Appendix C: Health and Safety Checklist
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Safety Factor |
Yes |
No |
Description/Comments |
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All areas are safe, clean, and free from pests. |
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Exits are clearly marked, and emergency evacuation routes and other safety procedures are posted in the classroom and in appropriate locations throughout the site. |
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Lighting is sufficient and adequate for all classroom activities. |
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Emergency lighting is available in case of a power failure. |
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Fire extinguishers are available, accessible, tested, and serviced regularly. |
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Smoke, carbon monoxide, and as necessary, radon detectors are installed, properly located, and tested regularly. |
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Current child care, health, fire, and other applicable licenses and inspection certificates are present on site. |
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All indoor and outdoor spaces meet minimum square footage requirements per most stringent regulations. |
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All playground areas are visible to supervising adults. |
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Necessary accommodations and modifications are made to ensure the safety, comfort, and full participation of all children including those with disabilities. |


8.4: Energy diagrams and equilibria
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We can write the mechanical energy of an object as:
\[\begin{aligned} E = K + U\end{aligned}\]
which will be a constant if there are no non-conservative forces doing work on the object. This means that if the potential energy of the object increases, then its kinetic energy must decrease by the same amount, and vice-versa.
Consider a block that can slide on a frictionless horizontal surface and that is attached to a spring, as is shown in Figure \(\PageIndex{1}\) (left side), where \(x=0\) is chosen as the position corresponding to the rest length of the spring. If you push on the block so as to compress the spring by a distance \(D\) and then release it, the block will initially accelerate because of the spring force in the positive \(x\) direction until the block reaches the rest position of the spring ( \(x=0\) on the diagram). When it passes that point, the spring will exert a force in the opposite direction. The block will continue in the same direction and decelerate until it stops and turns around. It will then accelerate again towards the rest position of the spring, and then decelerate once the spring starts being compressed again, until the block stops and the motion repeats. We say that the block “oscillates” back and forth about the rest position of the spring.
We can describe the motion of the block in terms of its total mechanical energy, \(E\) . Its potential energy is given by:
\[\begin{aligned} U(x)=\frac{1}{2}kx^2\end{aligned}\]
On the right of Figure \(\PageIndex{1}\) is an “Energy Diagram” for the block, which allows us to examine how the total energy, \(E\) , of the block is divided between kinetic and potential energy depending on the position of the block. The vertical axis corresponds to energy and the horizontal axis corresponds to the position of the block.
The total mechanical energy, \(E=25\text{J}\) , is shown by the horizontal red line. Also illustrated are the potential energy function ( \(U(x)\) in blue), and the kinetic energy, ( \(K=E-U(x)\) , in dotted black).
The energy diagram allows us to describe the motion of the object attached to the spring in terms of energy. A few things to note:
- At \(x=\pm D\) , the potential energy is equal to \(E\) , so the kinetic energy is zero. The block is thus instantaneously at rest at those positions.
- At \(x=0\) , the potential energy is zero, and the kinetic energy is maximal. This corresponds to where the block has the highest speed.
- The kinetic energy of the block can never be negative 1 , thus, the block cannot be located outside the range \([-D,+D]\) , and we would say that the motion of the block is “bound”. The points between which the motion is bound are called “turning points”.
An analysis of the energy diagram tells us that the block is bound between the two turning points, which themselves are equidistant from the origin. When we initially compress the spring, we are “giving” the block “spring potential energy”. As the block starts to move, the potential energy of the block is converted into kinetic energy as it accelerates and then back into potential energy as it decelerates.
Exercise \(\PageIndex{1}\)
Calculate the positions of the turning points for the situation shown in Figure \(\PageIndex{1}\). The total energy is \(25\text{ J}\) and the spring constant is \(k=1 \text{ N/m}\) .
- Answer
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\(7.1\text{ m}\)
By looking at only the potential energy function, without knowing that it is related to a spring, we can come to the same conclusions; namely that the motion is bound as long as the total mechanical energy is not infinite. We call the point \(x=0\) a “stable equilibrium”, because it is a local minimum of the potential energy function. If the object is displaced from the equilibrium point, it will want to move back towards that point. This can also be understood in terms of the force associated with the potential energy function:
\[\begin{aligned} F = -\frac{d}{dx}U(x)\end{aligned}\]
The local minimum occurs where the derivative of the potential function is equal to zero. Thus, the equilibrium point is given by the condition that the force associated with the potential is zero ( \(x=0\) in the case of the potential energy from a spring). The equilibrium is a stable equilibrium because the force associated with the potential energy function ( \(F(x)=-kx\) for the spring) points towards the equilibrium point.
The potential energy function for an object with total mechanical energy, \(E\) , can be thought of as a little “roller coaster”, on which you place a marble and watch it “roll down” the potential energy function. You can think of placing a marble where \(U(x)=E\) and releasing it. The marble would then roll down the potential energy function, just as an actual marble would roll down a real slope, mimicking the motion of the object along the \(x\) axis. This is illustrated in Figure \(\PageIndex{2}\) which shows an arbitrary potential energy function and a marble being placed at a location where the potential energy is equal to \(E\) .
The motion of the marble will be bound between the two points where the potential energy function is equal to \(E\) . When the marble is placed as shown, it will roll towards the left, just as if it were a real marble on a track. Since the potential energy is increasing as a function of \(x\) at the point where we placed the marble, the force is in the negative \(x\) direction (remember, the force is the negative of the derivative of the potential energy function). With the given energy, the marble would never be able to make it to point \(D\) , as it does not have enough energy to “climb up the hill”. It would roll down, through point \(C\) , up to point \(B\) , down to point \(A\) , and then turn around where \(U(x)=E\) and return to where it started.
Locations \(A\) and \(C\) on the diagram are stable equilibria, because if a marble is placed in one of those locations and nudged slightly, it will come back to the equilibrium point (or oscillate about that point). Points \(B\) and \(D\) are “unstable equilibria”, because if the marble is placed there and nudged, it will not immediately come back to those points. Note that if the marble were placed at point \(D\) and nudged towards the right, the motion of the marble would be unbound on the right, and it would keep going in that direction.
Now, say an object’s potential energy is described by the function in Figure \(\PageIndex{2}\), and the object has total energy \(E\) . The object’s motion along the \(x\) axis will be exactly the same as the projection of the marble’s motion on the \(x\) axis.
Exercise \(\PageIndex{2}\)
A force, \(F(x)\) , acts on an object. The potential energy function, \(U(x)\) , associated with the force is given by \(U(x)=a(x-6)^2(x-1)(x-3)+{20}~\text{J}\) , where \(a\) is a positive constant. \(U(x)\) is plotted in Figure \(\PageIndex{3}\). Use the “marble” method to determine the direction of the force at \(x=5\) . Confirm your answer by finding the value of the force , \(F(x)\) , at \(x=5\) .
- \(F(x=5)=-10a\)
- \(F(x=5)=10a\)
- \(F(x=5)=20a\)
- \(F(x=5)=-20a\)
- Answer
- B.
Footnotes
1. Remember, the kinetic energy is given by \(K = \frac{1}{2}mv^{2}\). Since neither mass nor the value of \(v^{2}\) can be negative, the kinetic energy of an object can never be negative.


<IP_ADDRESS>: Faulty Sentence Structure
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Have you ever had a teacher say, “That sentence starts one way and ends another”? If you have, don’t feel badly. Others make this same mistake all the time. Sometimes, we simply lose our train of thought when we write, and we literally start a sentence one way and end it another, as illustrated in the following example:
It’s tough to make sense of this sentence, right?
Of course, this is not the only faulty sentence structure error you might make. It’s important to pay close attention to your sentence structure to make sure you have both a subject and a verb. And, if you are listing items or phrases in a sentence, make sure you keep the items in your list in the same form. Doing so keeps your sentences “parallel,” which is a good thing.
Here is an example of a sentence without parallel structure:
In the sentence above, "breakfast" is a noun, and "exercising" and "checking" begin verb phrases. The items on the list are not in the same form. Here is a correction where the items listed are in the same form:


16.5: Glossary
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Key Terms
- organizational structure
- The system of task and reporting relationships that control and motivate colleagues to achieve organizational goals.
- organizational design
- The process by which managers define organizational structure and culture so that the organization can achieve its goals.
- Organizational change
- The movement that organizations take as they move from one state to a future state.
- managed change
- How leaders in an organization intentionally shape shifts that occur in the organization when market conditions shift, supply sources change, or adaptations are introduced in the processes for accomplishing work over time.
- organization development (OD)
- Techniques and methods that managers can use to increase the adaptability of their organization.
- formal organization
- A fixed set of rules of organizational procedures and structures.
- informal organization
- The connecting social structure in organizations that denotes the evolving network of interactions among its employees, unrelated to the firm's formal authority structure.
- bureaucratic model
- Max Weber’s model that states that organizations will find efficiencies when they divide the duties of labor, allow people to specialize, and create structure for coordinating their differentiated efforts within a hierarchy of responsibility.
- specialization
- The degree to which people are organized into subunits according to their expertise—for example, human resources, finance, marketing, or manufacturing.
- command-and-control
- The way in which people report to one another or connect to coordinate their efforts in accomplishing the work of the organization.
- span of control
- The scope of the work that any one person in the organization will be accountable for.
- centralization
- The concentration of control of an activity or organization under a single authority.
- formalization
- The process of making a status formal for the practice of formal acceptance.
- mechanistic bureaucratic structure
- Describes organizations characterized by (1) centralized authority, (2) formalized procedures and practices, and (3) specialized functions. They are usually resistant to change.
- vertical organizational structure
- Organizational structures found in large mechanistic organizations; also called “tall” structures due to the presence of many levels of management.
- organic bureaucratic structure
- Used in organizations that face unstable and dynamic environments and need to quickly adapt to change.
- horizontal organizational structure
- Flat organizational structure in which many individuals across the whole system are empowered to make organizational decisions.
- flat organization
- A horizontal organizational structure in which many individuals across the whole system are empowered to make organizational decisions.
- product structures
- Occurs when businesses organize their employees according to product lines or lines of business.
- geographic structures
- Occur when organizations are set up to deliver a range of products within a geographic area or region.
- matrix structure
- An organizational structure that groups people by function and by product team simultaneously.
- structural change
- Changes in the overall formal relationships, or the architecture of relationships, within an organization.
- technological change
- Implementation of new technologies often forces organizations to change.
- culture change
- Involves reshaping and reimagining the core identity of the organization.
- differentiation
- The process of organizing employees into groups that focus on specific functions in the organization.
- entrepreneurship
- The process of designing, launching, and running a new business.
- scope of change
- The degree to which the required change will disrupt current patterns and routines.
- incremental change
- Small refinements in current organizational practices or routines that do not challenge, but rather build on or improve, existing aspects and practices within the organization.
- transformational change
- Significant shifts in an organizational system that may cause significant disruption to some underlying aspect of the organization, its processes, or its structures.
- strategic change
- A change, either incremental or transformational, that helps align an organization’s operations with its strategic mission and objectives.
- level of organization
- The breadth of the systems that need to be changed within an organization.
- individual-level change
- Focuses on how to help employees to improve some active aspect of their performance or the knowledge they need to continue to contribute to the organization in an effective manner.
- group-level change
- Centers on the relationships between people and focuses on helping people to work more effectively together.
- organization-level change
- A change that affects an entire organizational system or several of its units.
- intentionality
- The degree to which the change is intentionally designed or purposefully implemented.
- planned change
- An intentional activity or set of intentional activities that are designed to create movement toward a specific goal or end.
- unplanned change
- An unintentional activity that is usually the result of informal organizing.
- change management
- The process of designing and implementing change.
- organizational development (OD)
- Specialized field that focuses on how to design and manage change.
- OD consultant
- Someone who has expertise in change management processes.
- deficit-based change
- Leaders assume that employees will change if they know they will otherwise face negative consequences.
- abundance-based change
- Leaders assume that employees will change if they can be inspired to aim for greater degrees of excellence in their work.
- top-down change
- Relies on mechanistic assumptions about the nature of an organization.
- emergent or bottom-up approach
- Organizations exist as socially constructed systems in which people are constantly making sense of and enacting an organizational reality as they interact with others in a system.
- participatory management
- Includes employees in deliberations about key business decisions.
- conventional mindset
- Leaders assume that most people are inclined to resist change and therefore need to be managed in a way that encourages them to accept change.
- positive or appreciative mindset
- Leaders assume that people are inclined to embrace change when they are respected as individuals with intrinsic worth, agency, and capability.
- Lewin’s change model
- Explains a very basic process that accompanies most organizational changes.
- Kotter’s change model
- An overall framework for designing a long-term change process.
- Appreciative Inquiry model
- A model specifically designed as an abundance-based, bottom-up, positive approach.
- appreciative conversations
- Intense, positively framed discussions that help people to develop common ground as they work together to cocreate a positive vision of an ideal future for their organization.
- Complex Adaptive Systems (CAS)
- A model that views organizations as constantly developing and adapting to their environment, much like a living organism.
- change agents
- People in the organization who view themselves as agents who have discretion to act.
- boundary conditions
- Define the degree of discretion that is available to employees for self-directed action.
- disturbances
- Can cause tension amongst employees, but can also be positive and a catalyst for change.
Summary of Learning Outcomes
16.1 Organizational Structures and Design
- What are mechanistic versus organic organizational structures?
The organizational structure is designed from both the mechanistic and the organic points of view, and the structure depends upon the extent to which it is rigid or flexible. Flexible structures are also viewed as more humanistic than mechanistic structures. The mechanistic organizational structure is similar to Max Weber’s bureaucratic organization. Organic structures are more flexible in order to cope with rapidly changing environments. These structures are more effective if the environment is dynamic, requiring frequent changes within the organization in order to adjust to change. It is also considered to be a better form of organization when employees seek autonomy, openness, change, support for creativity and innovation, and opportunities to try new approaches.
All organizations need structures to accomplish their work, and they need an ability to change in order to sustain and renew themselves over time
16.2 Organizational Change
- What are the fundamental dimensions of change?
It is often said that the only constant is change. Managers need to have the ability to understand the dimensions of change, know what drives change, and know how to implement changes to meet and exceed organizational goals. The three types of change are structural, technological, and culture changes. Managers need to understand change as organizations evolve and grow over time.
One of the key responsibilities of management is to design organizational structures that will allow an organization to accomplish its primary objectives. The structure should always match the need for coordination. Often, managers cannot tell what form the organization should take until they experience the informal organization that determines how work is actually accomplished. Only then can they understand how to draw on the concepts of bureaucracy to appropriately design a structure that will maximize the likelihood of organizational success.
16.3 Managing Change
- How do managers deal with change?
As an organization grows and matures, change becomes necessary to its sustained viability. Thus, another key responsibility for most leaders is the task of designing and managing change. We have reviewed several questions that should be considered when designing a change process, and we have explored several approaches that may be used to guide the development of organizational change.
The field of knowledge about how to change and develop organizations is vast and can be somewhat confusing to the novice learner. The material presented in this chapter provides an overview of key ideas, but there is so much more to learn. Should you wish to become an influential leader of change, it is important to learn more about this very important field of research and practice.
Chapter Review Questions
- What is an organizational structure?
- What are different types of organizational structures?
- What is organizational design?
- What concepts should guide decisions about how to design structures?
- What is organizational change?
- What are the fundamental dimensions of change?
- What are organizational development (OD) and change management?
- What questions may be used to guide OD and change management?
- What are the common models of OD and change management?
Management Skills Application Exercises
- Refer to Exhibit 16.1.1, Exhibit 16.1.2, and Exhibit 16.2.1 for this exercise. Pick a business that you are familiar with, and draw their existing organizational chart. You may be able to infer much of the information from their website or through a short interview with someone in their organization. After completing this task, construct an alternative organizational chart and comment on why it may be more effective than the current organizational structure and what risks that new structure may have.
- You have been assigned the task of working with a company that had a traditional, functional organizational structure with sales, marketing, product development, finance and accounting, and operations teams each reporting to a VP, who then reported to the CEO. The company wants to move to a matrix organization that will retain the efficiencies of the functional organization but also groups employees by product teams. You have been asked to comment on how to manage this change and how to communicate and respond to employee concerns. Specifically, you need to address: What are the desired impacts or benefits of this project on the organization? What are the emotions that your employees may have about this organizational change? How could the employee emotions impact the organization or its operations? How can the organization manage these emotions, or in what ways do you think they should manage these emotions to get desired outcome?
Managerial Decision Exercises
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Place yourself in the position of a CEO who is contemplating a reorganization of your company and has received conflicting opinions from two of your trusted reports. Presently you are a wholesaler with 45 regional warehouses who acquires products from manufacturers and distributes them to retailers and service establishments. You have over 100,000 SKUs (stock keeping unit) ranging from ACE bandages to Ziploc bags. You have 825 field-based sales representatives who represent all the products within a geographic area.
One of the ideas that has been brought up by the vice president of marketing is to specialize the salesforce into three groups, fashion retail, general retail, and services. Basically, individual sales representatives would be able to specialize with greater expertise and product knowledge to better serve customers. The vice president of sales fears that many of her salespeople will leave due to the expanded geography that this change would require.
What process would you take to address the concerns of your managers? How would you implement the plan? What customer considerations would you need to address?
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You have recently accepted the position of director for a full-service retirement home that has three components. The first component is for retired individuals and married couples who can still manage on their own but appreciate the amenities such as medical care and having other residents that they interact with through planned activities. The second is for residents who are still relatively healthy but do need assistance for specific tasks such as mobility and the like. The third section is for individuals with chronic health issues and palliative care patients.
You have learned during the interview process that the facility has performance and morale issues and that the previous director had a rigid structure, did not allow workers from different roles to interact, and wanted all decisions to be directed to her. This has led to dramatic staff turnover and a larger number of empty units compared to other facilities.
As the incoming new director, you will need to address the staff, and your new assistant asks whether you would like to address the staff in one large room or in smaller meeting rooms with employees from the different functional units. She also asks how to handle the workers who are from different shifts. Make your communication decisions, and write up an opening statement to make to the employees before you open the meeting to questions.
Critical Thinking Case
Danny Meyer Leads His Company through the Challenges of Eliminating Tips
What happens when your CEO wants to remove the tip structure from your restaurant? Do you complain about the new prices as a customer? Do you worry about your paychecks as a server?
Danny Meyer, CEO of Union Square Hospitality (home to some of the most successful New York restaurants), discovered these answers when he began eliminating the tip structure in most of his restaurants. He had seen firsthand the largest negative impact of a tipping culture: employees stuck in front-line positions with no chance to advance to management without taking significant pay cuts.
Meyer began by first involving the affected employees in town-hall talks. These town halls happened months before any publicity was released. Meyer then hosted town halls with customers to explain the importance of fair wages for all his employees at the restaurant, not just the few who served the food. The transition period for each restaurant to eliminate tips was usually three to six months.
As a result of eliminating the tip structure in most of his restaurants, Meyer has been able to increase the pay structure for cooks at those locations, which enables him to fill more cook positions and address a common industry shortage. Meyer has also been able to hire employees with a purpose to deliver exceptional hospitality. Meyer encourages his employees to take care of each other first, and to then take care of the customer, which creates a virtuous cycle of hospitality.
Meyer constantly uses feedback from his employees even after the tip structure was eliminated. He wants to ensure that each employee feels their voice is heard and understood. Employees continue to have access to town-hall meetings and internal feedback channels to offer honest feedback.
Critical Thinking Questions
- What type of change is this: transformational or incremental? Why?
- What level(s) of change is Meyer aiming for in this case?
- What models are consistent with Meyer’s process for designing and implementing change?
Sources:
Mark Matousek, Dannu Meyer Banned Tipping at his Restaurants- But Employees Say it has Led to Lower Pay and High Turnover,” Business Insider , October 20, 2017, http://www.businessinsider.com/danny...uggles-2017-10
Loren Feldman, “Danny Meyer On Eliminating Tipping: “It Takes a Year to Get The Math Right,” Forbes , January 14, 2018, https://www.forbes.com/sites/lorenfe.../#189bd5c8431f ;
Elizabeth Dunn, “The Limitations of American Restaurants’ No-Tipping Experience,” The New Yorker , February 24, 2018, https://www.newyorker.com/culture/an...ing-experiment .


2.2: Solve Equations using the Division and Multiplication Properties of Equality
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Learning Objectives
By the end of this section, you will be able to:
- Solve equations using the Division and Multiplication Properties of Equality
- Solve equations that require simplification
- Translate to an equation and solve
- Translate and solve applications
Note
Before you get started, take this readiness quiz.
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Simplify: \(−7(\frac{1}{-7})\).
If you missed this problem, review Exercise 1.6.13 . -
Evaluate \(9x+2\) when \(x=−3\).
If you missed this problem, review Exercise 1.5.34 .
Solve Equations Using the Division and Multiplication Properties of Equality
You may have noticed that all of the equations we have solved so far have been of the form \(x+a=b\) or \(x−a=b\). We were able to isolate the variable by adding or subtracting the constant term on the side of the equation with the variable. Now we will see how to solve equations that have a variable multiplied by a constant and so will require division to isolate the variable.
Let’s look at our puzzle again with the envelopes and counters in Figure \(\PageIndex{1}\).
In the illustration there are two identical envelopes that contain the same number of counters. Remember, the left side of the workspace must equal the right side, but the counters on the left side are “hidden” in the envelopes. So how many counters are in each envelope?
How do we determine the number? We have to separate the counters on the right side into two groups of the same size to correspond with the two envelopes on the left side. The 6 counters divided into 2 equal groups gives 3 counters in each group (since \(6\div 2=3\)).
What equation models the situation shown in Figure \(\PageIndex{2}\)? There are two envelopes, and each contains xx counters. Together, the two envelopes must contain a total of 6 counters.
| If we divide both sides of the equation by 2, as we did with the envelopes and counters, | |
| we get: |
We found that each envelope contains 3 counters. Does this check? We know \(2\cdot 3=6\), so it works! Three counters in each of two envelopes does equal six!
This example leads to the Division Property of Equality .
THE DIVISION PROPERTY OF EQUALITY
For any numbers a , b , and c , and \(c\neq 0\),
\[\begin{array} {llll} {\text { If }} &{a} &{=} &{b} \\ {\text {then}} & {\frac { a } { c }} &{=} &{\frac { b } { c }} \end{array}\]
When you divide both sides of an equation by any non-zero number, you still have equality.
Note
Doing the Manipulative Mathematics activity “ Division Property of Equality ” will help you develop a better understanding of how to solve equations by using the Division Property of Equality.
The goal in solving an equation is to ‘undo’ the operation on the variable. In the next example, the variable is multiplied by 5, so we will divide both sides by 5 to ‘undo’ the multiplication.
Example \(\PageIndex{1}\)
Solve: \(5x=−27\).
Solution
| To isolate x, “undo” the multiplication by 5. | ||
| Divide to ‘undo’ the multiplication. | ||
| Simplify. | ||
| Check: | ||
| Substitute \(-\frac{27}{5}\) for x. | ||
|
Since this is a true statement, \(x = -\frac{27}{5}\)
is the solution to \(5x=−27\). |
Try It \(\PageIndex{2}\)
Solve: \(3y=−41\).
- Answer
-
\(y = -\frac{41}{3}\)
Try It \(\PageIndex{3}\)
Solve: \(4z=−55\).
- Answer
-
\(y = -\frac{55}{4}\)
Consider the equation \(\frac{x}{4} = 3\). We want to know what number divided by 4 gives 3. So to “undo” the division, we will need to multiply by 4. The Multiplication Property of Equality will allow us to do this. This property says that if we start with two equal quantities and multiply both by the same number, the results are equal.
THE MULTIPLICATION PROPERTY OF EQUALITY
For any numbers a , b , and c ,
\[\begin{array} {llll} {\text {If}} &{a} & {=} &{b} \\ {\text {then}} &{a c} &{=} &{b c} \end{array}\]
If you multiply both sides of an equation by the same number, you still have equality.
Example \(\PageIndex{4}\)
Solve: \(\frac{y}{-7} = -14\)
Solution
Here y is divided by −7. We must multiply by −7 to isolate y.
| \(\frac{y}{-7} = -14\) | ||
| Multiply both sides by −7. | ||
| Multiply. | ||
| Simplify. | ||
| Check: \(\frac{y}{-7} = -14\) | ||
| Substitute y=98. | \(\dfrac{\color{red}{98}}{-7} \stackrel{?}{=} -14\) | |
| Divide. |
Try It \(\PageIndex{5}\)
Solve: \(\frac{a}{-7} = -42\)
- Answer
-
\(a = 294\)
Try It \(\PageIndex{6}\)
Solve: \(\frac{b}{-6} = -24\)
- Answer
-
\(b = 144\)
Example \(\PageIndex{7}\)
Solve: \(-n = 9\)
Solution
| Remember −n is equivalent to −1n. | ||
| Divide both sides by −1. | ||
| Divide. | ||
| Notice that there are two other ways to solve −n=9. We can also solve this equation by multiplying both sides by −1 and also by taking the opposite of both sides. | ||
| Check: | ||
| Substitute n=−9. | ||
| Simplify. |
Try It \(\PageIndex{8}\)
Solve: \(−k=8\).
- Answer
-
\(k = -8\)
Try It \(\PageIndex{9}\)
Solve: \(−g=3\).
- Answer
-
\(g = -3\)
Example \(\PageIndex{10}\)
Solve: \(\frac{3}{4}x = 12\)
Solution
Since the product of a number and its reciprocal is 1, our strategy will be to isolate x by multiplying by the reciprocal of \(\frac{3}{4}\).
| Multiply by the reciprocal of \(\frac{3}{4}\). | ||
| Reciprocals multiply to 1. | ||
| Multiply. | ||
| Notice that we could have divided both sides of the equation \(\frac{3}{4}x = 12\) by \(\frac{3}{4}\) to isolate x. While this would work, most people would find multiplying by the reciprocal easier. | ||
| Check: | ||
| Substitute \(x=16\). | ||
Try It \(\PageIndex{11}\)
Solve: \(\frac{2}{5}n=14\).
- Answer
-
\(n = 35\)
Try It \(\PageIndex{12}\)
Solve: \(\frac{5}{6}y=15\).
- Answer
-
\(y = 18\)
In the next example, all the variable terms are on the right side of the equation. As always, our goal in solving the equation is to isolate the variable.
Example \(\PageIndex{13}\)
Solve: \(\frac{8}{15} = -\frac{4}{5}x\)
Solution
| Multiply by the reciprocal of \(-\frac{4}{5}\). | ||
| Reciprocals multiply to 1. | ||
| Multiply. | ||
| Check: | ||
| Let \(x = -\frac{2}{3}\). | ||
Try It \(\PageIndex{14}\)
Solve: \(\frac{9}{25} = -\frac{4}{5}z\)
- Answer
-
\(z = - \frac{9}{5}\)
Try It \(\PageIndex{15}\)
\(\frac{5}{6} = -\frac{8}{3}r\)
- Answer
-
\(r = -\frac{5}{16}\)
Solve Equations That Require Simplification
Many equations start out more complicated than the ones we have been working with.
With these more complicated equations the first step is to simplify both sides of the equation as much as possible. This usually involves combining like terms or using the distributive property.
Example \(\PageIndex{16}\)
Solve: \(14−23=12y−4y−5y\).
Solution
Begin by simplifying each side of the equation.
| Simplify each side. | ||
| Divide both sides by 3 to isolate y. | ||
| Divide. | ||
| Check: | ||
| Substitute \(y=−3\). | ||
Try It \(\PageIndex{17}\)
Solve: \(18−27=15c−9c−3c\).
- Answer
-
\(c=−3\)
Try It \(\PageIndex{18}\)
Solve: \(18−22=12x−x−4x\).
- Answer
-
\(x = -\frac{4}{7}\)
Example \(\PageIndex{19}\)
Solve: \(−4(a−3)−7=25\).
Solution
Here we will simplify each side of the equation by using the distributive property first.
| Distribute. | ||
| Simplify. | ||
| Simplify. | ||
| Divide both sides by \(-4\) to isolate a. | ||
| Divide. | ||
| Check: | ||
| Substitute \(a = -5\) | ||
Try It \(\PageIndex{20}\)
Solve: \(−4(q−2)−8=24\).
- Answer
-
\(q=−6\)
Try It \(\PageIndex{21}\)
Solve: \(−6(r−2)−12=30\).
- Answer
-
\(r=−5\)
Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference.
PROPERTIES OF EQUALITY
When you add, subtract, multiply, or divide the same quantity from both sides of an equation, you still have equality.
\[\begin{array} {ll} {\textbf {Subtraction Property of Equality}} &{\textbf{Addition Property of Equality}} \\
{\text{For any real numbers a, b, and c,}} &{\text{For any real numbers a, b, and c,}}
\\ {\text{if }a = b,} &{\text{if }a = b,}
\\ {\text{then }a - c = b - c} &{\text{then }a + c = b + c}
\\ {\textbf {Divition Property of Equality}} &{\textbf{Multiplication Property of Equality}} \\
{\text{For any real numbers a, b, and c,}} &{\text{For any real numbers a, b, and c,}}
\\ {\text{if }a = b,} &{\text{if }a = b,}
\\ {\text{then }a - c = b - c} &{\text{then }a + c = b + c}
\end{array}\]
When you add, subtract, multiply, or divide the same quantity from both sides of an equation, you still have equality.
Translate to an Equation and Solve
In the next few examples, we will translate sentences into equations and then solve the equations. You might want to review the translation table in the previous chapter.
Example \(\PageIndex{22}\)
Translate and solve: The number 143 is the product of −11 and y .
Solution
| Translate. | |
| Divide by −11. | |
| Simplify. | |
|
Check:
\[\begin{array} {lll} {143} &{=} &{-11y} \\ {143} &{\stackrel{?}{=}} &{-11(-13)} \\ {143} &{=} &{143\checkmark} \end{array}\] |
Try It \(\PageIndex{23}\)
Translate and solve: The number 132 is the product of −12 and y .
- Answer
-
132=−12y;y=−11
Try It \(\PageIndex{24}\)
Translate and solve: The number 117 is the product of −13 and z .
- Answer
-
117=−13z;z=−9
Example \(\PageIndex{25}\)
Translate and solve: n divided by 8 is −32.
Solution
|
Begin by translating the sentence into an equation.
Translate. |
||
| Multiple both sides by 8. | ||
| Simplify. | ||
| Check: | Is nn divided by 8 equal to −32? | |
| Let \(n=−256\). | Is −256 divided by 88 equal to −32? | |
| Translate. | \(\frac{-256}{8} \stackrel{?}{=} -32\) | |
| Simplify. | \(−32=−32\checkmark\) |
Try It \(\PageIndex{26}\)
Translate and solve: nn divided by 7 is equal to −21.
- Answer
-
\(\frac{n}{7}=−21; n=−147\)
Try It \(\PageIndex{27}\)
Translate and solve: n divided by 8 is equal to −56.
- Answer
-
\(\frac{n}{8}=−56;n=−448\)
Example \(\PageIndex{28}\)
Translate and solve: The quotient of yy and −4 is 68.
Solution
Begin by translating the sentence into an equation.
| Translate. | ||
| Multiply both sides by -4. | ||
| Simplify. | ||
| Check: | Is the quotient of y and −4 equal to 68? | |
| Let y=−272. | Is the quotient of −272 and −4 equal to 68? | |
| Translate. | \(\frac{-272}{-4} \stackrel{?}{=} 68\) | |
| Simplify. | \(68 = 68\checkmark\) |
Try It \(\PageIndex{29}\)
Translate and solve: The quotient of q and −8 is 72.
- Answer
-
\(\frac{q}{-8}=72;q=−576\)
Try It \(\PageIndex{30}\)
Translate and solve: The quotient of p and −9 is 81.
- Answer
-
\(\frac{p}{-9}=81;p=−729\)
Example \(\PageIndex{31}\)
Translate and solve: Three-fourths of p is 18.
Solution
Begin by translating the sentence into an equation. Remember, “of” translates into multiplication.
| Translate. | ||
| Multiply both sides by \(\frac{4}{3}\). | ||
| Simplify. | ||
| Check: | Is three-fourths of p equal to 18? | |
| Let p = 24. | Is three-fourths of 24 equal to 18? | |
| Translate. | \(\frac{3}{4}\cdot 24 \stackrel{?}{=} 18\) | |
| Simplify. | \(18=18\checkmark\) |
Try It \(\PageIndex{32}\)
Translate and solve: Two-fifths of f is 16.
- Answer
-
\(\frac{2}{5}f=16; f=40\)
Try It \(\PageIndex{33}\)
Translate and solve: Three-fourths of f is 21.
- Answer
-
\(\frac{3}{4}f=21; f=28\)
Example \(\PageIndex{34}\)
Translate and solve: The sum of three-eighths and x is one-half.
Solution
Begin by translating the sentence into an equation.
| Translate. | ||
| Subtract \(\frac{3}{8}\) from each side. | ||
| Simplify and rewrite fractions with common denominators. | ||
| Simplify. | ||
| Check: | Is the sum of three-eighths and x equal to one-half? | |
| Let \(x=\frac{1}{8}\). | Is the sum of three-eighths and one-eighth equal to one-half? | |
| Translate. | \(\frac{3}{8} + \frac{1}{8} \stackrel{?}{=} \frac{1}{2}\) | |
| Simplify. | \(\frac{4}{8} \stackrel{?}{=} \frac{1}{2}\) | |
| Simplify. | \(\frac{1}{2} = \frac{1}{2} \checkmark\) |
Try It \(\PageIndex{35}\)
Translate and solve: The sum of five-eighths and x is one-fourth.
- Answer
-
\(\frac{5}{8} + x = \frac{1}{4}; x = -\frac{3}{8}\)
Try It \(\PageIndex{36}\)
Translate and solve: The sum of three-fourths and x is five-sixths.
- Answer
-
\(\frac{3}{4} + x = \frac{5}{6}; x = \frac{1}{12}\)
Translate and Solve Applications
To solve applications using the Division and Multiplication Properties of Equality, we will follow the same steps we used in the last section. We will restate the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve.
Example \(\PageIndex{37}\)
Denae bought 6 pounds of grapes for $10.74. What was the cost of one pound of grapes?
Solution
\[\begin{array} {ll} {\text{What are you asked to find?}} &{\text{The cost of 1 pound of grapes}} \\\\ {\text{Assign a variable.}} &{\text{Let c = the cost of one pound.}} \\\\ {\text{Write a sentence that gives the}} &{\text{The cost of 6 pounds is }$10.74} \\ {\text{information to find it.}} &{} \\\\ {\text{Translate into an equation.}} &{6c = 10.74} \\ {\text{Solve.}} &{\frac{6c}{c} = \frac{10.74}{6}} \\ {} &{c = 1.79} \\\\ {} &{\text{The grapes cost }$ 1.79 \text{ per pound.}} \\ \\ {\text{Check: If one pound costs }$1.79, do} &{} \\ {\text{6 pounds cost }$ 10.74?} &{} \\\\ {6(1.79) \stackrel{?}{=} 10.74} &{} \\ {10.74 = 10.74\checkmark} &{} \end{array}\]
Try It \(\PageIndex{38}\)
Translate and solve:
Arianna bought a 24-pack of water bottles for $9.36. What was the cost of one water bottle?
- Answer
-
$0.39
Try It \(\PageIndex{39}\)
Translate and solve:
At JB’s Bowling Alley, 6 people can play on one lane for $34.98. What is the cost for each person?
- Answer
-
$5.83
Example \(\PageIndex{40}\)
Andreas bought a used car for $12,000. Because the car was 4-years old, its price was \(\frac{3}{4}\) of the original price, when the car was new. What was the original price of the car?
Solution
\[\begin{array} {ll} {\text{What are you asked to find?}} &{\text{The original price of the car}} \\\\ {\text{Assign a variable.}} &{\text{Let p = the original price.}} \\\\ {\text{Write a sentence that gives the}} &{$12000\text{ is }\frac{3}{4} \text{ of the original price.}} \\ {\text{information to find it.}} &{} \\\\ {\text{Translate into an equation.}} &{12000 = \frac{3}{4}p} \\ {} &{\frac{3}{4}(12000) = \frac{4}{3}\cdot \frac{3}{4}p}\\ {}&{16000 = p} \\{\text{Solve.}} &{} \\\\ {} &{\text{The original cost of the car was }$ 16000.} \\ \\ {\text{Check: Is }\frac{3}{4} \text{ of }$16000 \text{ equal to }$12000} &{} \\\\ {\frac{3}{4}\cdot 16000 \stackrel{?}{=} 12000} &{} \\ {12000 = 12000\checkmark} &{} \end{array}\]
Try It \(\PageIndex{41}\)
Translate and solve:
The annual property tax on the Mehta’s house is $1,800, calculated as \(\frac{15}{1000}\) of the assessed value of the house. What is the assessed value of the Mehta’s house?
- Answer
-
$120000
Try It \(\PageIndex{42}\)
Translate and solve:
Stella planted 14 flats of flowers in \(\frac{2}{3}\) of her garden. How many flats of flowers would she need to fill the whole garden?
- Answer
-
21 flats
Key Concepts
-
The Division Property of Equality
—For any numbers
a
,
b
, and
c
, and \(c\neq 0\), if \(a=b\), then \(\frac{a}{c} = \frac{b}{c}\).
When you divide both sides of an equation by any non-zero number, you still have equality. -
The Multiplication Property of Equality
—For any numbers
a
,
b
, and
c
, if \(a=b\), then \(ac = bc\).
If you multiply both sides of an equation by the same number, you still have equality.


3.2: Industrial and Technological Innovation
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The railroads created the first great concentrations of capital, spawned the first massive corporations, made the first of the vast fortunes that would define the Gilded Age, unleashed labor demands that united thousands of farmers and immigrants, and linked many towns and cities. National railroad mileage tripled in the twenty years after the outbreak of the Civil War, and tripled again over the four decades that followed. Railroads impelled the creation of uniform time zones across the country, gave industrialists access to remote markets, and opened the American West. Railroad companies were the nation’s largest businesses. Their vast national operations demanded the creation of innovative new corporate organization, advanced management techniques, and vast sums of capital. Their huge expenditures spurred countless industries and attracted droves of laborers. And as they crisscrossed the nation, they created a national market, a truly national economy, and, seemingly, a new national culture. 3
The railroads were not natural creations. Their vast capital requirements required the use of incorporation, a legal innovation that protected shareholders from losses. Enormous amounts of government support followed. Federal, state, and local governments offered unrivaled handouts to create the national rail networks. Lincoln’s Republican Party—which dominated government policy during the Civil War and Reconstruction—passed legislation granting vast subsidies. Hundreds of millions of acres of land and millions of dollars’ worth of government bonds were freely given to build the great transcontinental railroads and the innumerable trunk lines that quickly annihilated the vast geographic barriers that had so long sheltered American cities from one another.
As railroad construction drove economic development, new means of production spawned new systems of labor. Many wage earners had traditionally seen factory work as a temporary stepping-stone to attaining their own small businesses or farms. After the war, however, new technology and greater mechanization meant fewer and fewer workers could legitimately aspire to economic independence. Stronger and more organized labor unions formed to fight for a growing, more-permanent working class. At the same time, the growing scale of economic enterprises increasingly disconnected owners from their employees and day-to-day business operations. To handle their vast new operations, owners turned to managers. Educated bureaucrats swelled the ranks of an emerging middle class.
Industrialization also remade much of American life outside the workplace. Rapidly growing industrialized cities knit together urban consumers and rural producers into a single, integrated national market. Food production and consumption, for instance, were utterly nationalized. Chicago’s stockyards seemingly tied it all together. Between 1866 and 1886, ranchers drove a million head of cattle annually overland from Texas ranches to railroad depots in Kansas for shipment by rail to Chicago. After travelling through modern “disassembly lines,” the animals left the adjoining slaughterhouses as slabs of meat to be packed into refrigerated rail cars and sent to butcher shops across the continent. By 1885, a handful of large-scale industrial meatpackers in Chicago were producing nearly five hundred million pounds of “dressed” beef annually. 4 The new scale of industrialized meat production transformed the landscape. Buffalo herds, grasslands, and old-growth forests gave way to cattle, corn, and wheat. Chicago became the Gateway City, a crossroads connecting American agricultural goods, capital markets in New York and London, and consumers from all corners of the United States.
Technological innovation accompanied economic development. For April Fool’s Day in 1878, the New York Daily Graphic published a fictitious interview with the celebrated inventor Thomas A. Edison. The piece described the “biggest invention of the age”—a new Edison machine that could create forty different kinds of food and drink out of only air, water, and dirt. “Meat will no longer be killed and vegetables no longer grown, except by savages,” Edison promised. The machine would end “famine and pauperism.” And all for $5 or $6 per machine! The story was a joke, of course, but Edison nevertheless received inquiries from readers wondering when the food machine would be ready for the market. Americans had apparently witnessed such startling technological advances—advances that would have seemed far-fetched mere years earlier—that the Edison food machine seemed entirely plausible. 5
In September 1878, Edison announced a new and ambitious line of research and development—electric power and lighting. The scientific principles behind dynamos and electric motors—the conversion of mechanical energy to electrical power, and vice versa—were long known, but Edison applied the age’s bureaucratic and commercial ethos to the problem. Far from a lone inventor gripped by inspiration toiling in isolation, Edison advanced the model of commercially minded management of research and development. Edison folded his two identities, business manager and inventor, together. He called his Menlo Park research laboratory an “invention factory” and promised to turn out “a minor invention every ten days and a big thing every six months or so.” He brought his fully equipped Menlo Park research laboratory and the skilled machinists and scientists he employed to bear on the problem of building an electric power system—and commercializing it.
By late fall 1879, Edison exhibited his system of power generation and electrical light for reporters and investors. Then he scaled up production. He sold generators to businesses. By the middle of 1883, Edison had overseen construction of 330 plants powering over sixty thousand lamps in factories, offices, printing houses, hotels, and theaters around the world. He convinced municipal officials to build central power stations and run power lines. New York’s Pearl Street central station opened in September 1882 and powered a square mile of downtown Manhattan. Electricity revolutionized the world. It not only illuminated the night, it powered the Second Industrial Revolution. Factories could operate anywhere at any hour. Electric rail cars allowed for cities to build out and electric elevators allowed for them to build up.
Economic advances, technological innovation, social and cultural evolution, demographic changes: the United States was a nation transformed. Industry boosted productivity, railroads connected the nation, more and more Americans labored for wages, new bureaucratic occupations created a vast “white collar” middle class, and unprecedented fortunes rewarded the owners of capital. These revolutionary changes, of course, would not occur without conflict or consequence (see Chapter 16), but they demonstrated the profound transformations remaking the nation. Change was not confined to economics alone. Change gripped the lives of everyday Americans and fundamentally reshaped American culture. 6


6.6: Job Search Skills
Visible and Hidden Job Market
Experts estimate that only 20% of jobs are ever advertised through a newspaper ad, job posting, or an employment agency. This 20% is the visible job market. It takes relatively little time to apply for advertised jobs. Be sure to search the visible job market as thoroughly as possible. An employer will not be impressed if you phone looking for a job, unaware of an advertisement.
Job Advertisements
Online job boards, classified ads, and newspapers are the main channels through which jobs are advertised. A number of hospitality-specific websites cover all aspects of the tourism industry enabling employers to advertise positions in specific fields. Useful hospitality and tourism websites include go2HR , Hcareers , and Workopolis (hospitality and tourism sections). The B.C. government has also recently invested a great deal of resources into a provincial job board and other tools such as an apprentice job match at its WorkBC website .
In smaller communities, check the local newspapers, or newspapers from the nearest metropolitan area as well as classified ad websites like Craigslist and Kijiji. Each ad usually includes the job title, qualifications, and experience needed, and how and where to apply for the job.
A job advertisement may also give you information about the company, the salary and some of the benefits offered. Read the ad very carefully to see what the job requires. Compare the requirements with your own qualifications. In the current job market, you will need to meet all of the requirements in order to be interviewed for the job.
Job Postings
Most companies will have a career section on their website that may include available positions and how to submit resumés. Many of these will also include an online application form that must be completed.
Job ads may also be posted in trade publications, websites, and newsletters. Professional associations and special interest groups such as trade associations and organizations are a helpful source and often have job postings.
Colleges, institutes, and universities often have student employment or placement offices that provide information about job postings and advice on job searches.
Positions with the provincial government (for example, a cook in a provincial penitentiary or provincially run hospital) may be listed on the B.C. government’s MyHR website .
Private Employment Agencies
If you choose to do your search through a private employment agency, be sure that it deals with employers in the hotel, restaurant, and food service industry. Employment agencies must register with the Employment Standards Branch. They are not permitted to charge you a fee for finding you a job or identifying possible employers. They receive payment from the employer who uses their services.
Hidden Job Market
Many jobs are filled without ever having been advertised. You can find these jobs by contacting employers directly or using your personal and business contacts to develop job leads. You can contact employers directly by email, mail, or in person. Usually, personal contacts are preferable.
To identify employers who might hire someone with your qualifications, consult an online search engine or a local business directory.
A good way to find job leads is to become involved in the local branch of the restaurant association or the Canadian Culinary Federation. Some associations offer student or junior membership. Volunteer to assist with fundraising and other events. When a chef or restaurant manager knows that you are interested in the trade and willing to volunteer, he or she may be more willing to help you find a job. Attend trade fairs, restaurant, and food shows. This will help you find out more about the businesses in which you could work and can be a source of job contacts.
Watch for news items in your local media that may identify job opportunities. For example, a local travel magazine may have an article on a resort that will be opening soon or a hotel that is expanding its dining room. You might also come across a website that caters to foodies that lists upcoming restaurant openings.
Another source of direct contacts for job leads comes from your family, friends, colleagues, and co-workers, as well as from former employers, customers, and suppliers. The wider the range of your contacts, the greater the likelihood of a successful lead. Let people know that you are looking for a job so that they can be on the lookout for possible opportunities. Social media is an increasingly popular way to identify the hidden job market. Posting your profile on LinkedIn or asking your contacts via social networking sites like Facebook or Twitter for leads can pay dividends.
Conducting an Information Interview
The process of exploring possible job leads is called information interviewing. There are five steps to keep in mind:
1. Make initial contact
Speak with people directly. Have a specific person’s name if possible. If not, ask to speak to the restaurant manager or the chef. In large hotels, ask to speak to the executive chef. If you are contacting someone you know well, you may leave a message if that person is out.
If you don’t know the contact well, try and find out the best time to reach that person. Take into account the schedule of the business that you are contacting. If a restaurant is open for dinner only, try in the early afternoon. If they are open for lunch, try before lunch service or after it ends. Be polite and thank the person to whom you are speaking.
Once you have reached the manager or chef, ask the person to meet with you briefly about job leads. Explain that you would like to speak with him or her even though the company is not hiring right now. Be clear and to the point as you describe your background, training, and experience. Prepare yourself for the call so that you can speak clearly and concisely. If the person agrees to meet with you, confirm the person’s name, address, and the time of the meeting. Thank the person for his or her time.
If the person is unwilling to meet with you, ask whether he or she has any suggestions for other persons or companies you can contact. Get the necessary information. Thank him or her for speaking with you and go on to the next contact.
2. Request assistance
Arrive for the meeting on time. Dress neatly and be well groomed. Treat this meeting as though it were a job interview. Be polite. Bring a copy of your resumé and references. Be prepared with the necessary information to fill out an application. Be prepared with specific questions. Some questions that you might ask include:
- How are new employees usually recruited and what strategies work best (e.g., dropping in on employers, going through an employment agency)?
- Where might a person with my skills and background fit in this field of employment?
- Who would you recommend I talk to about job opportunities?
Ask whether the person might be willing to make an introduction for you. Having your contact write or phone other prospects is always more effective than you making the contact. If you make the contacts yourself, use the initial contact’s name when you call. For example, you might say, “This is Peter Jones calling. I was referred to you by Mr. Goodhost, the manager of Jerry’s Good Eats.”
3. Record information
Make notes during each contact interview for later follow-up. When you contact any of the leads given to you by another source, make notes for each person. Be sure to update your files each time you contact someone. Also be sure that you keep copies of all letters, job ads, and other job search information.
4. Encourage contact
Leave your name, email, and telephone number to make sure that everyone you contact knows how to get in contact with you. Request that the person contact you if he or she thinks of a lead after you leave. Leave a copy of your resumé. This provides the person with the necessary information for talking about you to others. Write a brief thank-you note after the meeting. This provides you with another opportunity to remind the person of your name and goal.
5. Keep checking
Check regularly and briefly with all of the people on your list of contacts. This acts as a reminder for them and may also provide new information. It is especially important to check with those people who have agreed to put you in touch with someone else. You want to be able to make that new contact as soon as possible. When you get a job, send a brief thank-you note to the contacts who assisted you in finding the job.


11.E: Fluid Statics (Exercises)
Conceptual Questions
11.1: What Is a Fluid?
1. What physical characteristic distinguishes a fluid from a solid?
2. Which of the following substances are fluids at room temperature: air, mercury, water, glass?
3. Why are gases easier to compress than liquids and solids?
4. How do gases differ from liquids?
11.2: Density
5. Approximately how does the density of air vary with altitude?
6. Give an example in which density is used to identify the substance composing an object. Would information in addition to average density be needed to identify the substances in an object composed of more than one material?
7. Figure shows a glass of ice water filled to the brim. Will the water overflow when the ice melts? Explain your answer.
11.3: Pressure
8. How is pressure related to the sharpness of a knife and its ability to cut?
9. Why does a dull hypodermic needle hurt more than a sharp one?
10. The outward force on one end of an air tank was calculated in Example. How is this force balanced? (The tank does not accelerate, so the force must be balanced.)
11. Why is force exerted by static fluids always perpendicular to a surface?
12. In a remote location near the North Pole, an iceberg floats in a lake. Next to the lake (assume it is not frozen) sits a comparably sized glacier sitting on land. If both chunks of ice should melt due to rising global temperatures (and the melted ice all goes into the lake), which ice chunk would give the greatest increase in the level of the lake water, if any?
13. How do jogging on soft ground and wearing padded shoes reduce the pressures to which the feet and legs are subjected?
14. Toe dancing (as in ballet) is much harder on toes than normal dancing or walking. Explain in terms of pressure.
15. How do you convert pressure units like millimeters of mercury, centimeters of water, and inches of mercury into units like newtons per meter squared without resorting to a table of pressure conversion factors?
11.4: Variation of Pressure with Depth in a Fluid
16. Atmospheric pressure exerts a large force (equal to the weight of the atmosphere above your body—about 10 tons) on the top of your body when you are lying on the beach sunbathing. Why are you able to get up?
17. Why does atmospheric pressure decrease more rapidly than linearly with altitude?
18. What are two reasons why mercury rather than water is used in barometers?
19. Figure shows how sandbags placed around a leak outside a river levee can effectively stop the flow of water under the levee. Explain how the small amount of water inside the column formed by the sandbags is able to balance the much larger body of water behind the levee.
Because the river level is very high, it has started to leak under the levee. Sandbags are placed around the leak, and the water held by them rises until it is the same level as the river, at which point the water there stops rising.
20. Why is it difficult to swim under water in the Great Salt Lake?
21. Is there a net force on a dam due to atmospheric pressure? Explain your answer.
22. Does atmospheric pressure add to the gas pressure in a rigid tank? In a toy balloon? When, in general, does atmospheric pressure not affect the total pressure in a fluid?
23. You can break a strong wine bottle by pounding a cork into it with your fist, but the cork must press directly against the liquid filling the bottle—there can be no air between the cork and liquid. Explain why the bottle breaks, and why it will not if there is air between the cork and liquid.
11.5: Pascal’s Principle
24. Suppose the master cylinder in a hydraulic system is at a greater height than the slave cylinder. Explain how this will affect the force produced at the slave cylinder.
11.6: Gauge Pressure, Absolute Pressure, and Pressure Measurement
25. Explain why the fluid reaches equal levels on either side of a manometer if both sides are open to the atmosphere, even if the tubes are of different diameters.
26. Figure shows how a common measurement of arterial blood pressure is made. Is there any effect on the measured pressure if the manometer is lowered? What is the effect of raising the arm above the shoulder? What is the effect of placing the cuff on the upper leg with the person standing? Explain your answers in terms of pressure created by the weight of a fluid.
27. Considering the magnitude of typical arterial blood pressures, why are mercury rather than water manometers used for these measurements?
11.7: Archimedes’ Principle
28. More force is required to pull the plug in a full bathtub than when it is empty. Does this contradict Archimedes’ principle? Explain your answer.
29. Do fluids exert buoyant forces in a “weightless” environment, such as in the space shuttle? Explain your answer.
30. Will the same ship float higher in salt water than in freshwater? Explain your answer.
31. Marbles dropped into a partially filled bathtub sink to the bottom. Part of their weight is supported by buoyant force, yet the downward force on the bottom of the tub increases by exactly the weight of the marbles. Explain why.
11.8: Cohesion and Adhesion in Liquids: Surface Tension and Capillary Action
32. The density of oil is less than that of water, yet a loaded oil tanker sits lower in the water than an empty one. Why?
33. Is surface tension due to cohesive or adhesive forces, or both?
34. Is capillary action due to cohesive or adhesive forces, or both?
35. Birds such as ducks, geese, and swans have greater densities than water, yet they are able to sit on its surface. Explain this ability, noting that water does not wet their feathers and that they cannot sit on soapy water.
36. Water beads up on an oily sunbather, but not on her neighbor, whose skin is not oiled. Explain in terms of cohesive and adhesive forces.
37. Could capillary action be used to move fluids in a “weightless” environment, such as in an orbiting space probe?
38. What effect does capillary action have on the reading of a manometer with uniform diameter? Explain your answer.
39. Pressure between the inside chest wall and the outside of the lungs normally remains negative. Explain how pressure inside the lungs can become positive (to cause exhalation) without muscle action.
Problems & Exercises
11.2: Density
40. Gold is sold by the troy ounce (31.103 g). What is the volume of 1 troy ounce of pure gold?
Solution
\(1.610cm^3\)
41. Mercury is commonly supplied in flasks containing 34.5 kg (about 76 lb). What is the volume in liters of this much mercury?
42. (a) What is the mass of a deep breath of air having a volume of 2.00 L?
(b) Discuss the effect taking such a breath has on your body’s volume and density.
Solution
(a) 2.58 g
(b) The volume of your body increases by the volume of air you inhale. The average density of your body decreases when you take a deep breath, because the density of air is substantially smaller than the average density of the body before you took the deep breath.
43. A straightforward method of finding the density of an object is to measure its mass and then measure its volume by submerging it in a graduated cylinder. What is the density of a 240-g rock that displaces \(89.0cm^3\) of water? (Note that the accuracy and practical applications of this technique are more limited than a variety of others that are based on Archimedes’ principle.)
Solution
\(2.70g/cm^3\)
44. Suppose you have a coffee mug with a circular cross section and vertical sides (uniform radius). What is its inside radius if it holds 375 g of coffee when filled to a depth of 7.50 cm? Assume coffee has the same density as water.
45. (a) A rectangular gasoline tank can hold 50.0 kg of gasoline when full. What is the depth of the tank if it is 0.500-m wide by 0.900-m long?
(b) Discuss whether this gas tank has a reasonable volume for a passenger car.
Solution
(a) 0.163 m
(b) Equivalent to 19.4 gallons, which is reasonable
46. A trash compactor can reduce the volume of its contents to 0.350 their original value. Neglecting the mass of air expelled, by what factor is the density of the rubbish increased?
47. A 2.50-kg steel gasoline can holds 20.0 L of gasoline when full. What is the average density of the full gas can, taking into account the volume occupied by steel as well as by gasoline?
Solution
\(7.9×10^2kg/m^3\)
48. What is the density of 18.0-karat gold that is a mixture of 18 parts gold, 5 parts silver, and 1 part copper? (These values are parts by mass, not volume.) Assume that this is a simple mixture having an average density equal to the weighted densities of its constituents.
Solution
\(15.6g/cm^3\)
49. There is relatively little empty space between atoms in solids and liquids, so that the average density of an atom is about the same as matter on a macroscopic scale—approximately \(10^3kg/m^3\). The nucleus of an atom has a radius about \(10^{−5}\) that of the atom and contains nearly all the mass of the entire atom.
(a) What is the approximate density of a nucleus?
(b) One remnant of a supernova, called a neutron star, can have the density of a nucleus. What would be the radius of a neutron star with a mass 10 times that of our Sun (the radius of the Sun is \(7×10^8m\))?
Solution
(a) \(10^{18}kg/m^3\)
(b) \(2×10^4m\)
11.3: Pressure
50. As a woman walks, her entire weight is momentarily placed on one heel of her high-heeled shoes. Calculate the pressure exerted on the floor by the heel if it has an area of \(1.50cm^2\) and the woman’s mass is 55.0 kg. Express the pressure in Pa. (In the early days of commercial flight, women were not allowed to wear high-heeled shoes because aircraft floors were too thin to withstand such large pressures.)
Solution
\(3.59×10^6Pa\); or \(521lb/in^2\)
51. The pressure exerted by a phonograph needle on a record is surprisingly large. If the equivalent of 1.00 g is supported by a needle, the tip of which is a circle 0.200 mm in radius, what pressure is exerted on the record in \(N/m^2\)?
52. Nail tips exert tremendous pressures when they are hit by hammers because they exert a large force over a small area. What force must be exerted on a nail with a circular tip of 1.00 mm diameter to create a pressure of \(3.00×10^9N/m^2\)?(This high pressure is possible because the hammer striking the nail is brought to rest in such a short distance.)
Solution
\(2.36×10^3N\)
11.4: Variation of Pressure with Depth in a Fluid
53. What depth of mercury creates a pressure of 1.00 atm?
Solution
0.760 m
54. The greatest ocean depths on the Earth are found in the Marianas Trench near the Philippines. Calculate the pressure due to the ocean at the bottom of this trench, given its depth is 11.0 km and assuming the density of seawater is constant all the way down.
55. Verify that the SI unit of \(hρg\) is \(N/m^2\).
\((hρg)_{units}=(m)(kg/m^3)(m/s^2)=(kg⋅m^2)/(m^3⋅s^2)\)
\(=(kg⋅m/s^2)(1/m^2)\)
\(=N/m^2\)
56. Water towers store water above the level of consumers for times of heavy use, eliminating the need for high-speed pumps. How high above a user must the water level be to create a gauge pressure of \(3.00×10^5N/m^2\)?
57. The aqueous humor in a person’s eye is exerting a force of 0.300 N on the \(1.10-cm^2\) area of the cornea.
(a) What pressure is this in mm Hg?
(b) Is this value within the normal range for pressures in the eye?
Solution
(a) 20.5 mm Hg
(b) The range of pressures in the eye is 12–24 mm Hg, so the result in part (a) is within that range
58. How much force is exerted on one side of an 8.50 cm by 11.0 cm sheet of paper by the atmosphere? How can the paper withstand such a force?
59. What pressure is exerted on the bottom of a 0.500-m-wide by 0.900-m-long gas tank that can hold 50.0 kg of gasoline by the weight of the gasoline in it when it is full?
Solution
\(1.09×10^3N/m^2\)
60. Calculate the average pressure exerted on the palm of a shot-putter’s hand by the shot if the area of contact is \(50.0cm^2\) and he exerts a force of 800 N on it. Express the pressure in \(N/m^2\) and compare it with the \(1.00×10^6Pa\) pressures sometimes encountered in the skeletal system.
61. The left side of the heart creates a pressure of 120 mm Hg by exerting a force directly on the blood over an effective area of \(15.0cm^2\). What force does it exert to accomplish this?
Solution
24.0 N
62. Show that the total force on a rectangular dam due to the water behind it increases with the square of the water depth. In particular, show that this force is given by \(F=ρgh^2L/2\), where \(ρ\) is the density of water, \(h\) is its depth at the dam, and \(L\) is the length of the dam. You may assume the face of the dam is vertical. (Hint: Calculate the average pressure exerted and multiply this by the area in contact with the water. (See Figure.)
11.5: Pascal’s Principle
63. How much pressure is transmitted in the hydraulic system considered in Example? Express your answer in pascals and in atmospheres.
Solution
\(2.55×10^7Pa\); or 251 atm
64. What force must be exerted on the master cylinder of a hydraulic lift to support the weight of a 2000-kg car (a large car) resting on the slave cylinder? The master cylinder has a 2.00-cm diameter and the slave has a 24.0-cm diameter.
65. A crass host pours the remnants of several bottles of wine into a jug after a party. He then inserts a cork with a 2.00-cm diameter into the bottle, placing it in direct contact with the wine. He is amazed when he pounds the cork into place and the bottom of the jug (with a 14.0-cm diameter) breaks away. Calculate the extra force exerted against the bottom if he pounded the cork with a 120-N force.
Solution
\(5.76×10^3N\) extra force
66. A certain hydraulic system is designed to exert a force 100 times as large as the one put into it.
(a) What must be the ratio of the area of the slave cylinder to the area of the master cylinder?
(b) What must be the ratio of their diameters?
(c) By what factor is the distance through which the output force moves reduced relative to the distance through which the input force moves? Assume no losses to friction.
67. (a) Verify that work input equals work output for a hydraulic system assuming no losses to friction. Do this by showing that the distance the output force moves is reduced by the same factor that the output force is increased. Assume the volume of the fluid is constant.
(b) What effect would friction within the fluid and between components in the system have on the output force? How would this depend on whether or not the fluid is moving?
Solution
(a) \(V=d_iA_i=d_oA_o⇒d_o=d_i(\frac{A_i}{A_o})\).
Now, using equation:
\(\frac{F_1}{A_1}=\frac{F_2}{A_2}⇒F_o=F_i(\frac{A_o}{A_i})\).
Finally,
\(W_o=F_od_o=(\frac{F_iA_o}{A_i})(\frac{d_iA_i}{A_o})=F_id_i=W_i\).
In other words, the work output equals the work input.
(b) If the system is not moving, friction would not play a role. With friction, we know there are losses, so that \(W_{out}=W_{in}−W_f\); therefore, the work output is less than the work input. In other words, with friction, you need to push harder on the input piston than was calculated for the nonfriction case.
11.6: Gauge Pressure, Absolute Pressure, and Pressure Measurement
68. Find the gauge and absolute pressures in the balloon and peanut jar shown in Figure, assuming the manometer connected to the balloon uses water whereas the manometer connected to the jar contains mercury. Express in units of centimeters of water for the balloon and millimeters of mercury for the jar, taking \(h=0.0500 m\) for each.
Solution
Balloon:
Pg=5.00 cmH2O,
Pabs=1.035×103cmH2O.
Jar:
Pg=−50.0 mm Hg,
Pabs=710 mm Hg.
69. (a) Convert normal blood pressure readings of 120 over 80 mm Hg to newtons per meter squared using the relationship for pressure due to the weight of a fluid (\(P=hρg\)) rather than a conversion factor.
(b) Discuss why blood pressures for an infant could be smaller than those for an adult. Specifically, consider the smaller height to which blood must be pumped.
70. How tall must a water-filled manometer be to measure blood pressures as high as 300 mm Hg?
Solution
4.08 m
71. Pressure cookers have been around for more than 300 years, although their use has strongly declined in recent years (early models had a nasty habit of exploding). How much force must the latches holding the lid onto a pressure cooker be able to withstand if the circular lid is \(25.0 cm\) in diameter and the gauge pressure inside is 300 atm? Neglect the weight of the lid.
72. Suppose you measure a standing person’s blood pressure by placing the cuff on his leg 0.500 m below the heart. Calculate the pressure you would observe (in units of mm Hg) if the pressure at the heart were 120 over 80 mm Hg. Assume that there is no loss of pressure due to resistance in the circulatory system (a reasonable assumption, since major arteries are large).
Solution
\(ΔP=38.7 mm Hg\), Leg blood pressure\(=\frac{159}{119}.\)
73. A submarine is stranded on the bottom of the ocean with its hatch 25.0 m below the surface. Calculate the force needed to open the hatch from the inside, given it is circular and 0.450 m in diameter. Air pressure inside the submarine is 1.00 atm.
74. Assuming bicycle tires are perfectly flexible and support the weight of bicycle and rider by pressure alone, calculate the total area of the tires in contact with the ground. The bicycle plus rider has a mass of 80.0 kg, and the gauge pressure in the tires is \(3.50×10^5Pa\).
Solution
\(22.4cm^2\)
11.7: Archimedes’ Principle
75. What fraction of ice is submerged when it floats in freshwater, given the density of water at 0°C is very close to \(1000 kg/m^3\)?
Solution
\(91.7%\)
76. Logs sometimes float vertically in a lake because one end has become water-logged and denser than the other. What is the average density of a uniform-diameter log that floats with 20.0% of its length above water?
77. Find the density of a fluid in which a hydrometer having a density of \(0.750 g/mL\) floats with \(92.0%\) of its volume submerged.
Solution
\(815 kg/m^3\)
78. If your body has a density of \(995 kg/m^3\), what fraction of you will be submerged when floating gently in:
(a) Freshwater?
(b) Salt water, which has a density of \(1027 kg/m^3\)?
79. Bird bones have air pockets in them to reduce their weight—this also gives them an average density significantly less than that of the bones of other animals. Suppose an ornithologist weighs a bird bone in air and in water and finds its mass is \(45.0 g\) and its apparent mass when submerged is \(3.60 g\) (the bone is watertight).
(a) What mass of water is displaced?
(b) What is the volume of the bone?
(c) What is its average density?
Solution
(a) \(41.4 g\)
(b) \(41.4cm^3\)
(c) \(1.09 g/cm^3\)
80. A rock with a mass of 540 g in air is found to have an apparent mass of 342 g when submerged in water.
(a) What mass of water is displaced?
(b) What is the volume of the rock?
(c) What is its average density? Is this consistent with the value for granite?
81. Archimedes’ principle can be used to calculate the density of a fluid as well as that of a solid. Suppose a chunk of iron with a mass of 390.0 g in air is found to have an apparent mass of 350.5 g when completely submerged in an unknown liquid.
(a) What mass of fluid does the iron displace?
(b) What is the volume of iron, using its density as given in [link]
(c) Calculate the fluid’s density and identify it.
Solution
(a) 39.5 g
(b) \(50cm^3\)
(c) \(0.79g/cm^3\)
It is ethyl alcohol.
82. In an immersion measurement of a woman’s density, she is found to have a mass of 62.0 kg in air and an apparent mass of 0.0850 kg when completely submerged with lungs empty.
(a) What mass of water does she displace?
(b) What is her volume?
(c) Calculate her density.
(d) If her lung capacity is 1.75 L, is she able to float without treading water with her lungs filled with air?
83. Some fish have a density slightly less than that of water and must exert a force (swim) to stay submerged. What force must an 85.0-kg grouper exert to stay submerged in salt water if its body density is \(1015kg/m^3\)?
Solution
8.21 N
84. (a) Calculate the buoyant force on a 2.00-L helium balloon.
(b) Given the mass of the rubber in the balloon is 1.50 g, what is the net vertical force on the balloon if it is let go? You can neglect the volume of the rubber.
85. (a) What is the density of a woman who floats in freshwater with \(4.00%\) of her volume above the surface? This could be measured by placing her in a tank with marks on the side to measure how much water she displaces when floating and when held under water (briefly).
(b) What percent of her volume is above the surface when she floats in seawater?
Solution
(a) \(960kg/m^3\)
(b) \(6.34%\)
She indeed floats more in seawater.
86. A certain man has a mass of 80 kg and a density of \(955kg/m^3\) (excluding the air in his lungs).
(a) Calculate his volume.
(b) Find the buoyant force air exerts on him.
(c) What is the ratio of the buoyant force to his weight?
87. A simple compass can be made by placing a small bar magnet on a cork floating in water.
(a) What fraction of a plain cork will be submerged when floating in water?
(b) If the cork has a mass of 10.0 g and a 20.0-g magnet is placed on it, what fraction of the cork will be submerged?
(c) Will the bar magnet and cork float in ethyl alcohol?
Solution
(a) \(0.24\)
(b) \(0.68\)
(c) Yes, the cork will float because \(ρ_{obj}<ρ_{\text{ethyl alcohol}}(0.678g/cm^3<0.79g/cm^3)\)
88. What fraction of an iron anchor’s weight will be supported by buoyant force when submerged in saltwater?
89. Scurrilous con artists have been known to represent gold-plated tungsten ingots as pure gold and sell them to the greedy at prices much below gold value but deservedly far above the cost of tungsten. With what accuracy must you be able to measure the mass of such an ingot in and out of water to tell that it is almost pure tungsten rather than pure gold?
Solution
The difference is 0.006%.
90 . A twin-sized air mattress used for camping has dimensions of 100 cm by 200 cm by 15 cm when blown up. The weight of the mattress is 2 kg. How heavy a person could the air mattress hold if it is placed in freshwater?
91. Referring to Figure, prove that the buoyant force on the cylinder is equal to the weight of the fluid displaced (Archimedes’ principle). You may assume that the buoyant force is \(F_2−F_1\) and that the ends of the cylinder have equal areas \(A\). Note that the volume of the cylinder (and that of the fluid it displaces) equals \((h_2−h_1)A\).
Solution
\(F_{net}=F_2−F_1=P_2A−P_1A=(P_2−P_1)A\)
\(=(h_2ρ_{fl}g−h_1ρ_{fl}g)A\)
\(=(h_2−h_1)ρ_{fl}gA\)
where \(ρ_{fl}\) = density of fluid. Therefore,
\(F_{net}=(h_2−h_1)Aρ_{fl}g=V_{fl}ρ_{fl}g=m_{fl}g=w_{fl}\)
where is \(w_{fl}\) the weight of the fluid displaced.
92. (a) A 75.0-kg man floats in freshwater with \(3.00%\) of his volume above water when his lungs are empty, and \(5.00%\) of his volume above water when his lungs are full. Calculate the volume of air he inhales—called his lung capacity—in liters.
(b) Does this lung volume seem reasonable?
11.8: Cohesion and Adhesion in Liquids: Surface Tension and Capillary Action
93. What is the pressure inside an alveolus having a radius of \(2.50×10^{−4}m\) if the surface tension of the fluid-lined wall is the same as for soapy water? You may assume the pressure is the same as that created by a spherical bubble.
Solution
\(592N/m^2\)
94. (a) The pressure inside an alveolus with a \(2.00×10^{−4}\)-m radius is \(1.40×10^3Pa\), due to its fluid-lined walls.
(b) Identify the likely fluid. (You may need to extrapolate between values in Table.)
95. What is the gauge pressure in millimeters of mercury inside a soap bubble 0.100 m in diameter?
Solution
\(2.23×10^{−2}mm Hg\)
96. Calculate the force on the slide wire in Figure if it is 3.50 cm long and the fluid is ethyl alcohol.
97. Figure(a) shows the effect of tube radius on the height to which capillary action can raise a fluid.
(a) Calculate the height \(h\) for water in a glass tube with a radius of 0.900 cm—a rather large tube like the one on the left.
(b) What is the radius of the glass tube on the right if it raises water to 4.00 cm?
Solution
(a) \(1.65×10^{−3}m\)
(b) \(3.71×10^{–4}m\)
98. We stated in Example that a xylem tube is of radius \(2.50×10^{−5}m\). Verify that such a tube raises sap less than a meter by finding h for it, making the same assumptions that sap’s density is \(1050kg/m^3\), its contact angle is zero, and its surface tension is the same as that of water at \(20.0º C\).
99. What fluid is in the device shown in Figure if the force is \(3.16×10^{−3}N\) and the length of the wire is 2.50 cm? Calculate the surface tension \(γ\) and find a likely match from Table.
Solution
\(6.32×10^{−2}N/m\)
Based on the values in table, the fluid is probably glycerin.
100. If the gauge pressure inside a rubber balloon with a 10.0-cm radius is 1.50 cm of water, what is the effective surface tension of the balloon?
101. Calculate the gauge pressures inside 2.00-cm-radius bubbles of water, alcohol, and soapy water. Which liquid forms the most stable bubbles, neglecting any effects of evaporation?
Solution
\(P_w=14.6N/m^2,\)
\(P_a=4.46N/m^2,\)
\(P_{sw}=7.40N/m^2.\)
Alcohol forms the most stable bubble, since the absolute pressure inside is closest to atmospheric pressure.
102. Suppose water is raised by capillary action to a height of 5.00 cm in a glass tube.
(a) To what height will it be raised in a paraffin tube of the same radius?
(b) In a silver tube of the same radius?
103. Calculate the contact angle \(θ\) for olive oil if capillary action raises it to a height of 7.07 cm in a glass tube with a radius of 0.100 mm. Is this value consistent with that for most organic liquids?
Solution
\(5.1º\)
This is near the value of \(θ=0º\) for most organic liquids.
104. When two soap bubbles touch, the larger is inflated by the smaller until they form a single bubble.
(a) What is the gauge pressure inside a soap bubble with a 1.50-cm radius?
(b) Inside a 4.00-cm-radius soap bubble?
(c) Inside the single bubble they form if no air is lost when they touch?
105. Calculate the ratio of the heights to which water and mercury are raised by capillary action in the same glass tube.
Solution
\(−2.78\)
The ratio is negative because water is raised whereas mercury is lowered.
106. What is the ratio of heights to which ethyl alcohol and water are raised by capillary action in the same glass tube?
11.9: Pressures in the Body
107. During forced exhalation, such as when blowing up a balloon, the diaphragm and chest muscles create a pressure of 60.0 mm Hg between the lungs and chest wall. What force in newtons does this pressure create on the \(600cm^2\) surface area of the diaphragm?
Solurion
479 N
108. You can chew through very tough objects with your incisors because they exert a large force on the small area of a pointed tooth. What pressure in pascals can you create by exerting a force of \(500 N\) with your tooth on an area of \(1.00mm^2\)?
109. One way to force air into an unconscious person’s lungs is to squeeze on a balloon appropriately connected to the subject. What force must you exert on the balloon with your hands to create a gauge pressure of 4.00 cm water, assuming you squeeze on an effective area of \(50.0cm^2\)?
Solution
1.96 N
110. Heroes in movies hide beneath water and breathe through a hollow reed (villains never catch on to this trick). In practice, you cannot inhale in this manner if your lungs are more than 60.0 cm below the surface. What is the maximum negative gauge pressure you can create in your lungs on dry land, assuming you can achieve \(−3.00 cm\) water pressure with your lungs 60.0 cm below the surface?
Solution
\(−63.0 cm\) \(H_2O\)
111. Gauge pressure in the fluid surrounding an infant’s brain may rise as high as 85.0 mm Hg (5 to 12 mm Hg is normal), creating an outward force large enough to make the skull grow abnormally large.
(a) Calculate this outward force in newtons on each side of an infant’s skull if the effective area of each side is \(70.0cm^2\).
(b) What is the net force acting on the skull?
112. A full-term fetus typically has a mass of 3.50 kg.
(a) What pressure does the weight of such a fetus create if it rests on the mother’s bladder, supported on an area of \(90.0cm^2\)?
(b) Convert this pressure to millimeters of mercury and determine if it alone is great enough to trigger the micturition reflex (it will add to any pressure already existing in the bladder).
Solution
(a) \(3.81×10^3N/m^2\)
(b) \(28.7 mm Hg\), which is sufficient to trigger micturition reflex
113. If the pressure in the esophagus is \(−2.00 mm Hg\) while that in the stomach is \(+20.0 mm Hg\), to what height could stomach fluid rise in the esophagus, assuming a density of 1.10 g/mL? (This movement will not occur if the muscle closing the lower end of the esophagus is working properly.)
114. Pressure in the spinal fluid is measured as shown in Figure. If the pressure in the spinal fluid is 10.0 mm Hg:
(a) What is the reading of the water manometer in cm water?
(b) What is the reading if the person sits up, placing the top of the fluid 60 cm above the tap? The fluid density is 1.05 g/mL.
A water manometer used to measure pressure in the spinal fluid. The height of the fluid in the manometer is measured relative to the spinal column, and the manometer is open to the atmosphere. The measured pressure will be considerably greater if the person sits up.
Solution
(a) 13.6 m water
(b) 76.5 cm water
115. Calculate the maximum force in newtons exerted by the blood on an aneurysm, or ballooning, in a major artery, given the maximum blood pressure for this person is 150 mm Hg and the effective area of the aneurysm is \(20.0cm^2\). Note that this force is great enough to cause further enlargement and subsequently greater force on the ever-thinner vessel wall.
116. During heavy lifting, a disk between spinal vertebrae is subjected to a 5000-N compressional force.
(a) What pressure is created, assuming that the disk has a uniform circular cross section 2.00 cm in radius?
(b) What deformation is produced if the disk is 0.800 cm thick and has a Young’s modulus of \(1.5×10^9N/m^2\) ?
Solution
(a) \(3.98×10^6Pa\)
(b) \(2.1×10^{−3}cm\)
117. When a person sits erect, increasing the vertical position of their brain by 36.0 cm, the heart must continue to pump blood to the brain at the same rate.
(a) What is the gain in gravitational potential energy for 100 mL of blood raised 36.0 cm?
(b) What is the drop in pressure, neglecting any losses due to friction?
(c) Discuss how the gain in gravitational potential energy and the decrease in pressure are related.
118. (a) How high will water rise in a glass capillary tube with a 0.500-mm radius?
(b) How much gravitational potential energy does the water gain?
(c) Discuss possible sources of this energy.
Solution
(a) 2.97 cm
(b) \(3.39×10^{−6}J\)
(c) Work is done by the surface tension force through an effective distance \(h/2\) to raise the column of water.
119. A negative pressure of 25.0 atm can sometimes be achieved with the device in Figure before the water separates.
(a) To what height could such a negative gauge pressure raise water?
(b) How much would a steel wire of the same diameter and length as this capillary stretch if suspended from above?
(a) When the piston is raised, it stretches the liquid slightly, putting it under tension and creating a negative absolute pressure \(P=−F/A\) (b) The liquid eventually separates, giving an experimental limit to negative pressure in this liquid.
120. Suppose you hit a steel nail with a 0.500-kg hammer, initially moving at \(15.0 m/s\) size 12{"15" "." 0`"m/s"} {} and brought to rest in 2.80 mm. (a) What average force is exerted on the nail? (b) How much is the nail compressed if it is 2.50 mm in diameter and 6.00-cm long? (c) What pressure is created on the 1.00-mm-diameter tip of the nail?
Solution
(a) \(2.01×10^4N\)
(b) \(1.17×10^{−3}m\)
(c) \(2.56×10^{10}N/m^2\)
121. Calculate the pressure due to the ocean at the bottom of the Marianas Trench near the Philippines, given its depth is \(11.0 km\) and assuming the density of sea water is constant all the way down.
(b) Calculate the percent decrease in volume of sea water due to such a pressure, assuming its bulk modulus is the same as water and is constant.
(c) What would be the percent increase in its density? Is the assumption of constant density valid? Will the actual pressure be greater or smaller than that calculated under this assumption?
122. The hydraulic system of a backhoe is used to lift a load as shown in Figure.
(a) Calculate the force \(F\) the slave cylinder must exert to support the 400-kg load and the 150-kg brace and shovel.
(b) What is the pressure in the hydraulic fluid if the slave cylinder is 2.50 cm in diameter?
(c) What force would you have to exert on a lever with a mechanical advantage of 5.00 acting on a master cylinder 0.800 cm in diameter to create this pressure?
Hydraulic and mechanical lever systems are used in heavy machinery such as this back hoe.
Solution
(a) \(1.38×10^4N\)
(b) \(2.81×10^7N/m^2\)
(c) 283 N
123. Some miners wish to remove water from a mine shaft. A pipe is lowered to the water 90 m below, and a negative pressure is applied to raise the water.
(a) Calculate the pressure needed to raise the water.
(b) What is unreasonable about this pressure?
(c) What is unreasonable about the premise?
124. You are pumping up a bicycle tire with a hand pump, the piston of which has a 2.00-cm radius.
(a) What force in newtons must you exert to create a pressure of \(6.90×10^5Pa\)
(b) What is unreasonable about this (a) result?
(c) Which premises are unreasonable or inconsistent?
Solution
(a) 867 N
(b) This is too much force to exert with a hand pump.
(c) The assumed radius of the pump is too large; it would be nearly two inches in diameter—too large for a pump or even a master cylinder. The pressure is reasonable for bicycle tires.
125. Consider a group of people trying to stay afloat after their boat strikes a log in a lake. Construct a problem in which you calculate the number of people that can cling to the log and keep their heads out of the water. Among the variables to be considered are the size and density of the log, and what is needed to keep a person’s head and arms above water without swimming or treading water.
126. The alveoli in emphysema victims are damaged and effectively form larger sacs. Construct a problem in which you calculate the loss of pressure due to surface tension in the alveoli because of their larger average diameters. (Part of the lung’s ability to expel air results from pressure created by surface tension in the alveoli.) Among the things to consider are the normal surface tension of the fluid lining the alveoli, the average alveolar radius in normal individuals and its average in emphysema sufferers.
Contributors and Attributions
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Paul Peter Urone (Professor Emeritus at California State University, Sacramento) and Roger Hinrichs (State University of New York, College at Oswego) with Contributing Authors: Kim Dirks (University of Auckland) and Manjula Sharma (University of Sydney). This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0) .


3.1.13: Wilfred Owen (1893-1918)
Selected Poems British Modernism Wilfred Edward Salter Owen is one of the most respected poets of World War I, which is an impressive feat for someone who wrote almost all of his poems between August 1917 and September 1918. Many previous British poets had focused on the glory of war. Owen's experiences in the trenches, however, shaped his much grimmer view of war. Along with his friend and fellow British poet Siegfried Sassoon, Owen wrote about the realities of war in a way that was unfamiliar to audiences of the time. The title of his poem "Dulce et Decorum est" is a reference to a quotation by the Latin writer Horace: "Dulce et decorum est pro patria mori" (roughly translated, "It is sweet and proper to die for your country"). In the poem, Owen challenges that view with a description of a poisonous gas attack on soldiers in the trenches. In "Anthem for Doomed Youth," Owen replaces heroic songs and music with the terrifying sounds of battle, and he explores the dread of waiting for something to happen in the poem "Exposure." In "A Parable of the Old Men and the Young," Owen rewrites the ending of the Bible story of Abraham and Isaac, implicitly questioning the motives of the older men who send young men to their deaths. Owen's poetry is remarkable not only for its content, but also for its use of half rhyme and assonance instead of full rhyme: a style that he is credited with popularizing. His rejection of traditional poetic form and reaction to the horrors of World War I are textbook examples of Modernism in poetry. Despite both his feelings about the war and being wounded badly enough that he was sent home for recovery, Owen insisted that it was his duty to return to the fighting, in part to continue to record the experiences of the common soldier. One week before the signing of the Armistice that ended the war, Owen was killed on the battlefield. Consider while reading:
- Compare Owen's poetry to Alfred, Lord Tennyson's "The Charge of the Light Brigade" and Rupert Brooks' "The Soldier." How does each poet write about war?
- Compare Owen's poems to the works of his friend Siegfried Sassoon. Are there differences in their approaches to describing the war?
Written by Laura Getty