) \\ A(L)&=80L−2L^2 \end{align*}\nonumber \]
This formula represents the area of the fence in terms of the variable length \(L\). The function, written in general form, is
\[A(L)=−2L^2+80L\nonumber \]
The quadratic has a negative leading coefficient, so the graph will open downward, and the vertex will be the maximum value for the area. In finding the vertex, we must be careful because the equation is not written in standard polynomial form with decreasing powers. This is why we rewrote the function in general form above. Since \(a\) is the coefficient of the squared term, \(a=−2\), \(b=80\), and \(c=0\).
To find the vertex:
\[\begin{align*} h& =−\dfrac{80}{2(−2)} \\ &=20 \end{align*}\nonumber \]
and
\[\begin{align*}k&=A(20) \\ &=80(20)−2(20)^2 \\ &=800 \end{align*}\nonumber \]
The maximum value of the function is an area of 800 square feet, which occurs when \(L=20\) feet. When the shorter sides are 20 feet, there is 40 feet of fencing left for the longer side. To maximize the area, she should enclose the garden so the two shorter sides have length 20 feet and the longer side parallel to the existing fence has length 40 feet.
Analysis
This problem also could be solved by graphing the quadratic function. We can see where the maximum area occurs on a graph of the quadratic function in Figure \(\PageIndex{5}\).
A ball is thrown upwards from the top of a 40-foot-tall building at a speed of 80 feet per second. The ball’s height above ground can be modeled by the equation \(H(t)=-16t^{2} +80t+40\).
What is the maximum height of the ball?
When does the ball hit the ground? Round to the nearest thousandth.
Solution
To find the maximum height of the ball, we would need to know the vertex of the quadratic.
\[h=-\dfrac{80}{2(-16)} =\dfrac{80}{32} =\dfrac{5}{2} , k=H\left(\dfrac{5}{2} \right)=-16\left(\dfrac{5}{2} \right)^{2} +80\left(\dfrac{5}{2} \right)+40=140\ \nonumber \]
The ball reaches a maximum height of 140 feet after 2.5 seconds.
To find when the ball hits the ground, we need to determine when the height is zero – when \(H(t) = 0\). While we could do this using the transformation form of the quadratic, we can also use the quadratic formula:
\[t=\dfrac{-80\pm \sqrt{80^{2} -4(-16)(40)} }{2(-16)} =\dfrac{-80\pm \sqrt{8960} }{-32}\ \nonumber \]
Since the square root does not simplify nicely, we can use a calculator to approximate the values of the solutions:
\[t=\dfrac{-80-\sqrt{8960} }{-32} \approx 5.458\text{ or }t=\dfrac{-80+\sqrt{8960} }{-32} \approx -0.458\ \nonumber \]
The second answer is outside the reasonable domain of our model, so we conclude the ball will hit the ground after about 5.458 seconds.


3.12: Procedural Law
Phases of the Criminal Justice Process
Investigative Phase
affidavit (sworn statement). When an individual is arrested without a warrant, judges will need to promptly review whether there is probable cause exists to hold them in custody before trial.
Pretrial Phase
Trial Phase
petit jurors (trial jurors); the rules of evidence (statutory and common law rules governing the admissibility of certain types of evidence such as hearsay or character evidence, the competency and impeachment of witnesses, the existence of any privilege, and the exclusion of witnesses during the testimony of other witnesses); the right of the defendant compulsory process (to secure favorable testimony and evidence); the right of the defendant to cross-examine any witnesses or evidence presented by the government against him; fair trials free of prejudicial adverse pre-trial or trial publicity; fair trials which are open to the public; and the continued right of the defendant to have the assistance of counsel and be present during his or her trial.
Sentencing Phase
right of allocution (right to make a statement to the court before the judge imposes sentence); any victims’ rights to appear and make statements at sentencing; the defendant’s rights to present mitigation evidence and witnesses; and the defendant’s continued rights to the assistance of counsel at sentencing. In capital cases in which the state is seeking the death penalty, the trial will be bifurcated (a trial split into the “guilt/innocence phase” and the “penalty phase”) and the sentencing hearing will be more like a mini-trial.
Post-Conviction Phase (Appeals Phase)
appeal of right (the initial appeal which must be reviewed by an appellate court) and right to file a discretionary appeal; the defendant’s right to have the assistance of counsel in helping to file either the appeal of right or a discretionary appeal. The post-conviction phase is also governed by rules and laws concerning the defendant’s ability to file a writ of habeas corpus (a civil suit against the entity who is currently holding the defendant in custody) or a post-conviction relief suit (a civil suit similar to a habeas corpus suit but one which can be filed by the defendant regardless if he or she is in custody). The post-conviction phase would also include any probation and parole revocation hearings.


13.7: Effects of Organizational Downsizing
- Describe the effects of terminating employees during organizational downsizing
Terminating employees is never a fun task. Even the word “termination” sounds doomful, final, and permanent. People work to support their families and their lifestyles. Without a job, providing for yourself and those that depend on you is extremely challenging. Therefore, layoffs put former employees in a tough position and should never be taken lightly. Although downsizing may be deemed necessary, deciding to make large cuts in the company’s workforce is going to impact a large number of people, including those getting terminated and those who are being retained.
It is expected for employees getting laid off to be upset and surprised. Upon announcing company layoffs, employers may experience backlash from employees. Although these are normal short-term reactions and consequences of downsizing, there are a wide variety of long-term effects companies can expect to experience. Read the information below to learn more.
Loss of Knowledge and Experience
Companies invest a lot of time and resources into hiring and training employees. In theory, every employee should have the essential skills and knowledge base required to complete their job functions. Also, the longer an employee is with a company, the more information and skills they can acquire. Therefore, when employees are laid off during the downsizing process, the company loses more than just a warm body; they lose the employee’s knowledge, experience, and expertise. In addition, downsizing companies typically do not hire new employees, which means there will be a void instead of a replacement.
Increased Workloads and Pressure on Remaining Employees
When companies downsize, gaps within the workload are created. This adds pressure to the remaining employees to not only complete their current job functions, but also to pick up additional work to compensate for the terminated employees. The additional responsibilities, on top of their current ones, can create stress and pressure on remaining employees and negatively affect the morale and motivation of the team. In addition, when employees are bogged down in the day-to-day work tasks, there is little time left to learn a new skill or expand their current knowledge base. This can create a stagnant work environment and stunt current employees’ growth potential.
Loss of Trust and Security
Layoffs tend to put people on edge. For obvious reasons, terminated employees are feeling a sense of insecurity as they are being forced to leave the company and find other work. The remaining employees may also struggle with downsizing. Even if their job has not been cut, knowing that their company is willing to lay off employees to cut costs can be an unsettling realization. Although the remaining employees may be relieved to be spared from termination, there is oftentimes a loss of trust with management and a loss of comfort in job security.
Legal Issues
Although downsizing is legal in the United States, companies that choose to downsize may experience legal backlash, specifically from terminated employees. If employees that were laid off feel as though they were laid off due to discriminatory reasons, they may have grounds for a lawsuit. Legal consequences are not guaranteed in all downsizing situations; however, they are a definite possibility.
Client Relations
When a large number of employees are laid off, it may impact client relationships. If there are fewer people employed to handle clients and their accounts, the customer experience will suffer. Also, if clients had a great working relationship with an employee that was then terminated, they may be frustrated to have a new person assigned to their account. These frustrations may lead to a loss in profitability and customer satisfaction.
As you can see, terminations during the downsizing process have a wide variety of consequences for an organization. Terminating employees is never easy and understanding how to handle downsizing with care is essential for maintaining a strong and effective work environment. Let’s move on to the next section to discuss how to downsize without downsizing morale.
Contributors and Attributions
- Effects of Organizational Downsizing. Authored by : Freedom Learning Group. Provided by : Lumen Learning. License : CC BY: Attribution
- Authored by : myrfa. Provided by : Pixabay. Located at : https://pixabay.com/photos/files-paper-office-paperwork-stack-1614223/ . License : CC0: No Rights Reserved . License Terms : Pixabay License
- Authored by : mohamed hassan. Provided by : pxhere . Located at : https://pxhere.com/en/photo/1457525 . License : CC0: No Rights Reserved . License Terms : pxhere License


9.14: Employee Training Methods
- Describe employee training methods
As is true with almost every other aspect of human resources (indeed, business), employee training methods are on the verge of transformation. However, new methods have not yet been documented, so articles on the topic are dated and, in some cases, underlying learning concepts are disputed. For example, one of the articles recommended in a recent article titled “The Most Effective Training Methods” [1] is a review and analysis of training methods published in February 2013 [2] . The article reviews 13 training methods based on seven criteria: learning modality, learning environment, trainer presence, proximity, interaction level, cost considerations, and time demands.
If those criteria are driving your choice of training, here’s the link: Training Methods: A Review & Analysis.
What is perhaps more valuable than the analysis is the prompt to look beyond the usual employee training method “suspects,” which generally include apprenticeships, internships, on-the-job training, lectures, job rotation, simulations and e-learning. To those, author Tony Lam adds: case study, games-based training, job shadowing, mentoring, programmed instruction, role-modeling, role play, stimulus-based training, and team training. The methods are defined, as in Lam’s review, briefly below:
- Case Study: Provides the participants an opportunity to develop skills by presenting a problem, without a solution, for them to solve, or with a solution, as an example of how to solve it.
- Games-Based Training: Trainees compete in a series of decision-making tasks which allows them to explore a variety of strategic alternatives and experience the consequences which affect the other players, but with without risk to the individuals or the organization.
- Internship: Involves supervised, practical training while on the job where the trainee is permitted to work in the position for which they are training, but with some restrictions and with substantially less pay or no pay.
- Job Rotation: Involves training for a job by working in the job for a limited duration, while still maintaining the original job.
- Job Shadowing: Involves a trainee closely observing someone perform a specific job in the natural job environment for the purpose of witnessing first-hand the details of the job.
- Lecture: Involves the dissemination of training material by a trainer to a group of trainees, by means of verbal instruction.
- Mentoring & Apprenticeship: Involves a one-on-one partnership between a novice employee with a senior employee. Mentorship aims to provide support and guidance to less experienced employees whereas apprenticeship is for the development of job skills.
- Programmed Instruction: Involves the delivery of training through instruction that is delivered by a program via some electronic device without the presence of an instructor (think: language training).
- Role-Modeling: Involves the live presentation of skill(s) to an audience of trainees.
- Role Play: Requires trainees to assume a character and act out the role in a make-believe scenario or series of scenarios; learning comes by way of reflection on the play.
- Simulation: Involves the use of a simulator where specific skills are developed through repeated practice with a multisensory experience of imitated conditions. A special form of simulation training is Virtual Reality Training which entails total sensory immersion.
- Stimulus-Based Training: Using some type of stimulus (i.e., music, works of art, narratives, etc.) to motivate the learner to learn. The training induces a state of being (e.g., relaxation or awareness) in the participants to achieve learning.
- Team Training: Intended exclusively for groups of individuals that behave interactively, to either improve mutual knowledge within a team or to train the team on a team-specific skill.
To the disputed learning theory point: one of the frameworks that is often cited and, in fact, used as the basis for training method/program design or selection is referred to as the Learning Pyramid or “cone of learning,” which purports to reflect the rate of learning retention after a 24-hour period based on 7 different types of learning ranging in effectiveness from lecture (5%) at the top of the pyramid to teaching others (90%) at the base. Specifically, the model states that “We do better as teachers and trainers if we let learners see, hear, experience, try and teach it to others.” [3]
In a WorkSMART blog post titled The Myth and Mystique of the Learning Pyramid , the author notes that “the model could never be substantiated by research findings” and cites research commissioned by Cisco that concluded “ the optimum design depends on the content, context, and the learner [and that] most effective designs for learning adapt to include a variety of media, combinations of modalities, levels of interactivity, learner characteristics, and pedagogy.” [4] Further research published in Education concluded that “for successful learning experiences, students need to experience a variety of instructional methods and that direct instruction needs to be accompanied by methods that further student understanding and recognize why what they are learning is useful.” [5]
Key takeaway: check your assumptions—and evaluate learning results—to arrive at your own conclusions about what works for different learners, learning retention and, ultimately, employee performance.
Training with Technology
Technology and AI is not only changing the way we live and experience the world, it will change the way we learn, including employee training. In a HRTechologist article [6] , staff writer Chiradeep BasuMalick observes that
AI will have three powerful impacts on learning and development:
- Personalizing the learning experience—AI allows organizations to leverage employee data for insights and develop customized learning experiences. These insights and user data can also be used to develop predictive capabilities and create content that “adaptive, intuitive, and responsive to a learner’s personal journey.”
- Virtual mentoring—Intelligent mentors utilize AI techniques to evaluate and track learner progress, estimating subject matter understanding and retooling the program as necessary. These systems can offer feedback and guidance to support learning and recommend targeted training to continue the learning process. In addition, virtual tutors can be used in conjunction with face-to-face mentoring, allowing for greater personal and strategic impact at a lower cost.
- Advanced analytics—Providing human resource management with insights into learner progress, retention and related metrics, and allowing for effective evaluation, identification or any gaps and redesign is/as needed.
Coming full circle, BasuMalick invites the reader to “imagine a future where AI instructional designers are taking charge of virtual classrooms, powered by real-time case studies and simulations.”
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Raynolds, Jeremy. "
The Most Effective Training Methods
." HR Daily Advisor. May 22, 2019. Accessed July 18, 2019. ↵
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Ostrowski Martin, Barbara & Kolomitro, Klodiana & Lam, Tony. "Training Methods: A Review and Analysis. Human Resource Development Review." 2013. Accessed July 18, 2019. ↵
-
"
The Myth and Mystique of the Learning Pyramid.
" WorkSMART. Accessed July 18, 2019. ↵
-
Ibid. ↵
-
Ibid. ↵
-
Basumallick, Chiradeep. "3 Incredible Ways AI Will Transform Learning and Development." HR Technologist. November 20, 2018. Accessed July 18, 2019. ↵
Contributors and Attributions
- Untitled. Authored by : rawpixel. Provided by : Pixabay. Located at : pixabay.com/photos/brainstorm-brainstorming-business-2449730/. License : CC0: No Rights Reserved . License Terms : Pixabay License
- Women In Tech - 59. Authored by : WOCinTech Chat. Provided by : Flickr. Located at : https://flic.kr/p/zoc8sR . License : CC BY: Attribution


11.2: Operatic Terminology
- Libretto – the words of an opera. Some composers wrote their own librettos; some collaborated with their librettists.
- Orchestral overture – an instrumental opening to most of the operas. Some overtures are based on melodies from the arias.
- Aria – a formal song in which characters expresses their emotions.
- Recitative – the plot-driving passages sung in a style designed to imitate and emphasize the inflections of speech.
Listen to the recitative “ Ch’io mi scordi di te ” from The Marriage of Figaro by Mozart.
- Ensembles – duets, trios, other ensembles, and choruses.
Contributors and Attributions
All rights reserved content
- Gluck Overture to Iphigenie en Aulide. Authored by : mvvm18. Located at : https://youtu.be/ctko9P6IaMM . License : All Rights Reserved . License Terms : Standard YouTube License
- The Barber of Seville - Figaro's Aria. Authored by : Gioreto1. Located at : https://youtu.be/7qHZkkgowdY . License : All Rights Reserved . License Terms : Standard YouTube License
- Verdi: La Traviata - III.Act - Gypsy and Picadors Chorus Noi siamo zingarelle. Authored by : Kanu00e1l uu017eivatele AchillesValda. Located at : https://youtu.be/c3bHC8B5o-k . License : All Rights Reserved . License Terms : Standard YouTube License


28.4B: The DNA Double Helix
The DNA double helix looks like a twisted staircase, with the sugar and phosphate backbone surrounding complementary nitrogen bases.
- Describe the structure of DNA
Key Points
- The structure of DNA is called a double helix, which looks like a twisted staircase.
- The sugar and phosphate make up the backbone, while the nitrogen bases are found in the center and hold the two strands together.
- The nitrogen bases can only pair in a certain way: A pairing with T and C pairing with G. This is called base pairing .
- Due to the base pairing , the DNA strands are complementary to each other, run in opposite directions, and are called antiparallel strands.
Key Terms
- mutation : any error in base pairing during the replication of DNA
- sugar-phosphate backbone : The outer support of the ladder, forming strong covalent bonds between monomers of DNA.
- base pairing : The specific way in which bases of DNA line up and bond to one another; A always with T and G always with C.
DNA is a Double Helix : Native DNA is an antiparallel double helix. The phosphate backbone (indicated by the curvy lines) is on the outside, and the bases are on the inside. Each base from one strand interacts via hydrogen bonding with a base from the opposing strand.
A Double-Helix Structure
DNA has a double-helix structure, with sugar and phosphate on the outside of the helix, forming the sugar-phosphate backbone of the DNA. The nitrogenous bases are stacked in the interior in pairs, like the steps of a staircase; the pairs are bound to each other by hydrogen bonds. The two strands of the helix run in opposite directions. This antiparallel orientation is important to DNA replication and in many nucleic acid interactions.
Base Pairs
Only certain types of base pairing are allowed. This means Adenine pairs with Thymine, and Guanine pairs with Cytosine. This is known as the base complementary rule because the DNA strands are complementary to each other.
If the sequence of one strand is AATTGGCC, the complementary strand would have the sequence TTAACCGG.
Antiparallel Strands : In a double stranded DNA molecule, the two strands run antiparallel to one another so one is upside down compared to the other. The phosphate backbone is located on the outside, and the bases are in the middle. Adenine forms hydrogen bonds (or base pairs) with thymine, and guanine base pairs with cytosine.
DNA Replication
During DNA replication, each strand is copied, resulting in a daughter DNA double helix containing one parental DNA strand and a newly synthesized strand. At this time it is possible a mutation may occur. A mutation is a change in the sequence of the nitrogen bases. For example, in the sequence AATTGGCC, a mutation may cause the second T to change to a G. Most of the time when this happens the DNA is able to fix itself and return the original base to the sequence. However, sometimes the repair is unsuccessful, resulting in different proteins being created.


26.11: Changing With the Tide- The American Perspective
In Herman Melville’s Benito Cereno , we are moving into literature beyond the nation because we are beginning to see that American influences, and even ideological changes, are no longer bound to the physical country of the United States of America. In the story, a crew of American sailors stumble upon a confusing wreck of a Spanish ship called the San Dominick. Once the Americans arrive on board, their confusion escalates as they meet its passengers.
As Captain Delano interacts with the ship’s captain and the slaves (which he views as cargo,) we begin to see American ideologies make their way into the story. When noticing the strange behaviors of the slaves on board this statement is made: “. . . long-continued suffering seemed to have brought out the less good-natured qualities of the negroes, besides, at the same time, impairing the Spaniards authority over them” (167). Delany sees African “cargo” moving freely around the ship–not in complete obedience to Cereno–revealing his limited understanding of black subjectivity and behavior.
In Delano’s mind, captured slaves are supposed to cheerfully serve without difficulty, and assume that only some form of tragedy brought about by the storms they sailed through has unleashed some latent, internal, “bad qualities.” This idea of how a slave on this ship should is supposed to behave is reflected in the way that Cereno’s main servant, Babo, is discussed and approved of: “But the good conduct of Babo, hardly more than the ill-behavior of others, seemed to withdraw the half-lunatic Don Benito from his cloudy languor” (168). Babo is differentiated from others on the ship. He appears entirely subservient to his master Don Benito, referring to him at all times as Master and refusing any level of compliment on his own work with the survival of the captain.
For this service, the American captain Delano states that Don Benito should see the faithful Babo as a friend, not as a slave (172). I think that it is very interesting for the American to have this kind of an epiphany upon boarding the ship. On one hand, it could be a case for an indication that America was slowly trying to move into a place of more racial acceptance, but on the other, we have to look at what behavior is and isn’t accepted by Delano. Babo may be faithful to Don Benito, but the others on board are not. Even though they claim that Babo should be rewarded as a friend and not simply as a slave they do not feel the same about the seemingly less obedient others onboard.


6.1: Introduction
Introduction
Cells vary in size. With few exceptions, individual cells cannot be seen with the naked eye, so scientists use microscopes (micro- = “small”; -scope = “to look at”) to study them. A microscope is an instrument that magnifies an object. Most photographs of cells are taken with a microscope; these images can also be called micrographs.
The optics of a microscope’s lenses change the orientation of the image that the user sees. A specimen that is right-side up and facing right on the microscope slide will appear upside-down and facing left when viewed through a microscope, and vice versa. Similarly, if the slide is moved left while looking through the microscope, it will appear to move right, and if moved down, it will seem to move up. This occurs because microscopes use two sets of lenses to magnify the image. Because of the manner by which light travels through the lenses, this system of two lenses produces an inverted image (binocular, or dissecting microscopes, work in a similar manner, but they include an additional magnification system that makes the final image appear to be upright).
A. Light Microscopes
To give you a sense of cell size, a typical human red blood cell is about eight millionths of a meter or eight micrometers (abbreviated as eight μm) in diameter; the head of a pin of is about two thousandths of a meter (two mm) in diameter. That means about 250 red blood cells could fit on the head of a pin.
Most student microscopes are classified as light microscopes. Visible light passes and is bent through the lens system to enable the user to see the specimen. Light microscopes are advantageous for viewing living organisms, but since individual cells are generally transparent, their components are not distinguishable unless they are colored with special stains. Staining, however, usually kills the cells.
Light microscopes, commonly used in undergraduate college laboratories, magnify up to approximately 400 times. Two parameters that are important in microscopy are magnification and resolving power. Magnification is the process of enlarging an object in appearance. Resolving power is the ability of a microscope to distinguish two adjacent structures as separate: the higher the resolution, the better the clarity and detail of the image. When oil immersion lenses are used for the study of small objects, magnification is usually increased to 1,000 times. In order to gain a better understanding of cellular structure and function, scientists typically use electron microscopes.
B. Electron Microscopes
In contrast to light microscopes, electron microscopes use a beam of electrons instead of a beam of light. Not only does this allow for higher magnification and, thus, more detail, it also provides higher resolving power. The method used to prepare the specimen for viewing with an electron microscope kills the specimen. Electrons have short wavelengths (shorter than photons) that move best in a vacuum, so living cells cannot be viewed with an electron microscope.
In a scanning electron microscope, a beam of electrons moves back and forth across a cell’s surface, creating details of cell surface characteristics. In a transmission electron microscope, the electron beam penetrates the cell and provides details of a cell’s internal structures. As you might imagine, electron microscopes are significantly more bulky and expensive than light microscopes.
C. Staining
Staining is a technique used in microscopy to enhance contrast in a microscopic image. Stains and dyes are frequently used to highlight structures in microbes for viewing, often with the aid of different microscopes. Stains may be used to define and examine different types of microbes, various stages of cellular life (e.g., the mitotic cycle), and even organelles within individual cells (e.g., mitochondria or chloroplasts).
In-vivo staining is the process of dyeing living tissue — in vivo means “in life” (as contrasted to in-vitro staining). When a certain cell or structure takes on contrasting color(s), its form (morphology) or position within a cell or tissue can be readily seen and studied. The usual purpose is to reveal cytological details that might otherwise not be apparent; however, staining can also reveal where certain chemicals or specific chemical reactions are taking place within cells. In-vitro staining involves coloring cells or structures that have been removed from their biological context. Certain stains are often combined to reveal more details and features than a single stain could reveal alone, and a counterstain is a stain that increases visibility of cells or structures when the principal stain is not sufficient. Scientists and physicians can combine staining with specific protocols for fixation and sample preparation and can use these standard techniques as consistent, repeatable diagnostic tools.
There are an incredible number of stains that can be used in a variety of different methods. What follows here are some common aspects of the process of preparing for in-vitro staining.
- Fixation: This can itself consist of several steps. Fixation aims to preserve the shape of the cells (in this case, microbes) as much as possible. Sometimes heat fixation is used to kill, adhere, and alter the cells so they will accept stains. Most chemical fixatives generate chemical bonds between proteins and other substances within the sample, increasing their rigidity. Common fixatives include formaldehyde, ethanol, methanol, and picric acid.
- Permeabilization: This involves treatment of the cells with (usually) a mild surfactant. This treatment dissolves cell membranes, allowing larger dye molecules to enter the cell’s interior.
- Mounting: This step usually involves attaching the samples to a glass microscope slide for observation and analysis. In some cases, cells may be grown directly on a slide. For samples of loose cells the sample can be directly applied to a slide.
At its simplest, the actual staining process may involve immersing the sample (before or after fixation and mounting) in dye solution, followed by rinsing and observation. Many dyes, however, require the use of a mordant — a chemical compound that reacts with the stain to form an insoluble colored precipitate. When the excess dye solution is washed away, the mordanted stain remains. There is an incredible array of stains that can be used at this step, from those that stain specific microbial types (see the figure below) to those that highlight sub-compartments or organelles of a cell, such as the nucleus or endoplasmic reticulum. Alternatively, negative staining can be employed. This is a simple staining method for bacteria, performed by smearing the cells onto the slide and then applying nigrosin (a black synthetic dye) or Indian ink (an aqueous suspension of carbon particles). After drying, the microorganisms may be viewed in bright field microscopy as lighter inclusions contrast well against the dark environment surrounding them
Live, in-vivo staining microscopy shares many of these steps, with the exception of fixation, which invariably kills the microbe to be examined.
D. Dark-Field Microscopy
Dark-field microscopy is ideally used to illuminate unstained samples causing them to appear brightly lit against a dark background. This type of microscope contains a special condenser that scatters light and causes it to reflect off the specimen at an angle. Rather than illuminating the sample with a filled cone of light, the condenser is designed to form a hollow cone of light. The light at the apex of the cone is focused at the plane of the specimen; as this light moves past the specimen plane it spreads again into a hollow cone. The objective lens sits in the dark hollow of this cone; although the light travels around and past the objective lens, no rays enter it.
The entire field appears dark when there is no sample on the microscope stage; thus the name dark-field microscopy. When a sample is on the stage, the light at the apex of the cone strikes it. The rays scattered by the sample and captured in the objective lens thus make the image.
Samples observed under dark-field microscopy should be carefully prepared since dust and other particles also scatter the light and are easily detected. Glass slides need to be thoroughly cleaned of extraneous dust and dirt. It may be necessary to filter sample media (agar, water, saline) to exclude confusing contaminants. Sample materials need to be spread thinly; too much material on the slide creates many overlapping layers and edges, making it difficult to interpret structures.
Dark-field microscopy has many applications in microbiology. It allows the visualization of live bacteria, and distinguishes some structure (rods, curved rods, spirals, or cocci) and movement.
E. Phase-Contrast Microscopy
Phase-contrast microscopy visualizes differences in the refractive indexes of different parts of a specimen relative to unaltered light.
Phase-contrast microscopy is a method of manipulating light paths through the use of strategically placed rings in order to illuminate transparent objects. Dutch physicist Fritz Zernike developed the technique in the 1930s; for his efforts he was awarded the Nobel Prize in 1953.
In phase-contrast microscopy, parallel beams of light are passed through objects of different densities. The microscope contains special condensers that throw light “out of phase” causing it to pass through the object at different speeds. Internal details and organelles of live, unstained organisms (e.g. mitochondria, lysosomes, and the Golgi body) can be seen clearly with this microscope.
A phase ring in condenser allows a cylinder of light to pass through it while still in phase. Unaltered light hits the phase ring in the lens and is excluded. Light that is slightly altered by passing through a different refractive index is allowed to pass through. Light passing through cellular structures, such as chromosomes or mitochondria is retarded because they have a higher refractive index than the surrounding medium. Elements of lower refractive index advance the wave. Much of the background light is removed and light that constructively or destructively interfered is let through with enhanced contrast.
Phase-contrast microscopy allows the visualization of living cells in their natural state with high contrast and high resolution. This tool works best with a thin specimen and is not ideal for a thick specimen. Phase-contrast images have a characteristic grey background with light and dark features found across the sample. One disadvantage of phase-contrast microscopy is halo formation called halo-light ring.
F. Key Points
- Light microscopes allow for magnification of an object approximately up to 400-1000 times depending on whether the high power or oil immersion objective is used.
- Light microscopes use visible light which passes and bends through the lens system.
- Electron microscopes use a beam of electrons, opposed to visible light, for magnification.
- Electron microscopes allow for higher magnification in comparison to a light microscope thus, allowing for visualization of cell internal structures.
- In-vivo staining, which visualizes cells that are alive, and in-vitro staining, which visualizes fixed cells, both have important uses.
- There is a vast array of stains that can be used on microbes that can highlight almost any characteristic of a cell, even organelles within a cell.
- Staining protocols can be complex, but they share some basic steps: preparation, fixation, staining, and mounting.
- In dark-field microscopy, the light reaches the specimen from an angle with the help of an opaque disk.
- The specimen appears lit up against a dark background.
- Dark-field microscopy is most useful for extremely small living organisms that are invisible in the light microscope.
- A phase-contrast microscope splits a beam of light into 2 types of light, direct and refracted (reflected) and brings them together to form an image of the specimen.
- Where the lights are “in-phase” the image is brighter, where the lights are “out of phase” the image is darker, and by amplifying these differences in the light, it enhances contrast.
- Phase-contrast microscopy allows for the detailed observation of living organisms, especially the internal structures.
G. Key Terms
- resolution : The degree of fineness with which an image can be recorded or produced, often expressed as the number of pixels per unit of length (typically an inch).
- electron : The subatomic particle having a negative charge and orbiting the nucleus; the flow of electrons in a conductor constitutes electricity.
- surfactant : a surface active agent, or wetting agent, capable of reducing the surface tension of a liquid; typically organic compounds having a hydrophilic “head” and a hydrophobic “tail”
- organelle : a specialized structure found inside cells that carries out a specific life process (e.g., ribosomes, vacuoles)
- condenser : A lens (or combination of lenses) designed to gather light and focus it onto a specimen or part of a mechanism.
- refractive index : the ratio of the speed of light in air or vacuum to that in another medium.
LICENSES AND ATTRIBUTIONS
CC LICENSED CONTENT, SPECIFIC ATTRIBUTION
- OpenStax College, Biology. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44405/latest...ol11448/latest . License : CC BY: Attribution
- electron. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/electron. License : CC BY-SA: Attribution-ShareAlike
- resolution. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/resolution. License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, Studying Cells. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44405/latest...1_01ab_new.jpg . License : CC BY: Attribution
- Staining. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Staining. License : CC BY-SA: Attribution-ShareAlike
- Staining. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Staining. License : CC BY-SA: Attribution-ShareAlike
- Staining. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Staining. License : CC BY-SA: Attribution-ShareAlike
- organelle. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/organelle. License : CC BY-SA: Attribution-ShareAlike
- surfactant. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/surfactant. License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, Studying Cells. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44405/latest...1_01ab_new.jpg . License : CC BY: Attribution
- Chlamydia Geimsa Stain CDC. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/Fi..._Stain_CDC.jpg. License : Public Domain: No Known Copyright


6.S: Environmental Hazards & Human Health (Summary)
Summary
Environmental health is concerned with preventing disease, death and disability by reducing exposure to adverse environmental conditions and promoting behavioral change. It focuses on the direct and indirect causes of diseases and injuries, and taps resources inside and outside the health care system to help improve health outcomes. Environmental health risks can be grouped into two broad categories. Traditional hazards related to poverty and lack of development affect developing countries and poor people most. Modern hazards, caused by development that lacks environmental safeguards, such as urban (outdoor) air pollution and exposure to agro-industrial chemicals and waste, prevail in industrialized countries, where exposure to traditional hazards is low. Each year contaminated water and poor sanitation contribute to 5.4 billion cases of diarrhea worldwide and 1.6 million deaths, mostly among children under the age of five. Indoor air pollution—a much less publicized source of poor health—is responsible for more than 1.6 million deaths per year and for 2.7 percent of global burden of disease.
Emerging and reemerging diseases have been defined as infectious diseases of humans whose occurrence during the past two decades has substantially increased or threatens to increase in the near future relative to populations affected, geographic distribution, or magnitude of impacts. Antibiotic resistance is a global problem. New forms of antibiotic resistance can cross international boundaries and spread between continents. Environmental toxicology is the scientific study of the health effects associated with exposure to toxic chemicals and systems occurring in the natural, work, and living environments; the management of environmental toxins and toxicity; and the development of protections for humans, animals, and plants. Environmental contaminants are chemicals found in the environment in amounts higher than what would be there naturally. We can be exposed to these contaminants from a variety of residential, commercial, and industrial sources.
Review Questions
-
The pesticide DDT is an example of which one of the following?
- Biological agent
- Atomic agent
- Chemical agent
- Capricious agent
- Inorganic agent
-
From the perspective of human health, malnutrition is important because it…
- Causes the majority of birth defects
- Is the primary source of teratogens for most children
- is an important contributor to child mortality
- Is the second leading cause of death from emerging disease
- Increases the occurrence of infectious cancers
-
Antibiotic resistance in organisms is the result of what process?
- Differentiation
- Evolution
- Emergence
- Succession
- Fixation
-
Which one of the following is an example of an emerging disease?
- Malaria
- Ebola
- Cancer
- Heart disease
- Leukemia
-
“Effective dose – 50%” describes which one of the following?
- The dose that results in 50% mortality
- The dose the results in 50% survival
- The dose that is 50% less than the lethal dose
- The dose that results in a significant response in 50% of subjects
- The dose that is 50% more than the minimal dose
-
You are working as an environmental toxicologist for the government. Your results indicate that a particular chemical causes birth defects. Which one of the following best describes the effects of this chemical?
- Teratogen
- Carcinogen
- Mutagen
- Endocrine disruptor
- Neurotoxicant
-
Which one of the following is most directly associated with radon?
- Radiation
- Endocrine disruption
- Birth defects
- Biological agents
- Biomagnification
-
Which one of the following is an example of bioremediation?
- Adding chemical dispersants following an oil spill
- Growing large colonies of bacteria to produce antibiotic chemicals
- Mimicking the processes of natural in a laboratory or industrial context
- Removing invasive species that are overpopulating an ecosystem
- Using plants to remove toxic metals from soils following a mining operation
-
Which one of the following is not a naturally-occurring element that may be hazardous to human health?
- Lead
- Radon
- Phthalate
- Mercury
- Arsenic
-
Which one of the following is not true regarding the “Love Canal Disaster”?
- It involved the improper storage of chemical waste
- It was one of several incidents that led to the creation of the Superfund
- A school was built on the contaminated site
- Many homes had to be evacuated due to contamination from various biological agents
- Many people living in the year reported serious health problems
See Appendix for answers
Attributions
EPA. (n.d.). Attachment 6: Useful terms and definitions for explaining risk . Accessed August 31, 2015 at http://www.epa.gov/superfund/community/pdfs/toolkit/risk_communication-attachment6.pdf . Modified from original.
OSHA. (n.d.). Understanding chemical hazards. Accessed August 25, 2015 from https://www.osha.gov/dte/grant_mater...calHazards.pdf . Modified from original.
Theis, T. & Tomkin, J. (Eds.). (2015). Sustainability: A comprehensive foundation. Retrieved from<EMAIL_ADDRESS>. Available under Creative Commons Attribution 4.0 International License. ( CC BY 4.0 ). Modified from original.
University of California College Prep. (2012). AP environmental science . Retrieved from http://cnx.org/content/col10548/1.2/ . Available under Creative Commons Attribution 4.0 International License. ( CC BY 4.0 ). Modified from original.
World Bank. (2003). Environmental health. Washington, DC. World Bank. Retrieved from https://openknowledge.worldbank.org/handle/10986/9734 . Available under Creative Commons Attribution License 3.0 ( CC BY 3.0 ). Modified from Original.
World Bank. (2008). Environmental health and child survival : Epidemiology, economics, experiences. Washington, DC: World Bank. World Bank. Retrieved from https://openknowledge.worldbank.org/handle/10986/6534 . Available under Creative Commons Attribution License 3.0 ( CC BY 3.0 ). Modified from original.
World Bank . (2009). Environmental health and child survival. Retrieved from https://openknowledge.worldbank.org/...le/10986/11719 . Available under Creative Commons Attribution License 3.0 ( CC BY 3.0 ). Modified from original.
Page attribution: Essentials of Environmental Science by Kamala Doršner is licensed under CC BY 4.0 . Modified from the original by Matthew R. Fisher. “Review Questions” is licensed under CC BY 4.0 by Matthew R. Fisher.


13.2: Images by Reflection
Infants are always fascinated by the antics of the Baby in the Mirror. Now if you want to know something about mirror images that most people don't understand, try this. First bring this page closer and closer to your eyes, until you can no longer focus on it without straining. Then go in the bathroom and see how close you can get your face to the surface of the mirror before you can no longer easily focus on the image of your own eyes. You will find that the shortest comfortable eye-mirror distance is much less than the shortest comfortable eye-paper distance. This demonstrates that the image of your face in the mirror acts as if it had depth and existed in the space behind the mirror. If the image was like a flat picture in a book, then you wouldn't be able to focus on it from such a short distance.
In this chapter we will study the images formed by flat and curved mirrors on a qualitative, conceptual basis. Although this type of image is not as commonly encountered in everyday life as images formed by lenses, images formed by reflection are simpler to understand, so we discuss them first. In section 12.3 we will turn to a more mathematical treatment of images made by reflection. Surprisingly, the same equations can also be applied to lenses, which are the topic of section 12.4.
12.2.1 A virtual image
We can understand a mirror image using a ray diagram. Figure a shows several light rays, 1, that originated by diffuse reflection at the person's nose. They bounce off the mirror, producing new rays, 2. To anyone whose eye is in the right position to get one of these rays, they appear to have come from a behind the mirror, 3, where they would have originated from a single point. This point is where the tip of the image-person's nose appears to be. A similar analysis applies to every other point on the person's face, so it looks as though there was an entire face behind the mirror. The customary way of describing the situation requires some explanation:
- Customary description in physics: There is an image of the face behind the mirror.
- Translation: The pattern of rays coming from the mirror is exactly the same as it would be if there were a face behind the mirror. Nothing is really behind the mirror.
This is referred to as a virtual image, because the rays do not actually cross at the point behind the mirror. They only appear to have originated there.
self-check:
Imagine that the person in figure a moves his face down quite a bit --- a couple of feet in real life, or a few inches on this scale drawing. The mirror stays where it is. Draw a new ray diagram. Will there still be an image? If so, where is it visible from?
(answer in the back of the PDF version of the book)
The geometry of specular reflection tells us that rays 1 and 2 are at equal angles to the normal (the imaginary perpendicular line piercing the mirror at the point of reflection). This means that ray 2's imaginary continuation, 3, forms the same angle with the mirror as ray 1. Since each ray of type 3 forms the same angles with the mirror as its partner of type 1, we see that the distance of the image from the mirror is the same as that of the actual face from the mirror, and it lies directly across from it. The image therefore appears to be the same size as the actual face.
| Example 2: An eye exam |
|---|
|
Figure b shows a typical setup in an optometrist's examination room. The patient's vision is supposed to be tested at a distance of 6 meters (20 feet in the U.S.), but this distance is larger than the amount of space available in the room. Therefore a mirror is used to create an image of the eye chart behind the wall. |
| Example 3: The Praxinoscope |
|---|
|
Figure c shows an old-fashioned device called a praxinoscope, which displays an animated picture when spun. The removable strip of paper with the pictures printed on it has twice the radius of the inner circle made of flat mirrors, so each picture's virtual image is at the center. As the wheel spins, each picture's image is replaced by the next. |
Discussion Question
◊ The figure shows an object that is off to one side of a mirror. Draw a ray diagram. Is an image formed? If so, where is it, and from which directions would it be visible?
12.2.2 Curved mirrors
An image in a flat mirror is a pretechnological example: even animals can look at their reflections in a calm pond. We now pass to our first nontrivial example of the manipulation of an image by technology: an image in a curved mirror. Before we dive in, let's consider why this is an important example.  The reason this is an important example is not that curved mirrors are so important in and of themselves, but that the results we derive for curved bowl-shaped mirrors turn out to be true for a large class of other optical devices, including mirrors that bulge outward rather than inward, and lenses as well. A microscope or a telescope is simply a combination of lenses or mirrors or both. What you're really learning about here is the basic building block of all optical devices from movie projectors to octopus eyes.
Because the mirror in figure d is curved, it bends the rays back closer together than a flat mirror would: we describe it as converging . Note that the term refers to what it does to the light rays, not to the physical shape of the mirror's surface . (The surface itself would be described as concave . The term is not all that hard to remember, because the hollowed-out interior of the mirror is like a cave.) It is surprising but true that all the rays like 3 really do converge on a point, forming a good image. We will not prove this fact, but it is true for any mirror whose curvature is gentle enough and that is symmetric with respect to rotation about the perpendicular line passing through its center (not asymmetric like a potato chip). The old-fashioned method of making mirrors and lenses is by grinding them in grit by hand, and this automatically tends to produce an almost perfect spherical surface.
Bending a ray like 2 inward implies bending its imaginary continuation 3 outward, in the same way that raising one end of a seesaw causes the other end to go down. The image therefore forms deeper behind the mirror. This doesn't just show that there is extra distance between the image-nose and the mirror; it also implies that the image itself is bigger from front to back. It has been magnified in the front-to-back direction.
It is easy to prove that the same magnification also applies to the image's other dimensions. Consider a point like E in figure e. The trick is that out of all the rays diffusely reflected by E, we pick the one that happens to head for the mirror's center, C. The equal-angle property of specular reflection plus a little straightforward geometry easily leads us to the conclusion that triangles ABC and CDE are the same shape, with ABC being simply a scaled-up version of CDE. The magnification of depth equals the ratio BC/CD, and the up-down magnification is AB/DE. A repetition of the same proof shows that the magnification in the third dimension (out of the page) is also the same. This means that the image-head is simply a larger version of the real one, without any distortion. The scaling factor is called the magnification, \(M\). The image in the figure is magnified by a factor \(M=1.9\).
Note that we did not explicitly specify whether the mirror was a sphere, a paraboloid, or some other shape. However, we assumed that a focused image would be formed, which would not necessarily be true, for instance, for a mirror that was asymmetric or very deeply curved.
12.2.3 A real image
If we start by placing an object very close to the mirror, g/1, and then move it farther and farther away, the image at first behaves as we would expect from our everyday experience with flat mirrors, receding deeper and deeper behind the mirror. At a certain point, however, a dramatic change occurs. When the object is more than a certain distance from the mirror, g/2, the image appears upside-down and in front of the mirror.
Here's what's happened. The mirror bends light rays inward, but when the object is very close to it, as in g/1, the rays coming from a given point on the object are too strongly diverging (spreading) for the mirror to bring them back together. On reflection, the rays are still diverging, just not as strongly diverging. But when the object is sufficiently far away, g/2, the mirror is only intercepting the rays that came out in a narrow cone, and it is able to bend these enough so that they will reconverge.
Note that the rays shown in the figure, which both originated at the same point on the object, reunite when they cross. The point where they cross is the image of the point on the original object. This type of image is called a real image , in contradistinction to the virtual images we've studied before.
Definition: A real image is one where rays actually cross. A virtual image is a point from which rays only appear to have come.
The use of the word “real” is perhaps unfortunate. It sounds as though we are saying the image was an actual material object, which of course it is not.
The distinction between a real image and a virtual image is an important one, because a real image can be projected onto a screen or photographic film. If a piece of paper is inserted in figure g/2 at the location of the image, the image will be visible on the paper (provided the object is bright and the room is dark). Your eye uses a lens to make a real image on the retina.
self-check:
Sketch another copy of the face in figure g/1, even farther from the mirror, and draw a ray diagram. What has happened to the location of the image?
(answer in the back of the PDF version of the book)
12.2.4 Images of images
If you are wearing glasses right now, then the light rays from the page are being manipulated first by your glasses and then by the lens of your eye. You might think that it would be extremely difficult to analyze this, but in fact it is quite easy. In any series of optical elements (mirrors or lenses or both), each element works on the rays furnished by the previous element in exactly the same manner as if the image formed by the previous element was an actual object.
Figure h shows an example involving only mirrors. The Newtonian telescope, invented by Isaac Newton, consists of a large curved mirror, plus a second, flat mirror that brings the light out of the tube. (In very large telescopes, there may be enough room to put a camera or even a person inside the tube, in which case the second mirror is not needed.) The tube of the telescope is not vital; it is mainly a structural element, although it can also be helpful for blocking out stray light. The lens has been removed from the front of the camera body, and is not needed for this setup. Note that the two sample rays have been drawn parallel, because an astronomical telescope is used for viewing objects that are extremely far away. These two “parallel” lines actually meet at a certain point, say a crater on the moon, so they can't actually be perfectly parallel, but they are parallel for all practical purposes since we would have to follow them upward for a quarter of a million miles to get to the point where they intersect.
The large curved mirror by itself would form an image \(\text{I}\), but the small flat mirror creates an image of the image, \(\text{I}'\). The relationship between \(\text{I}\) and \(\text{I}'\) is exactly the same as it would be if \(\text{I}\) was an actual object rather than an image: \(\text{I}\) and \(\text{I}'\) are at equal distances from the plane of the mirror, and the line between them is perpendicular to the plane of the mirror.
One surprising wrinkle is that whereas a flat mirror used by itself forms a virtual image of an object that is real, here the mirror is forming a real image of virtual image \(\text{I}\). This shows how pointless it would be to try to memorize lists of facts about what kinds of images are formed by various optical elements under various circumstances. You are better off simply drawing a ray diagram.
Although the main point here was to give an example of an image of an image, figure i also shows an interesting case where we need to make the distinction between magnification and angular magnification . \(\text{I}\)f you are looking at the moon through this telescope, then the images \(\text{I}\) and \(\text{I}'\) are much smaller than the actual moon. Otherwise, for example, image \(\text{I}\) would not fit inside the telescope! However, these images are very close to your eye compared to the actual moon. The small size of the image has been more than compensated for by the shorter distance. The important thing here is the amount of angle within your field of view that the image covers, and it is this angle that has been increased. The factor by which it is increased is called the angular magnification , \(M_a\).
 What happens if the mirrors are not perfectly perpendicular?
◊ Locate the images formed by the periscope.
Contributor
Benjamin Crowell (Fullerton College). Conceptual Physics is copyrighted with a CC-BY-SA license.


6.4: Cyclical Component
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A cyclical pattern repeats with some regularity over several years. Cyclical patterns differ from seasonal patterns in that cyclical patterns occur over multiple years, whereas seasonal patterns occur within one year.
One example of a cyclical pattern, the business cycle, is from macroeconomics. Over time, economic expansions are followed by economic recessions followed again by economic expansions. There is not perfect regularity in the business cycle, as expansions and recessions differ in length. Nevertheless, this process has repeated itself over and over through time.
Cycles are also observed in some agricultural commodity prices and are most pronounced when there is a time lapse between a change in price and the producer’s response to this change. The time lapse is due to two things:
Price cycles emerge when future production decisions are based on current prices and when producers have little control over prices (i.e., when producers are price takers). Thus, cycles are more likely to emerge in industries where there are a large number of relatively small operations. Price cycles are also more likely when there is a a large degree of control over output. Otherwise, random shocks tend to disrupt the cyclical pattern and cause it to dissipate. For this reason, cycles are often more common for livestock than for crops.
One commonly mentioned price cycle is the cattle cycle. The cattle cycle lasts about 8 to 12 years from peak to peak or trough to trough. There is some evidence of cycles, albeit shorter cycles, in hogs and broilers. Demonstration \(\PageIndex{1}\) shows monthly feeder cattle prices from the nearby feeder cattle futures contract from the mid-1970s through 2016. The raw prices are plotted as red points and a centered 12-period moving average is superimposed in blue. Do you see any evidence of a cattle cycle in these data? The demonstration allows you to examine the data in both nominal and real terms. Evidence of the cattle cycle is easier to see if you look at the data in real terms.
Demonstration \(\PageIndex{1}\). Feeder Cattle Prices with 12-Month Centered Moving Average (Sept. 1973 through Dec. 2016)


11.1: Introduction
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- Ly-Huong T. Pham, Tejal Desai-Naik, Laurie Hammond, & Wael Abdeljabbar
- ASCCC Open Educational Resources Initiative (OERI)
In this chapter, we will look at how the internet has opened the world to globalization. We will look at where it began and fast forward to where we are today. We will be reviewing the influences of man, machine, and technology that enables globalization. It is now just as simple to communicate with someone on the other side of the world as to talk to someone next door. In this chapter, we will look at the implications of globalization and its impact on the world.
What Is Globalization?
Globalization is found in economics and refers to the integration of goods, services, and culture among the people and nations of the world. Globalization has accelerated since the turn of the 18 th century due to mass improvement in transportation and technology. Globalization has its roots as far back as an exploration of finding the New World. G lobalization creates world markets. Places that were once limited to only providing goods and services to the immediate area now have open access to other countries worldwide. The expansion of global markets has increased economic activities in the exchange of goods, services, and funds, which has created global markets that are now readily feasible. Today the ease of the connectivity of people has accelerated the speed of globalization. People no longer have to sail for a year to share goods or services.
The internet has connected nations together. From its initial beginnings in the United States in the 1970s to the World Wide Web development, it has crept into home use with the introduction of the personal computer by the 1980s. The 90’s then introduced social networks and e-commerce of today; the Internet has continued to increase the integration between countries, making globalization a fact of life for citizens worldwide. The Internet is truly a worldwide phenomenon. By Q3 of 2020, approximately 4.9 billion people, or more than half of the world’s population, use the internet. For more details, please view the data at internetworldstats.com/stats.htm.
The Network Society
In 1996, social-sciences researcher Manuel Castells published The Rise of the Network Society. He identified new ways to organize economic activity around the networks that the new telecommunication technologies have provided. This new, global economic activity was different from the past because “it is an economy with the capacity to work as a unit in real-time on a planetary scale.” (Castells, 2000) We are now into this network society, where we are all connected on a global scale.
The World Is Flat
In Thomas Friedman’s seminal book, The World Is Flat (Friedman, 2005), he unpacks the impacts that the personal computer, the Internet, and communication software have had on business, specifically its impact on globalization. He begins the book by defining the three eras of globalization:
- “ Globalization 1.0″ occurred from 1492 until about 1800 . In this era, globalization was centered around countries. It was about how much horsepower, wind power, and steam power a country had and how creatively it was deployed. The world shrank from size “large” to size “medium.”
- “ Globalization 2.0″ occurred from about 1800 until 2000, interrupted only by the two World Wars . In this era, the dynamic force driving change was multinational companies. The world shrank from size “medium” to size “small.”
- “Globalization 3.0″ is our current era, beginning in the year 2000 . The convergence of the personal computer, fiber-optic Internet connections, and software has created a “flat-world platform” that allows small groups and even individuals to go global. The world has shrunk from size “small” to size “tiny.”
According to Friedman (2005), this third era of globalization was brought about, in many respects, by information technology. Some of the specific technologies he lists include:
- The graphical user interface for the personal computer popularized in the late 1980s . Before the graphical user interface, using a computer was relatively difficult. By making the personal computer something that anyone could use, it became commonplace very quickly. Friedman points out that this digital storage of content made people much more productive and, as the Internet evolved, made it simpler to communicate content worldwide.
- The build-out of the Internet infrastructure during the dot-com boom during the late-1990s. During the late 1990s, telecommunications companies laid thousands of miles of fiber-optic cable worldwide, turning network communications into a commodity. At the same time, the Internet protocols, such as SMTP (e-mail), HTML (web pages), and TCP/IP (network communications), became standards that were available for free and used by everyone.
- The introduction of software to automate and integrate business processes . As the Internet continued to grow and become the dominant form of communication, it became essential to build on the standards developed earlier so that the websites and applications running on the Internet would work well together. Friedman calls this “workflow software,” by which he means software that allows people to work together more easily and allows different software packages and databases to integrate easily. Examples include payment-processing systems and shipping calculators.
These three technologies came together in the late 1990s to create a “platform for global collaboration.” Once these technologies were in place, they continued to evolve. Friedman also points out a couple more technologies that have contributed to the flat-world platform – the open-source movement (see chapter 10) and the advent of mobile technologies.
The World Is Flat was published in 2005. Since then, we have seen even more growth in information technologies that have contributed to global collaborations. We will discuss current and future trends in chapter 13.
References
Castells, Manuel (2000). The Rise of the Network Society (2nd ed.). Blackwell Publishers, Inc., Cambridge, MA, USA.
Friedman, T. L. (2005). The world is flat: A brief history of the twenty-first century. New York: Farrar, Straus and Giroux.
Q3 2020 Internet usage . Retrieved December 5, 2020, from https://internetworldstats.com/stats.htm .


14.3: Details on Newton's Second Law
Newton’s second law is closely related to his first law. It mathematically gives the cause-and-effect relationship between force and changes in motion. Newton’s second law is quantitative and is used extensively to calculate what happens in situations involving a force. Before we can write down Newton’s second law as a simple equation that gives the exact relationship of force, mass, and acceleration, we need to sharpen some ideas we mentioned earlier.
Force and Acceleration
First, what do we mean by a change in motion? The answer is that a change in motion is equivalent to a change in velocity. A change in velocity means, by definition, that there is acceleration. Newton’s first law says that a net external force causes a change in motion; thus, we see that a net external force causes nonzero acceleration .
We defined external force as force acting on an object or system that originates outside of the object or system. Let’s consider this concept further. An intuitive notion of external is correct—it is outside the system of interest. For example, in Figure \(\PageIndex{1a}\), the system of interest is the car plus the person within it. The two forces exerted by the two students are external forces. In contrast, an internal force acts between elements of the system. Thus, the force the person in the car exerts to hang on to the steering wheel is an internal force between elements of the system of interest. Only external forces affect the motion of a system, according to Newton’s first law. (The internal forces cancel each other out, as explained in the next section.) Therefore, we must define the boundaries of the system before we can determine which forces are external. Sometimes, the system is obvious, whereas at other times, identifying the boundaries of a system is more subtle. The concept of a system is fundamental to many areas of physics, as is the correct application of Newton’s laws. This concept is revisited many times in the study of physics.
From this example, you can see that different forces exerted on the same mass produce different accelerations. In Figure \(\PageIndex{1a}\), the two students push a car with a driver in it. Arrows representing all external forces are shown. The system of interest is the car and its driver. The weight \(\vec{w}\) of the system and the support of the ground \(\vec{N}\) are also shown for completeness and are assumed to cancel (because there was no vertical motion and no imbalance of forces in the vertical direction to create a change in motion). The vector \(\vec{f}\) represents the friction acting on the car, and it acts to the left, opposing the motion of the car. (We discuss friction in more detail in an upcoming chapter.) In Figure \(\PageIndex{1b}\), all external forces acting on the system add together to produce the net force \(\vec{F}_{net}\). The free-body diagram shows all of the forces acting on the system of interest. The dot represents the center of mass of the system. Each force vector extends from this dot. Because there are two forces acting to the right, the vectors are shown collinearly. Finally, in Figure \(\PageIndex{1c}\), a larger net external force produces a larger acceleration (\(\vec{a}' > \vec{a}\)) when the tow truck pulls the car.
It seems reasonable that acceleration would be directly proportional to and in the same direction as the net external force acting on a system. This assumption has been verified experimentally and is illustrated in Figure \(\PageIndex{1}\). To obtain an equation for Newton’s second law, we first write the relationship of acceleration \(\vec{a}\) and net external force \(\vec{F}_{net}\) as the proportionality
\[\vec{a}\; \propto\; \vec{F}_{net}\]
where the symbol \(\alpha\) means “proportional to.” (Recall from the first section of this chapter that the net external force is the vector sum of all external forces and is sometimes indicated as \(\sum \vec{F}\).) This proportionality shows what we have said in words—acceleration is directly proportional to net external force. Once the system of interest is chosen, identify the external forces and ignore the internal ones. It is a tremendous simplification to disregard the numerous internal forces acting between objects within the system, such as muscular forces within the students’ bodies, let alone the myriad forces between the atoms in the objects. Still, this simplification helps us solve some complex problems.
It also seems reasonable that acceleration should be inversely proportional to the mass of the system. In other words, the larger the mass (the inertia), the smaller the acceleration produced by a given force. As illustrated in Figure \(\PageIndex{2}\), the same net external force applied to a basketball produces a much smaller acceleration when it is applied to an SUV. The proportionality is written as
\[a\; \propto\; \frac{1}{m},\]
where m is the mass of the system and a is the magnitude of the acceleration. Experiments have shown that acceleration is exactly inversely proportional to mass, just as it is directly proportional to net external force.
It has been found that the acceleration of an object depends only on the net external force and the mass of the object. Combining the two proportionalities just given yields Newton’s second law .
The acceleration of a system is directly proportional to and in the same direction as the net external force acting on the system and is inversely proportion to its mass. In equation form, Newton’s second law is
\[\vec{a} = \frac{\vec{F}_{net}}{m},\]
where \(\vec{a}\) is the acceleration, \(\vec{F}_{net}\) is the net force, and m is the mass. This is often written in the more familiar form
\[\vec{F}_{net} = \sum \vec{F} = m \vec{a}, \label{5.3}\]
but the first equation gives more insight into what Newton’s second law means.
The law is a cause-and-effect relationship among three quantities that is not simply based on their definitions. The validity of the second law is based on experimental verification. The free-body diagram, which you will learn to draw in Drawing Free-Body Diagrams , is the basis for writing Newton’s second law.
Suppose that the net external force (push minus friction) exerted on a lawn mower is 51 N (about 11 lb.) parallel to the ground (Figure \(\PageIndex{3}\)). The mass of the mower is 24 kg. What is its acceleration?
Strategy
This problem involves only motion in the horizontal direction; we are also given the net force, indicated by the single vector, but we can suppress the vector nature and concentrate on applying Newton’s second law. Since F net and m are given, the acceleration can be calculated directly from Newton’s second law as F net = ma.
Solution
The magnitude of the acceleration a is a = \(\frac{F_{net}}{m}\). Entering known values gives
\[a = \frac{51\; N}{24\; kg} \ldotp\]
Substituting the unit of kilograms times meters per square second for newtons yields
\[a = \frac{51\; kg \cdotp m/s^{2}}{24\; kg} = 2.1\; m/s^{2} \ldotp\]
Significance
The direction of the acceleration is the same direction as that of the net force, which is parallel to the ground. This is a result of the vector relationship expressed in Newton’s second law, that is, the vector representing net force is the scalar multiple of the acceleration vector. There is no information given in this example about the individual external forces acting on the system, but we can say something about their relative magnitudes. For example, the force exerted by the person pushing the mower must be greater than the friction opposing the motion (since we know the mower moved forward), and the vertical forces must cancel because no acceleration occurs in the vertical direction (the mower is moving only horizontally). The acceleration found is small enough to be reasonable for a person pushing a mower. Such an effort would not last too long, because the person’s top speed would soon be reached.
At the time of its launch, the HMS Titanic was the most massive mobile object ever built, with a mass of 6.0 x 10 7 kg. If a force of 6 MN (6 x 10 6 N) was applied to the ship, what acceleration would it experience?
In the preceding example, we dealt with net force only for simplicity. However, several forces act on the lawn mower. The weight pulls down on the mower, toward the center of Earth; this produces a contact force on the ground. The ground must exert an upward force on the lawn mower, known as the normal force \(\vec{N}\), which we define in Motion from Forces . These forces are balanced and therefore do not produce vertical acceleration. In the next example, we show both of these forces. As you continue to solve problems using Newton’s second law, be sure to show multiple forces.
- The car shown in Figure \(\PageIndex{4}\) is moving at a constant speed. Which force is bigger, \(\vec{F}_{engine}\) or \(\vec{F}_{friction}\)? Explain.
- The same car is now accelerating to the right. Which force is bigger, \(\vec{F}_{engine}\) or \(\vec{F}_{friction}\)? Explain.
Strategy
We must consider Newton’s first and second laws to analyze the situation. We need to decide which law applies; this, in turn, will tell us about the relationship between the forces.
Solution
- The forces are equal. According to Newton’s first law, if the net force is zero, the velocity is constant.
- In this case, \(\vec{F}_{engine}\) must be larger than \(\vec{F}_{friction}\). According to Newton’s second law, a net force is required to cause acceleration.
Significance
These questions may seem trivial, but they are commonly answered incorrectly. For a car or any other object to move, it must be accelerated from rest to the desired speed; this requires that the engine force be greater than the friction force. Once the car is moving at constant velocity, the net force must be zero; otherwise, the car will accelerate (gain speed). To solve problems involving Newton’s laws, we must understand whether to apply Newton’s first law (where \(\sum \vec{F}\) = \(\vec{0}\)) or Newton’s second law (where \(\sum \vec{F}\) is not zero). This will be apparent as you see more examples and attempt to solve problems on your own.
Before manned space flights, rocket sleds were used to test aircraft, missile equipment, and physiological effects on human subjects at high speeds. They consisted of a platform that was mounted on one or two rails and propelled by several rockets.
Calculate the magnitude of force exerted by each rocket, called its thrust T, for the four-rocket propulsion system shown in Figure \(\PageIndex{5}\). The sled’s initial acceleration is 49 m/s 2 , the mass of the system is 2100 kg, and the force of friction opposing the motion is 650 N.
Strategy
Although forces are acting both vertically and horizontally, we assume the vertical forces cancel because there is no vertical acceleration. This leaves us with only horizontal forces and a simpler one-dimensional problem. Directions are indicated with plus or minus signs, with right taken as the positive direction. See the free-body diagram in Figure \(\PageIndex{5}\).
Solution
Since acceleration, mass, and the force of friction are given, we start with Newton’s second law and look for ways to find the thrust of the engines. We have defined the direction of the force and acceleration as acting “to the right,” so we need to consider only the magnitudes of these quantities in the calculations. Hence we begin with
\[F_{net} = ma\]
where F net is the net force along the horizontal direction. We can see from the figure that the engine thrusts add, whereas friction opposes the thrust. In equation form, the net external force is
\[F_{net} = 4T − f \ldotp\]
Substituting this into Newton’s second law gives us
\[F_{net} = ma = 4T − f \ldotp\]
Using a little algebra, we solve for the total thrust 4T:
\[4T = ma + f \ldotp\]
Substituting known values yields
\[4T = ma + f = (2100\; kg)(49\; m/s^{2}) + 650\; N \ldotp\]
Therefore, the total thrust is
\[4T = 1.0 \times 10^{5}\; N \ldotp\]
Significance
The numbers are quite large, so the result might surprise you. Experiments such as this were performed in the early 1960s to test the limits of human endurance, and the setup was designed to protect human subjects in jet fighter emergency ejections. Speeds of 1000 km/h were obtained, with accelerations of 45 g’s. (Recall that g, acceleration due to gravity, is 9.80 m/s 2 . When we say that acceleration is 45 g’s, it is 45 x 9.8 m/s 2 , which is approximately 440 m/s 2 .) Although living subjects are not used anymore, land speeds of 10,000 km/h have been obtained with a rocket sled.
In this example, as in the preceding one, the system of interest is obvious. We see in later examples that choosing the system of interest is crucial—and the choice is not always obvious.
Newton’s second law is more than a definition; it is a relationship among acceleration, force, and mass. It can help us make predictions. Each of those physical quantities can be defined independently, so the second law tells us something basic and universal about nature.
A 550-kg sports car collides with a 2200-kg truck, and during the collision, the net force on each vehicle is the force exerted by the other. If the magnitude of the truck’s acceleration is 10 m/s 2 , what is the magnitude of the sports car’s acceleration?
Component Form of Newton’s Second Law
We have developed Newton’s second law and presented it as a vector equation in Equation \ref{5.3}. This vector equation can be written as three component equations:
\[\sum \vec{F}_{x} = m \vec{a}_{x}, \sum \vec{F}_{y} = m \vec{a}_{y}, \sum \vec{F}_{z} = m \vec{a}_{z} \ldotp \label{5.5}\]
The second law is a description of how a body responds mechanically to its environment. The influence of the environment is the net force \(\vec{F}_{net}\), the body’s response is the acceleration \(\vec{a}\), and the strength of the response is inversely proportional to the mass m. The larger the mass of an object, the smaller its response (its acceleration) to the influence of the environment (a given net force). Therefore, a body’s mass is a measure of its inertia, as we explained in Newton’s First Law .
A 0.400-kg soccer ball is kicked across the field by a player; it undergoes acceleration given by \(\vec{a}\) = 3.00 \(\hat{i}\) + 7.00 \(\hat{j}\) m/s 2 . Find (a) the resultant force acting on the ball and (b) the magnitude and direction of the resultant force.
Strategy
The vectors in \(\hat{i}\) and \(\hat{j}\) format, which indicate force direction along the x-axis and the y-axis, respectively, are involved, so we apply Newton’s second law in vector form.
Solution
- We apply Newton’s second law: \(\vec{F}_{net} = m \vec{a} = (0.400\; kg) \big(3.00\; \hat{i} + 7.00\; \hat{j}\; m/s^{2} \big) = 1.20\; \hat{i} + 2.80\; \hat{j}\; N \ldotp\)
- . Magnitude and direction are found using the components of \(\vec{F}_{net}\): \(F_{net} = \sqrt{(1.20\; N)^{2} + (2.80\; N)^{2}} = 3.05\; N\; and\; \theta = \tan^{-1} \left(\dfrac{2.80}{1.20}\right) = 66.8^{o} \ldotp\)
Significance
We must remember that Newton’s second law is a vector equation. In (a), we are multiplying a vector by a scalar to determine the net force in vector form. While the vector form gives a compact representation of the force vector, it does not tell us how “big” it is, or where it goes, in intuitive terms. In (b), we are determining the actual size (magnitude) of this force and the direction in which it travels.
Find the mass of a car if a net force of −600.0 \(\hat{j}\) N produces an acceleration of −0.2 \(\hat{j}\) m/s 2 .
Strategy
Vector division is not defined, so \(m = \frac{\vec{F}_{net}}{\vec{a}}\) cannot be performed. However, mass m is a scalar, so we can use the scalar form of Newton’s second law, \(m = \frac{F_{net}}{a}\).
Solution
We use m = \(\frac{F_{net}}{a}\) and substitute the magnitudes of the two vectors: F net = 600.0 N and a = 0.2 m/s 2 . Therefore,
\[m = \frac{F_{net}}{a} = \frac{600.0\; N}{0.2\; m/s^{2}} = 3000\; kg \ldotp \nonumber\]
Significance
Force and acceleration were given in the \(\hat{i}\) and \(\hat{j}\) format, but the answer, mass m, is a scalar and thus is not given in \(\hat{i}\) and \(\hat{j}\) form.
Several Forces on a Particle A particle of mass m = 4.0 kg is acted upon by four forces of magnitudes. F 1 = 10.0 N, F 2 = 40.0 N, F 3 = 5.0 N, and F 4 = 2.0 N, with the directions as shown in the free-body diagram in Figure \(\PageIndex{6}\). What is the acceleration of the particle?
Strategy
Because this is a two-dimensional problem, we must use a free-body diagram. First, \(\vec{F}_{1}\) must be resolved into x- and y-components. We can then apply the second law in each direction.
Solution
We draw a free-body diagram as shown in Figure \(\PageIndex{6}\). Now we apply Newton’s second law. We consider all vectors resolved into x- and y-components:
|
\[\sum F_{x} = m a_{x}\] \[F_{1x} - F_{3x} = m a_{x}\] \[F_{1} \cos 30^{o} - F_{3x} = m a_{x}\] \[(10.0\; N)(\cos 30^{o}) - 5.0\; N = (4.0\; kg) a_{x}\] \[a_{x} = 0.92\; m/s^{2} \ldotp\] |
\[\sum F_{y} = m a_{y}\] \[F_{1y} +F_{4y} - F_{2y} = m a_{y}\] \[F_{1} \sin30^{o} + F_{4y} - F_{2y} = m a_{y}\] \[(10.0\; N)(\sin 30^{o}) + 2.0\; N - 40.0\; N = (4.0\; kg) a_{y}\] \[a_{y} =-8.3\; m/s^{2} \ldotp\] |
Thus, the net acceleration is
\[\vec{a} = \big( 0.92\; \hat{i} - 8.3\; \hat{j} \big) m/s^{2},\]
which is a vector of magnitude 8.4 m/s 2 directed at 276° to the positive x-axis.
Significance
Numerous examples in everyday life can be found that involve three or more forces acting on a single object, such as cables running from the Golden Gate Bridge or a football player being tackled by three defenders. We can see that the solution of this example is just an extension of what we have already done.
A car has forces acting on it, as shown below. The mass of the car is 1000.0 kg. The road is slick, so friction can be ignored. (a) What is the net force on the car? (b) What is the acceleration of the car?
Explore the forces at work when pulling a cart or pushing a refrigerator, crate, or person. Create an applied force and see how it makes objects move. Put an object on a ramp and see how it affects its motion.