5.11: Linear Programming
- Compose an objective function to be minimized or maximized.
- Compose inequalities representing a system application.
- Apply linear programming to solve application problems.
Imagine you hear about some natural disaster striking a far-away country; it could be an earthquake, a fire, a tsunami, a tornado, a hurricane, or any other type of natural disaster. The survivors of this disaster need help—they especially need food, water, and medical supplies. You work for a company that has these supplies, and your company has decided to help by flying the needed supplies into the disaster area. They want to maximize the number of people they can help. However, there are practical constraints that need to be taken into consideration; the size of the airplanes, how much weight each airplane can carry, and so on. How do you solve this dilemma? This is where linear programming comes into play. Linear programming is a mathematical technique to solve problems involving finding maximums or minimums where a linear function is limited by various constraints.
As a field, linear programming began in the late 1930s and early 1940s. It was used by many countries during World War II; countries used linear programming to solve problems such as maximizing troop effectiveness, minimizing their own casualties, and maximizing the damage they could inflict upon the enemy. Later, businesses began to realize they could use the concept of linear programming to maximize output, minimize expenses, and so on. In short, linear programming is a method to solve problems that involve finding a maximum or minimum where a linear function is constrained by various factors.
On December 26, 2004, a massive earthquake occurred in the Indian Ocean. This earthquake, which scientists estimate had a magnitude of 9.0 or 9.1 on the Richter Scale, set off a wave of tsunamis across the Indian Ocean. The waves of the tsunami averaged over 30 feet (10 meters) high, and caused massive damage and loss of life across the coastal regions bordering the Indian Ocean.
Usama Kadri works as an applied mathematician at Cardiff University in Wales. His areas of research include fluid dynamics and non-linear phenomena. Lately, he has been focusing his research on the early detection and easing of the effects of tsunamis. One of his theories involves deploying a series of devices along coastlines which would fire acoustic-gravity waves (AGWs) into an oncoming tsunami, which in theory would lessen the force of the tsunami. Of course, this is all in theory, but Kadri believes it will work. There are issues with creating such a device: they would take a tremendous amount of electricity to generate an AGW, for instance, but if it would save lives, it may well be worth it.
Compose an Objective Function to Be Minimized or Maximized
An objective function is a linear function in two or more variables that describes the quantity that needs to be maximized or minimized.
Miriam starts her own business, where she knits and sells scarves and sweaters out of high-quality wool. She can make a profit of $8 per scarf and $10 per sweater. Write an objective function that describes her profit.
- Answer
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Let \(x\) represent the number of scarves sold, and let \(y\) represent the number of sweaters sold. Let \(P\) represent profit. Since each scarf has a profit of \(\$ 8\) and each sweater has a profit of \(\$ 10\), the objective function is \(P=8 x+10 y\).
For a fundraiser at school, the Robotics Club is selling bags of apples and bunches of bananas during lunch. They will make a profit of $4 per bag of apples and $6 per bunch of bananas. Write an objective function that describes the profit the Robotics Club will make.
William’s factory produces two products, widgets and wadgets. It takes 24 minutes for his factory to make 1 widget, and 32 minutes for his factory to make 1 wadget. Write an objective function that describes the time it takes to make the products.
- Answer
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Let \(x\) equal the number of widgets made; let \(y\) equal the number of wadgets made; let \(T\) represent total time. The objective function is \(T=24 x+32 y\).
Suppose William has a second factory that can make widgets in 20 minutes and wadgets in 28 minutes. Write an objective function that describes the time it takes to make the products.
Composing Inequalities Representing a System Application
For our two examples of profit and production, in an ideal world the profit a person makes and/or the number of products a company produces would have no restrictions. After all, who wouldn’t want to have an unrestricted profit? However in reality this is not the case; there are usually several variables that can restrict how much profit a person can make or how many products a company can produce. These restrictions are called constraints .
Many different variables can be constraints. When making or selling a product, the time available, the cost of manufacturing and the amount of raw materials are all constraints. In the opening scenario with the tsunami, the maximum weight on an airplane and the volume of cargo it can carry would be constraints. Constraints are expressed as linear inequalities; the list of constraints defined by the problem forms a system of linear inequalities that, along with the objective function, represent a system application.
Two friends start their own business, where they knit and sell scarves and sweaters out of high-quality wool. They can make a profit of $8 per scarf and $10 per sweater. To make a scarf, 3 bags of knitting wool are needed; to make a sweater, 4 bags of knitting wool are needed. The friends can only make 8 items per day, and can use not more than 27 bags of knitting wool per day. Write the inequalities that represent the constraints. Then summarize what has been described thus far by writing the objective function for profit and the two constraints.
- Answer
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Let \(x\) represent the number of scarves sold, and let \(y\) represent the number of sweaters sold. There are two constraints: the number of items the business can make in a day (a maximum of 8 ) and the number of bags of knitting wool they can use per day (a maximum of 27). The first constraint (total number of items in a day) is written as:
\[x+y \leq 8 \nonumber \]
Since each scarf takes 3 bags of knitting wool and each sweater takes 4 bags of knitting wool, the second constraint, total bags of knitting wool per day, is written as:
\[3 x+4 y \leq 27 \nonumber \]
In summary, here are the equations that represent the new business: \(P=8 x+10 y\); This is the profit equation: The business makes \(\$ 8\) per scarf and \(\$ 10\) per sweater.
\[\begin{aligned}
x+y & \leq 8 \\
3 x+4 y & \leq 27
\end{aligned} \nonumber \]
For a fudraiser at school, the Robotics Club is selling bags of apples and bunches of bananas during lunch. They will make a profit of $4 per bag of apples and $6 per bunch of bananas. Due to school health regulations, the club is allowed to have only 20 bags and bunches of fruit on school grounds each day to sell. Another regulation: the container where the Robotics Club keeps the fruit has a maximum weight capacity of 70 pounds.  Write the inequalities that represent these constraints. Then summarize the equations that represent this system.
A factory produces two products, widgets and wadgets. It takes 24 minutes for the factory to make 1 widget, and 32 minutes for the factory to make 1 wadget. Research indicates that long-term demand for products from the factory will result in average sales of 12 widgets per day and 10 wadgets per day. Because of limitations on storage at the factory, no more than 20 widgets or 17 wadgets can be made each day. Write the inequalities that represent the constraints. Then summarize what has been described thus far by writing the objective function for time and the two constraints.
- Answer
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Let \(x\) equal the number of widgets made; let \(y\) equal the number of wadgets made. Based on the long-term demand, we know the factory must produce a minimum of 12 widgets and 10 wadgets per day. We also know because of storage limitations, the factory cannot produce more than 20 widgets per day or 17 wadgets per day. Writing those as inequalities, we have:
\[\begin{array}{l}
x \geq 12 \\
y \geq 10 \\
x \leq 20 \\
y \leq 17
\end{array} \nonumber \]The number of widgets made per day must be between 12 and 20 , and the number of wadgets made per day must be between 10 and 17. Therefore, we have:
\[\begin{array}{l}
12 \leq x \leq 20 \\
10 \leq y \leq 17
\end{array} \nonumber \]The system is:
\[\begin{array}{l}
T=24 x+32 y \\
12 \leq x \leq 20 \\
10 \leq y \leq 17
\end{array} \nonumber \]\(T\) is the variable for time; it takes 24 minutes to make a widget and 32 minutes to make a wadget.
Suppose a second factory can make widgets in 20 minutes and wadgets in 28 minutes. Research for this factory indicates that long-term demand for products from this second factory will result in average sales of 15 widgets per day and 13 wadgets per day. Because of limitations on storage at his factory, no more than 22 widgets or 19 wadgets can be made each day. Write the inequalities that represent the constraints. Then summarize what has been described thus far by writing the objective function for time and the two constraints.
Applying Linear Programming to Solve Application Problems
There are four steps that need to be completed when solving a problem using linear programming. They are as follows:
Step 1: Compose an objective function to be minimized or maximized.
Step 2: Compose inequalities representing the constraints of the system.
Step 3: Graph the system of inequalities representing the constraints.
Step 4: Find the value of the objective function at each corner point of the graphed region.
The first two steps you have already learned. Let’s continue to use the same examples to illustrate Steps 3 and 4.
Three friends start their own business, where they knit and sell scarves and sweaters out of high-quality wool. They can make a profit of $8 per scarf and $10 per sweater. To make a scarf, 3 bags of knitting wool are needed; to make a sweater, 4 bags of knitting wool are needed. The friends can only make 8 items per day, and can use not more than 27 bags of knitting wool per day. Determine the number of scarves and sweaters they should make each day to maximize their profit.
- Answer
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Step 1 : Compose an objective function to be minimized or maximized. From Example 5.98, the objective function is \(P=8 x+10 y\).
Step 2 : Compose inequalities representing the constraints of the system. From Example 5.98, the constraints are \(x+y \leq 8\) and \(3 x+4 y \leq 27\).
Step 3 : Graph the system of inequalities representing the constraints. Using methods discussed in Graphing Linear Equations and Inequalities, the graphs of the constraints are shown below. Because the number of scarves \((x)\) and the number of sweaters \((y)\) both must be non-negative numbers (i.e., \(x \geq 0\) and \(y \geq 0\) ), we need to graph the system of inequalities in Quadrant I only. Figure
shows each constraint graphed on its own axes, while Figure shows the graph of the system of inequalities (the two constraints graphed together). In Figure , the large shaded region represents the area where the two constraints intersect. If you are unsure how to graph these regions, refer back to Graphing Linear Equations and Inequalities.Figure 5. Figure \(\PageIndex{3}\): Graph of the System of Inequalities
Step 4: Find the value of the objective function at each corner point of the graphed region. The “graphed region” is the area where both of the regions intersect; in Figure 5.112, it is the large shaded area. The “corner points” refer to each vertex of the shaded area. Why the corner points? Because the maximum and minimum of every objective function will occur at one (or more) of the corner points. Figure 5.112 shows the location and coordinates of each corner point.
Figure \(\PageIndex{4}\): Graph of Region with Corner Points
Three of the four points are readily found, as we used them to graph the regions; the fourth point, the intersection point of the two constraint lines, will have to be found using methods discussed in Systems of Linear Equations in Two Variables, either using substitution or elimination. As a reminder, set up the two equations of the constraint lines:
\[\begin{aligned}
3 x+4 y & =27 \\
x+y & =8
\end{aligned} \nonumber \]For this example, substitution will be used.
\[\begin{aligned}
x+y & =8 \\
y & =8-x .
\end{aligned} \nonumber \]Substituting \(8-x\) into the first equation for \(y\), we have
\[\begin{aligned}
3 x+4(8-x) & =27 \\
3 x+32-4 x & =27 \\
-x & =-5 \\
x & =5
\end{aligned} \nonumber \]Now, substituting the 5 in for \(x\) in either equation to solve for \(y\). Choosing the second equation, we have:
\[\begin{array}{r}
5+y=8 \\
y=3
\end{array} \nonumber \]Therefore, \(x=5\), and \(y=3\).
To find the value of the objective function, \(P=8 x+10 y\), put the coordinates for each corner point into the equation and solve. The largest solution found when doing this will be the maximum value, and thus will be the answer to the question originally posed: determining the number of scarves and sweaters the new business should make each day to maximize their profit.
Corner \((x, y)\) Objective Function \(P=8 x+10 y\) \((0,0)\) \(P=8(0)+10(0)=0\) \((0,6.75)\) \(P=8(0)+10(6.75)=40+30 = 70\) \((5,3)\) \(P=8(5)+10(3)=67.5\) \((8,0)\) \(P=8(8)+10(0)=64\) The maximum value for the profit \(P\) occurs when \(x=5\) and \(y=3\). This means that to maximize their profit, the new business should make 5 scarves and 3 sweaters every day.
For a fundraiser at school, the Robotics Club is selling bags of apples and bunches of bananas during lunch. They will make a profit of $4 per bag of apples and $6 per bunch of bananas. Due to school health regulations, the club is allowed to have only 20 bags and bunches of fruit on school grounds each day to sell. Another regulation: the container where the Robotics Club keeps the fruit has a maximum weight capacity of 70 pounds.  Determine the number of bags of apples and the number of bags of bananas the Robotics Club should sell each day to maximize their profit.
Leonid Vitalyevich Kantorovich was born January 19, 1912, in St. Petersburg, Russia. Two major events affected young Leonid’s life: when he was five, the Russian Revolution began, making life in St. Petersburg very difficult; so much so that Leonid’s family fled to Belarus for a year. When Leonid was 10, his father died, leaving his mother to raise five children on her own.
Despite the hardships, Leonid showed incredible mathematical ability at a young age. When he was only 14, he enrolled in Leningrad State University to study mathematics. Four years later, at age 18, he graduated with what would be equivalent to a Ph.D. in mathematics.
Although his primary interests were in pure mathematics, in 1938 he began working on problems in economics. Supposedly, he was approached by a local plywood manufacturer with the following question: how to come up with a work schedule for eight lathes to maximize output, given the five different kinds of plywood they had at the factory. By July 1939, Leonid had come up with a solution, not only to the lathe scheduling problem but to other areas as well, such as an optimal crop rotation schedule for farmers, minimizing waste material in manufacturing, and finding optimal routes for transporting goods. The technique he discovered to solve these problems eventually became known as linear programming. He continued to use this technique for solving many other problems involving optimization, which resulted in the book The Best Use of Economic Resources , which was published in 1959. His continued work in linear programming would ultimately result in him winning the Nobel Prize of Economics in 1975.
Check Your Understanding
1. Kellie makes tables and chairs. Kellie profits $20 from a table (\(t\)), and $10 from a chair (\(c\)). The objective function for profit in this situation is:
- \({P = 20t - 10c}\)
- \({P = 20c + 10t}\)
- \({P = 20t + 10c}\)
- \({P = 20c - 10t}\)
2. Dave grows wheat (\(w\)) and barley (\(b\)) on a farm. Dave expects to profit $150 per acre for wheat and $180 per acre for barley. The objective function for profit in this situation is:
- \({P = 150w + 180b}\)
- \({P = 150b + 180w}\)
- \({P = 180w + 150b}\)
- \({P = 150w - 180b}\)
3. An antique music store sells two types of vinyl records; 45 rpm records (\(f\)) and 33 rpm records (\(t\)). It makes a profit of $2.50 for each 45 rpm record and $6.75 for each 33 rpm record. The objective function for profit in this situation is:
- \({P = 2.50f + 6.75c}\)
- \({P = 2.50f + 6.75t}\)
- \({P = 2.50t + 6.75f}\)
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None of these
4. Kellie makes tables and chairs. Kellie profits $20 from a table (\(t\)), and $10 from a chair (\(c\)). A table requires 15 board feet of wood, while a chair requires 4 board feet of wood. Kellie has 70 board feet available. What is the constraint inequality in this situation?
- \({20t + 10c \leq 70}\)
- \({ t + c \leq 70}\)
- \({ 15t + 4c \geq 70}\)
- \({ 15t + 4c \leq 70}\)
5. Kellie makes tables and chairs. Kellie profits $20 from a table (\(t\)), and $10 from a chair (\(c\)). The maximum number of tables and chairs Kellie can make in any one day is 12. What is the constraint inequality in this situation?
- \({t + c \leq 12}\)
- \({ 20t + 10c \leq 12}\)
- \({ 20t + 10c \geq 12}\)
- \({ 20t + 10c \leq 70}\)
6. Dave grows wheat (\(w\)) and barley (\(b\)) on a farm. Dave expects to profit $150 per acre for wheat and $180 per acre for barley. The cost of seed is $10 per acre for wheat and $15 per acre for barley. Dave can only afford to spend $945 on seed. What is the constraint inequality in this situation?
- \({w + b \leq 945}\)
- \({ 10w + 15b \geq 945}\)
- \({ 10w + 15b \leq 945}\)
- \({ 150w + 180w \leq 945}\)
7. Dave grows wheat (\(w\)) and barley (\(b\)) on a farm. Dave expects to profit $150 per acre for wheat and $180 per acre for barley. The cost of raising each crop is $30 per acre for wheat and $25 per acre for barley. Dave budgets $1,635 for the raising of both crops. What is the constraint inequality in this situation?
- \({150w + 180b \leq \text{1,635}}\)
- \({25w + 30b \leq \text{1,635}}\)
- \({w + b \leq \text{1,635}}\)
- \({30w + 25b \leq \text{1,635}}\)
8. Kellie makes tables and chairs. Kellie profits $20 from a table (\(t\)), and $10 from a chair (\(c\)). A table requires 15 board feet of wood, while a chair requires 4 board feet of wood. Kellie has 70 board feet available. The maximum number of tables and chairs Kellie can make in any one day is 12. The graph of the system of inequalities representing the constraints is:
9. Kellie makes tables and chairs. Kellie profits $20 from a table (\(t\)), and $10 from a chair (\(c\)). A table requires 15 board feet of wood, while a chair requires 4 board feet of wood. Kellie has 70 board feet available. The maximum number of tables and chairs Kellie can make in any one day is 12. The four corner points of the system are:
- \({(0, 0), (0, 12), (10, 2), (12, 0)}\)
- \({(0, 0), (0, 12), (2, 10), (4\tfrac{2}{3} , 0)}\)
- \({(0, 0), (17.5, 0), (2, 10), (12, 0)}\)
10. Kellie makes tables and chairs. Kellie profits $20 from a table (\(t\)), and $10 from a chair (\(c\)). A table requires 15 board feet of wood, while a chair requires 4 board feet of wood. Kellie has 70 board feet available. The maximum number of tables and chairs Kellie can make in any one day is 12. The maximum profit Kellie can make in one day is:
\($220\quad\quad{$120}\quad\quad{$140}\quad\quad{$24}\)


5.5: Combining Linear Optimization with Markov Deterioration Models
The previous section illustrated the use of expected component conditions for use in optimization and decision making. It is also possible to include a distribution of possible conditions over a period of time. The most common means of making this synthesis is to combine optimization with Markov deterioration models. A number of bridge and pavement management systems are based on this synthesis; for example, see AASHTO (2016) or Golabi (1997). It is unlikely that an infrastructure manager would formulate an optimization problem synthesis such as these, but managers regularly use the software programs embedding the synthesized optimization.
Table 5.5.1 presents an example of a Markov process transition matrix and possible actions for a concrete component. Five condition states are defined, with 1 representing good condition and 5 representing failure of the component. For components in state 1, recommended management action is to do nothing. In each year in State 1, there is a 3% chance of deterioration to state 2. For components in states 2 and 3, do-nothing or patch are possible actions with the probability consequences as shown. There are substantial chances that the patching may not be effective, as the components might remain in their initial state or deteriorate further (although with a low probability of such deterioration). For components in state 4, do-nothing, rehabilitation or replacement are possible actions. With do-nothing action and initial state 4, there is a 13% chance of failure over the course of a year.
| Initial State | Action | \(Pr\text{(state 1)}\) | \(Pr\text{(state 2)}\) | \(Pr\text{(state 3)}\) | \(Pr\text{(state 4)}\) | \(Pr\text{(state 5)}\) |
|---|---|---|---|---|---|---|
| 1 | Do Nothing | 0.97 | 0.03 | 0.0 | 0.0 | 0.0 |
| 2 | Do Nothing | 0.0 | 0.97 | 0.03 | 0.0 | 0.0 |
| 2 | Patch | 0.62 | 0.34 | 0.04 | 0.0 | 0.0 |
| 3 | Do Nothing | 0.0 | 0.0 | 0.92 | 0.08 | 0.0 |
| 3 | Patch | 0.52 | 0.35 | 0.10 | 0.03 | 0.0 |
| 4 | Do Nothing | 0.0 | 0.0 | 0.0 | 0.87 | 0.13 |
| 4 | Rehabilitate | 0.68 | 0.27 | 0.05 | 0.0 | 0.0 |
| 4 | Replaca | 0.99 | 0.01 | 0.0 | 0.0 | 0.0 |
An initial analysis step with a table of transition probabilities such as this could be to minimize the long-term cost of maintaining the concrete component. Of course, doing nothing at each stage would minimize cost, except that eventually the component would fail. Presumably, a manager would attempt to minimize cost subject to avoiding transitioning to state 5. Decision variables would be a particular action given a state. The objective function would be to minimize expected condition (or perhaps the probability of failure). With a planning horizon and an initial state, the changes in probabilities over time can be traced as a linear function of the decision variables. A budge constraint might also be imposed (as in Eq. 5.2.4) added over all the bridge components being managed.
We will illustrate this optimization approach with a small problem. Suppose a set of identical components can have three possible condition states: 1 – good, 2 – average and 3 – poor. One maintenance activity can be undertaken, which will move the component from any state to state 1 at a cost of ci. Table 5.5.2 shows the transition probabilities for the component with do-nothing and maintenance. Finally, there is a budget available for the year and a known state si for each component.
| Initial State | Action | \(Pr\text{(state 1)}\) | \(Pr\text{(state 2)}\) | \(Pr\text{(state 3)}\) |
|---|---|---|---|---|
| 1 | Do Nothing | 0.8 | 0.2 | 0.0 |
| 1 | Maintenance | 1.0 | 0.0 | 0.0 |
| 2 | Do Nothing | 0.0 | 0.8 | 0.2 |
| 2 | Maintenance | 1.0 | 0.0 | 0.0 |
| 3 | Do Nothing | 0.0 | 0.0 | 1.0 |
| 3 | Maintenance | 1.0 | 0.0 | 0.0 |
Following the formulation approach discussed in the previous section:
- Let us define our decision variable as \(x_i = 0\) if do-nothing and \(x_i = 1\) if maintenance is performed. (If there were more than two actions possible, then we could add a subscript as in Eq. 5.2.2 for each component and each possible action, and the decision variable \(x|{ij}\) would be 0 if the action \(j\) was not undertaken on component \(I\) and one if action \(j\) was taken on component \(i\)).
- Let us assume that the objective is to minimize the average component condition.
- The only constraint is the budget constraint for all the actions. (If more than one action is possible, however, we would have to add a constraint similar to Eq. 5.2.3 for each component, however).
The resulting problem is:
\(\text{Minimize Average Condition State}\)
\[=\sum_{s=1}(.8+2 * .2) *\left(1-x_{i}\right)+\sum_{s=2}(2 * 0.8+3 * 0.2) *\left(1-x_{i}\right)+\sum_{s=3} 3 *\left(1-x_{i s}\right)+\sum_{i} x_{i}\]
\(\text {Subject to } \sum_{i} c_{i} * x_{i} \leq B, x_{i} \text { binary }\)
Where the objective function has four terms: (1) resulting condition of components in state 1 with no action, (2) resulting condition of components starting in state 2 with no action, (3) resulting condition of components in state 3 with no action (they stay in state 3), and (4) components with maintenance moving to state 1. The constraints are the overall budget and the restriction of the \(x_i\) to zero or one. As noted above, variation would add additional potential actions (using notation \(x_{iS}) and different resulting conditions.


10.1: Wound Care Introduction
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Wound healing is a complex physiological process that restores function to skin and tissue that have been injured. The healing process is affected by several external and internal factors that either promote or inhibit healing. When providing wound care to patients, nurses, in collaboration with other members of the health care team, assess and manage external and internal factors to provide an optimal healing environment. [1]
Complex wounds often require care by specialists. Certified wound care nurses assess, treat, and create care plans for patients with complex wounds, ostomies, and incontinence conditions. They act as educators and consultants to staff nurses and other healthcare professionals. This chapter will discuss wound care basics for entry-level nurses. Request a consultation by a certified wound care nurse when caring for patients with complex or nonhealing wounds.


11.2: Emerging Norms of Human Protection
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Since the end of the Cold War, the practice of human protection has evolved through at least eight interconnected streams of norms, rules, practices and institutional developments. Each of these emerged to address the problem of civilian suffering, especially during war and will be addressed in turn.
International humanitarian law
International humanitarian law had its origins in the nineteenth century with the development of the US Government’s ‘General Orders No. 100’ (better known as the Lieber code), which were military laws designed to limit the conduct of soldiers – and the emergence of the Red Cross movement. After the Second World War, international humanitarian law was developed and codified in a series of international treaties. In 1948, the newly established UN General Assembly approved the Genocide Convention, which prohibited the crime of genocide and assigned all states a legal duty to prevent it and punish the perpetrators. The International Court of Justice (ICJ) was established as the judicial arm of the United Nations and is responsible for adjudicating on disputes between states and other legal matters. It judged that as a result of this convention, all states have a legal responsibility to do what they can, within existing law, to prevent genocide.
The laws of war were further codified in the four Geneva Conventions (1949), two additional protocols (1977), and in a range of protocols covering the use of Certain Conventional Weapons. Of particular importance was Common Article 3 of the 1949 Geneva Conventions, which committed parties to respect the human rights of all non-combatants; and the Convention on the Protection of Civilian Persons, which offered legal protection to non-combatants in occupied territories. The Geneva Protocols (1977) extended the legal protection afforded to non-combatants to situations of non-international armed conflict. They also insisted that armed attacks be strictly limited to military objectives and forbade attacks on non-combatants or their property. These principals provided the legal and moral foundation for subsequent campaigns for conventions banning weapons, such as landmines and cluster munitions, that were considered inherently indiscriminate. International humanitarian law has thus created a normative standard of civilian protection that not only prohibits attacks on non-combatants and restricts the use of certain weapons but also calls for the prevention of particular crimes, such as genocide, and the punishment of perpetrators.
Protection of civilians
The UN Security Council’s formal engagement with this theme dates back to 1998 when, at Canada’s request, it adopted a presidential statement calling for the Secretary-General to submit periodic reports on how the UN might improve the protection of civilians. Since then, it has held a series of open meetings on the protection of civilians, establishing it as one of its major thematic interests. In 1999, the Security Council unanimously adopted Resolution 1265 expressing its ‘willingness’ to consider ‘appropriate measures’ in response ‘to situations of armed conflict where civilians are being targeted or where humanitarian assistance to civilians is being deliberately obstructed’. In addition, the Security Council expressed its willingness to explore how peacekeeping mandates might be reframed to afford better protection to endangered civilians. In 2006, it adopted Resolution 1674, which built further on this progress by demanding that parties to armed conflict grant unfettered humanitarian access to civilians.
As it has developed its thematic interest in the protection of civilians, the Security Council has also developed and strengthened its practices of protection. In doing so it has broken new ground. In Resolution 1973, passed in 2011, the Security Council authorised the use of force for human protection purposes in Libya. This was the first time in the history of the Security Council that such an action had been passed without the consent of the host state. Through this resolution, and the one that preceded it (Resolution 1970) the Security Council utilised the full range of the collective security powers granted to it by the UN Charter. Three years later, Resolution 2165 authorised the delivery of humanitarian assistance into Syria without the consent of the Syrian government – the first time that the Council has done this. Hence, two very important issues of precedent were established, built on a new understanding of the need to protect civilians.
Before the turn of this century, civilian protection was typically not considered a core part of peacekeeping. Starting in 1999 with the UN mission in Sierra Leone, the Security Council has invoked Chapter VII of the UN Charter with increasing regularity to authorise peacekeepers to use all means necessary to protect civilians. Chapter VII of the Charter gives the UN Security Council the authority to authorise whatever means it deems necessary, including the use of force, for the maintenance of international peace and security. By design, it was intended as a key deterrent to international aggression. Today, civilian protection and the authorisation of ‘all means necessary’ to that end are core aspects of UN peacekeeping and central to many of its new mandates. In the Democratic Republic of the Congo (DRC), the Security Council went even further by tasking a ‘Force Intervention Brigade’ to take the fight to non-state armed groups that were employing mass violence against civilians. Today, the bulk of the UN’s 120,000 peacekeepers are deployed with mandates to use all necessary means to protect civilians from harm.
Addressing specific vulnerabilities
Since the end of the Second World War, international society has periodically recognised groups that are exposed to particular vulnerabilities and has established mechanisms aimed at addressing or reducing those vulnerabilities. Of these, the best developed is the international refugee regime, which is governed by the 1951 Refugee Convention and subsequent 1967 Protocol. It is overseen by the UN High Commissioner for Refugees (UNHCR). This system grants people facing persecution the right to claim asylum and receive resettlement in third countries and mandates the UNHCR to ensure that refugees have access to protection and durable solutions to their displacement. During the 1990s, it became apparent that this system was unable to cope with a new displacement crisis – that of internal displacement. Internal displacement occurs when people are forced from their homes by mass violence and other ills but remain within their host country. As a largely domestic issue there was little appetite for an international convention governing the displaced. Instead, the UNHCR extended its mandate to cover the protection of all displaced persons and United Nations officials developed ‘guiding principles’ for their treatment.
Another longstanding facet of mass violence that gained political prominence only in the 1990s was sexual and gender-based violence. The use of rape as a weapon of war in various cases pushed the UN Security Council to establish the protection of women and girls as one of the principal elements of its ‘Women, Peace and Security’ agenda adopted in the year 2000 via Resolution 1325. Since then, the United Nations has created the post of Special Representative of the Secretary-General to give permanent focus to the issue, and has instituted a series of annual reports that identify where these crimes are committed and advocate for steps to be taken in response. The United Nations has also begun to ‘mainstream’ the protection of women and girls through, for example, the deployment of women’s protection advisers. Beyond the United Nations, the British government launched its Preventing Sexual Violence in Conflict Initiative which, amongst other things, has helped persuade two-thirds of the world’s states to support a ‘Declaration of Commitment to End Sexual Violence in Conflict’. These developments have been paralleled by a range of initiatives focused on protecting children in armed conflict. Also led by the Security Council, the United Nations has appointed a Special Representative for the protection of children, which reports on the unique protection challenges facing children and related issues such as the recruitment of child soldiers. In 2014, the UN’s ambassador for the promotion of education, former UK prime minister Gordon Brown, launched a global initiative to establish a contingency fund to support the provision of education to children during humanitarian crises, be they caused by natural disasters or mass violence.
Human rights
While human rights as a whole are subject to a great deal of questioning, their higher profile has undoubtedly made an important contribution to human protection. Two aspects in particular stand out, but they are illustrative rather than definitive since the overlap is extensive and complex. First, emerging principles and practices of peer-to-peer review, where states evaluate and comment on each other’s performance (mainly through the compulsory review process of the UN’s Human Rights Council), create expectations about the type of steps that states ought to take in order to protect their populations from various forms of abuse, including mass violence. While the most intransigent states remain largely unmoved, there is increasing evidence that peer review activities are influencing many states and pushing them towards greater compliance with their human rights obligations due to the pressure that being ‘watched’ places on them. Second, over the past two decades, international society has made increasing use of permanent and ad hoc arrangements for human rights monitoring and reporting in its decisionmaking on mass violence. Through a variety of different mechanisms, such as independent commissions and inquiries, special rapporteurs and fact-finding missions, international society is increasingly utilising human rights mechanisms to monitor and prevent mass violence. Most obviously, this reporting helps support decision-making on mass violence by furnishing key institutions with reliable information. It also encourages states to respect human rights by raising international awareness of domestic human rights practices.
International criminal justice
The idea that some crimes are so serious that the prosecution of perpetrators should be universal has advanced significantly in the past two decades through the activities of the International Criminal Court and a series of special tribunals. These institutions have proliferated since the mid-1990s and contribute to individual perpetrators being held accountable for their actions. Proponents argue that by ending impunity such institutions help deter wouldbe perpetrators and also give some legal protection to the victims. The first tentative steps were taken in the mid-1990s when the Security Council established tribunals to prosecute the perpetrators of grave crimes in Bosnia and Rwanda. The Rome Statute establishing the International Criminal Court in 1998 held that the Court’s jurisdiction could be invoked when a state party proved unwilling or unable to investigate evidence pointing to the commission of widespread and systematic war crimes, crimes against humanity and genocide. The Court’s prosecutor can initiate proceedings in cases where he or she is able to persuade a panel of judges that a case fell under the Court’s jurisdiction, where a complaint was made by a signatory state, or when a case was referred to the prosecutor by the Security Council. To date, the Court has indicted 39 individuals and counts 124 states as members – though importantly the United States, Russia and China have yet to join. While it is important to state that developments like the International Criminal Court are still embryonic, the evidence suggests that transitional justice measures make reoccurrence less likely and improve general human rights within states. It also has a deterrent effect that spills over into other countries, including those that are not (yet) members of the International Criminal Court.
Humanitarian action
The notion that civilians ought to receive humanitarian assistance in wartime dates back to the nineteenth century and was integral to the development of the humanitarian idea of providing lifesaving assistance to whomever needed it. Those rights and expectations were incorporated into international humanitarian law but their applicability gradually expanded during the 1990s. The UN Security Council began authorising peacekeeping missions to support the delivery of humanitarian aid and, in the cases of Somalia and Bosnia, authorised the use of force to achieve this end. Since then, the Security Council has regularly authorised force for these purposes. What is more, however, in successive resolutions on the protection of civilians and in substantive resolutions on crises, the Security Council has demanded that parties to armed conflict grant unfettered access to humanitarian agencies.
Regional initiatives
The foundations for Europe’s engagement with civilian protection were laid in the 1970s with the Helsinki Accords. Over time, these provided the basis for a Conference on Security and Cooperation in Europe mechanism that by the 1990s incorporated specific references to protection issues, including the protection of children and protection against torture. When this was transformed into the Organisation for Security and Cooperation in Europe in 1995, it was given additional responsibility and capacities to protect human rights including the post of High Commissioner for National Minorities.
As part of its common foreign and security policy the European Union also started to develop a civilian protection role, exemplified by the French-led multinational force in the Democratic Republic of the Congo in 2003 and a range of other operations. The African Union has established a comprehensive regional system for crisis management and response that includes a specific focus on the protection of civilians from mass violence. Article 4(h) of the Union’s Constitutive Act enshrines its right to intervene in the affairs of its member states in issues relating to genocide and mass atrocities. Although this article has not been formally acted upon, owing to African leaders’ continuing commitment to sovereignty, the African Union’s peacekeeping operation in Darfur included a civilian protection mandate and its missions in Mali, the Central African Republic and Somalia have also supported civilian protection. In Latin America, states have established a comprehensive regional human rights mechanism. Even the Southeast Asian region, which is formally committed to the principle of non-interference in the domestic affairs of states, has begun to develop its own mechanisms for promoting human rights and protection through the ASEAN Intergovernmental Commission on Human Rights. These mechanisms might not understand or pursue ‘rights’ in precisely the same fashion, but they do rest on a shared understanding of atrocity crimes as grave human wrongs and a commitment to the prevention of these crimes.
Responsibility to Protect
In late 2005, world leaders unanimously adopted the Responsibility to Protect (R2P) in paragraphs 138–140 of the UN World Summit Outcome Document. This commitment was subsequently reaffirmed by both the UN Security Council and the UN General Assembly, which also committed to ongoing consideration of its implementation. The Responsibility to Protect rests on three pillars. The first is the responsibility of each state to use appropriate and necessary means to protect its own populations from genocide, war crimes, ethnic cleansing and crimes against humanity (hereafter referred to collectively as ‘atrocity crimes’). The second pillar refers to the commitment of the international community to encourage and help states exercise this responsibility. The third pillar refers to the international responsibility to respond through the United Nations in a timely and decisive manner when national authorities are manifestly failing to protect their populations from the four atrocity crimes. The principle was initially considered to be controversial, as it countenanced the potential use of force and other transgressions of sovereignty. Over time, however, international consensus on the principle has widened and deepened.
More tellingly, the Responsibility to Protect has become part of the working language that frames international engagement with political crises and the Security Council has referred to it in more than forty resolutions. It has reminded governments of their protection responsibilities (e.g. Resolution 2014 on Yemen); demanded active steps to protect civilians (e.g. Resolution 2139 on Syria); tasked peacekeepers with assisting governments to protect their own populations (e.g. Resolution 2085 on Mali) and demanded that perpetrators of mass violence be held legally accountable (e.g. Resolution 2211 on the Democratic Republic of the Congo). The Security Council has also connected its work on the Responsibility to Protect with its international efforts focused on preventive diplomacy and conflict prevention through such measures as the control of small arms and light weapons, the prevention of genocide, counter-terrorism and international policing. With this changing focus, debate amongst states turned to focus less on the principle of the Responsibility to Protect and more on its implementation.


5.11: Formula Mass- The Mass of a Molecule or Formula Unit
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Learning Objectives
- To determine the formula mass of an ionic or molecular compound.
A necessary skill for future chapters is the ability to determine the mass of the formula of an ionic compound. This quantity is called the formula mass. The formula mass is obtained by adding the masses of each individual atom in the formula of the compound. Because a proper formula is electrically neutral (with no net electrons gained or lost), the ions can be considered atoms for the purpose of calculating the formula mass.
Let us start by calculating the formula mass of sodium chloride (NaCl). This formula mass is the sum of the atomic masses of one sodium atom and one chlorine atom, which we find from the periodic table; here, we use the masses to two decimal places:
Na: 22.99 amu
Cl: +35.34 amu
Total: 58.44 amu
To two decimal places, the formula mass of NaCl is 58.44 amu.
When an ionic compound has more than one anion or cation, you must remember to use the proper multiple of the atomic mass for the element in question. For the formula mass of calcium fluoride (CaF 2 ), we must multiply the mass of the fluorine atom by 2 to account for the two fluorine atoms in the chemical formula:
Ca: 1 x 40.08 = 40.08 amu
F: 2 x 19.00 = +38.00 amu
Total = 78.08 amu
The formula mass of CaF 2 is 78.08 amu.
For ionic compounds with polyatomic ions, the sum must include the number and mass of each atom in the formula for the polyatomic ion. For example, potassium nitrate (KNO 3 ) has one potassium atom, one nitrogen atom, and three oxygen atoms:
K: 1 x 39.10 = 39.10 amu
N: 1 x 14.00 = +14.00 amu
O: 3 x 16.00 = +48.00 amu
Total = 101.10 amu
The formula mass of KNO 3 is 101.10 amu.
Potassium nitrate is a key ingredient in gunpowder and has been used clinically as a diuretic.
When a formula contains more than one polyatomic unit in the chemical formula, as in Ca(NO 3 ) 2 , do not forget to multiply the atomic mass of every atom inside of the parentheses by the subscript outside of the parentheses. This is necessary because the subscript refers to the entire polyatomic ion . Thus, for Ca(NO 3 ) 2 , the subscript 2 implies two complete nitrate ions, so we must sum the masses of two (1 × 2) nitrogen atoms and six (3 × 2) oxygen atoms, along with the mass of a single calcium atom:
Ca: 1 x 40.08 = 40.08 amu
N: 2 x 14.00 = +28.00 amu
O: 6 x 16.00 = +96.00 amu
Total = 164.08 amu
The key to calculating the formula mass of an ionic compound is to correctly count each atom in the formula and multiply the atomic masses of its atoms accordingly.
Example \(\PageIndex{1}\)
Use the atomic masses (rounded to two decimal places) to determine the formula mass for each ionic compound.
- FeCl 3
- (NH 4 ) 3 PO 4
Solution
a.
Fe: 1 x 55.85 = 55.85 amu
Cl: 1 x 35.45 = +106. 3 5 amu
________________________
Total = 162. 2 0 amu
The formula mass of FeCl 3 is 162.2 amu.
b. When we distribute the subscript 3 through the parentheses containing the formula for the ammonium ion, we see that we have 3 nitrogen atoms and 12 hydrogen atoms. Thus, we set up the sum as follows:
N: 3 x 14.00 = 42.00 amu
H: 12 x 1.00 = +12. 0 0 amu
P: 1 x 30.97 = +30.97 amu
O: 4 x 16.00 = +64.00 amu
Total = 148. 9 7 amu
The formula mass for (NH 4 ) 3 PO 4 is 149.0 amu.
Exercise \(\PageIndex{1}\)
Use the atomic masses (rounded to two decimal places) to determine the formula mass for each ionic compound.
- TiO 2
- AgBr
- Au(NO 3 ) 3
- Fe 3 (PO 4 ) 2
- Answer
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- 79.87 amu
- 187.77 amu
- 383.0 amu
To Your Health: Hydrates
Some ionic compounds have water (\(\ce{H2O}\)) incorporated within their formula unit. These compounds, called hydrates , have a characteristic number of water units associated with each formula unit of the compound. Hydrates are solids, not liquids or solutions, despite the water they contain.
To write the chemical formula of a hydrate, write the number of water units per formula unit of compound after its chemical formula. The two chemical formulas are separated by a vertically centered dot. The hydrate of copper(II) sulfate has five water units associated with each formula unit, so it is written as \(\ce{CuSO4 \cdot 5H2O}\). The name of this compound is copper(II) sulfate pentahydrate, with the penta- prefix indicating the presence of five water units per formula unit of copper(II) sulfate.
Hydrates have various uses in the health industry. Calcium sulfate hemihydrate (\(\ce{CaSO4 \cdot 1/2 H2O}\)), known as plaster of Paris , is used to make casts for broken bones. Epsom salt (\(\ce{MgSO4 \cdot 7H2O}\)) is used as a bathing salt and a laxative. Aluminum chloride hexahydrate is an active ingredient in antiperspirants. Table \(\PageIndex{1}\) lists some useful hydrates.
| Formula | Name | Uses |
|---|---|---|
| AlCl 3 •6H 2 O | aluminum chloride hexahydrate | antiperspirant |
| CaSO 4 •½H 2 O | calcium sulfate hemihydrate (plaster of Paris) | casts (for broken bones and castings) |
| CaSO 4 •2H 2 O | calcium sulfate di hydrate (gypsum) | drywall component |
| CoCl 2 •6H 2 O | cobalt(II) chloride hexahydrate | drying agent, humidity indicator |
| CuSO 4 •5H 2 O | copper(II) sulfate pentahydrate | fungicide, algicide, herbicide |
| MgSO 4 •7H 2 O | magnesium sulfate heptahydrate (Epsom salts) | laxative, bathing salt |
| Na 2 CO 3 •10H 2 O | sodium carbonate decahydrate (washing soda) | laundry additive/cleaner |
Key Takeaway
- Formula masses of ionic compounds can be determined from the masses of the atoms in their formulas.


2.6: Ethical Research
Learning Objectives
- Discuss the importance of ethical research
Today’s advances in technology allow for collection of data, information and analysis of that information to depths and breadths like never before. As the technological world has changed, the business world has too. Business research and the ethical considerations in the collection, research and use of the results are now more important than ever. As the market calls for business to be conducted with more transparency and social responsibility, customers expect business research to be conducted ethically. This also means that if unethical research is used, technology makes it easier to discover and expose online which in turn diminishes social trust and may even decrease customer sales and retention for that business.
The world is now more connected than ever and even though a business may be in the United States, many customers or business partners may reside in countries all over the world. This increases the need to conduct ethical business research with this wider world connection in mind. Ethical research is one of the foundational components of business ethics to be aware of and to practice in order to create a competitive and sustainable business in today’s world.
Ethical Research Principles
As a manager, part of your responsibility to a company and its employees is to ensure ethical research principles are understood and used when conducting research. It’s important to know if those principles are clearly defined and stated within the company. If not, there are many resources to assist in defining ethical business research for you and your team.
Over time, various organizations have developed research principles as research guidelines for social, scientific, educational, and business research. Ethical research is important for fostering a trusting, relational connection between all parties connected with the research. So, what is the essence of these business research principles?
Ethical research principles include:
- Respect for Persons and Their Autonomy. A person’s rights and decisions are respected.
- Beneficence and Nonmaleficence. Everyone involved is safe, considered and research ethically conducted.
- Fidelity and Integrity. Research is well designed, planned, process, outcomes, clear to everyone.
- Justice. Making sure the research is fair and inclusive.
Try It
Watch this video explaining the meaning of these principles. This video is regarding international development, but it is just as relevant for ethical business research.
You can also view a transcript for the video “Principles and Guidelines for Ethical Research and Evaluation in Development” here (opens in new window) .
You still may be asking yourself why the need to conduct ethical business research? What is so important about following ethical standards? To answer those questions read through these next points and think through your own specific business scenario and how it benefits from conducting ethical business research.
It’s important to follow ethical business research guidelines because they:
- Promote research to be truthful, inclusive, and minimize error or misrepresentation.
- Promote guiding standards of conduct for all people involved in the research.
- Promote accountability to businesses, customers, clients, and parties connected to the research.
- Promote public trust which can lead to public support of business.
- Promote compliance to laws, social responsibility, moral and ethical values, public health, etc.
What are ways to conduct ethical business research? Let’s look at an example.
Example: Ethical Online Research
In today’s world, billions of pieces of information are easily searched for online, but search engines do not determine what information is correct, useful, accurate, valuable, etc. Online searches return organized online data and information when search terms match the information, but search engines do not qualify that information. It is up to the researcher to validate and reference the source of the information being used.
How does a business researcher conduct ethical online research?
- Double checks information and sources used in business to account for its accuracy.
- Correctly attributes and sites data sources, reports, information, etc.
- Understands the copyright and fair use of images, content, audio, etc.
- Avoids plagiarism and taking credit for someone else’s ideas
With all of the information available on the internet, it is especially important for business to maintain their ethical standards and not rely on incorrect information or misuse another business or person’s work without the correct siting or payment for using images, ideas, audio, etc.
Practice Question
But what potentially happens to a business if ethical methods aren’t used? Let’s look briefly at a different example.
Non-Ethical Research Consequences
What if a business fostered an environment where its research was not conducted with guiding ethical standards? Can you think of a few examples? Here are a few ways businesses use unethical activities with research:
- misrepresent research results;
- break respondent confidentiality;
- violate non-disclosure agreements;
- avoid legal liability;
- plagiarism;
- manipulate research data or participants for favorable outcomes.
Example: Tobacco Research
One very public and famous example of research manipulation occurred over decades in tobacco and cigarette companies. Even though the scientific research proved smoking caused lung cancer and did harm to those who used it and people who were around it, tobacco companies’ manipulated the research results and denied the scientific facts thus endangering people’s health in the pursuit of profit. Now millions of dollars have be spent in lawsuits and the true results have been exposed to the public.
Additional consequences for unethical business research mean faster exposure and customer reaction happen now because of the rise of the internet, citizen reporters, whistleblower laws, and informed customers, when shady research practices are discovered.
Try It
Technology and strengthened whistleblower laws have made it safer for customers or employees to expose unethical practices when discovered. Check out this article about one such instance: Duke University Agrees To Pay U.S. $112.5 Million To Settle False Claims Act Allegations Related To Scientific Research Misconduct .
The United States Department Of Labor also has a Whistleblower Protection Program , put in place to protect those who report.
As the world and business becomes more transparent, customers, clients and business partners can more easily find out if research has not been conducted ethically and the business backlash can be expensive and even cripple a company’s revenue and public image. Yet one more reason it is important for a business to conduct its research ethically.
Practice Question
Setting up ethical business research practices from the beginning and creating a company which adheres to those practices not only help a company to avoid social backlash for unethical practices, but continually foster customer trust and in turn can lead to better relations between business, customers and society.
Sources
Bryman, Alan, and Emma Bell. “Ethics in Business Research.” In Business Research Methods , 3rd ed., 122–46. Oxford University Press, 2011.
Contributors and Attributions
- Ethical Research. Authored by : Sherri Pendleton. Provided by : Lumen Learning. License : CC BY: Attribution


9.4: Signaling Molecules and Cellular Receptors - Signaling Molecules
- Compare and contrast the different types of signaling molecules: hydrophobic, water-soluble, and gas ligands
Signaling Molecules
Produced by signaling cells and the subsequent binding to receptors in target cells, ligands act as chemical signals that travel to the target cells to coordinate responses. The types of molecules that serve as ligands are incredibly varied and range from small proteins to small ions like calcium (Ca 2+ ).
Small Hydrophobic Ligands
Small hydrophobic ligands can directly diffuse through the plasma membrane and interact with internal receptors. Important members of this class of ligands are the steroid hormones. Steroids are lipids that have a hydrocarbon skeleton with four fused rings; different steroids have different functional groups attached to the carbon skeleton.  Other hydrophobic hormones include thyroid hormones and vitamin D. In order to be soluble in blood, hydrophobic ligands must bind to carrier proteins while they are being transported through the bloodstream.
Water-Soluble Ligands
Water-soluble ligands are polar and, therefore, cannot pass through the plasma membrane unaided; sometimes, they are too large to pass through the membrane at all. Instead, most water-soluble ligands bind to the extracellular domain of cell-surface receptors. Cell-surface receptors include: ion-channel, G-protein, and enzyme-linked protein receptors. The binding of these ligands to these receptors results in a series of cellular changes. These water soluble ligands are quite diverse and include small molecules, peptides, and proteins.
Other Ligands
Nitric oxide (NO) is a gas that also acts as a ligand. It is able to diffuse directly across the plasma membrane; one of its roles is to interact with receptors in smooth muscle and induce relaxation of the tissue. NO has a very short half-life; therefore, it only functions over short distances. Nitroglycerin, a treatment for heart disease, acts by triggering the release of NO, which causes blood vessels to dilate (expand), thus restoring blood flow to the heart.
Key Points
- Signaling molecules can range from small proteins to small ions and can be hydrophobic, water-soluble, or even a gas.
- Hydrophobic signaling molecules ( ligands ) can diffuse through the plasma membrane and bind to internal receptors.
- Water-soluble ligands are unable to pass freely through the plasma membrane due to their polarity and must bind to an extracellular domain of a cell -surface receptor.
- Other types of ligands can include gases, such as nitric oxide, which can freely diffuse through the plasma membrane and bind to internal receptors.
Key Terms
- ligand : an ion, molecule, or functional group that binds to another chemical entity to form a larger complex
- hydrophobic : lacking an affinity for water; unable to absorb, or be wetted by water


8.1: Newton’s Second Law for Rotation
- Calculate the torques on rotating systems about a fixed axis to find the angular acceleration
- Explain how changes in the moment of inertia of a rotating system affect angular acceleration with a fixed applied torque
In this subsection, we put together all the pieces learned so far in this chapter to analyze the dynamics of rotating rigid bodies. We have analyzed motion with kinematics and rotational kinetic energy but have not yet connected these ideas with force and/or torque. In this subsection, we introduce the rotational equivalent to Newton’s second law of motion and apply it to rigid bodies with fixed-axis rotation.
Newton’s Second Law for Rotation
We have thus far found many counterparts to the translational terms used throughout this text, most recently, torque, the rotational analog to force. This raises the question: Is there an analogous equation to Newton’s second law, \(\sum \vec{F}\) = m\(\vec{a}\), which involves torque and rotational motion? To investigate this, we start with Newton’s second law for a single particle rotating around an axis and executing circular motion. Let’s exert a force \(\vec{F}\) on a point mass m that is at a distance r from a pivot point (Figure \(\PageIndex{1}\)). The particle is constrained to move in a circular path with fixed radius and the force is tangent to the circle. We apply Newton’s second law to determine the magnitude of the acceleration a = \(\frac{F}{m}\) in the direction of \(\vec{F}\). Recall that the magnitude of the tangential acceleration is proportional to the magnitude of the angular acceleration by a = r\(\alpha\). Substituting this expression into Newton’s second law, we obtain
\[F = mr \alpha \ldotp\]
Multiply both sides of this equation by r,
\[rF = mr^{2} \alpha \ldotp\]
Note that the left side of this equation is the torque about the axis of rotation, where r is the lever arm and F is the force, perpendicular to r. Recall that the moment of inertia for a point particle is I = mr 2 . The torque applied perpendicularly to the point mass in Figure \(\PageIndex{1}\) is therefore
\[\tau = I \alpha \ldotp\]
The torque on the particle is equal to the moment of inertia about the rotation axis times the angular acceleration . We can generalize this equation to a rigid body rotating about a fixed axis.
If more than one torque acts on a rigid body about a fixed axis, then the sum of the torques equals the moment of inertia times the angular acceleration:
\[\sum_{i} \vec\tau_{i} = I \vec\alpha \ldotp \label{10.25}\]
The term I\(\alpha\) is a scalar quantity and can be positive or negative (counterclockwise or clockwise) depending upon the sign of the net torque. Remember the convention that counterclockwise angular acceleration is positive. Thus, if a rigid body is rotating clockwise and experiences a positive torque (counterclockwise), the angular acceleration is positive.
Equation \ref{10.25} is Newton’s second law for rotation and tells us how to relate torque, moment of inertia, and rotational kinematics. This is called the equation for rotational dynamics . With this equation, we can solve a whole class of problems involving force and rotation. It makes sense that the relationship for how much force it takes to rotate a body would include the moment of inertia, since that is the quantity that tells us how easy or hard it is to change the rotational motion of an object.
Deriving Newton’s Second Law for Rotation in Vector Form
We have seen before that we can express the tangential acceleration vector as a cross product of the angular acceleration and the position vector. This expression can be found by taking the time derivative of \(\vec{v} = \vec{\omega} \times \vec{r}\) and is left as an exercise:
\[\vec{a} = \vec{\alpha} \times \vec{r} \ldotp \label{10.7}\]
The vector relationships for the angular acceleration and tangential acceleration are shown in Figure \(\PageIndex{2}\).
Similarly, we may find the torque vector. The second law \(\sum \vec{F}\) = m\(\vec{a}\) tells us the relationship between net force and how to change the translational motion of an object. We have a vector rotational equivalent of this equation, which can be found by using Equation \ref{10.7} and (Figure \(\PageIndex{2}\)) . Equation \ref{10.7} relates the angular acceleration to the position and tangential acceleration vectors. We form the cross product of this equation with \(\vec{r}\) and use a cross product identity (note that \(\vec{r}\; \cdotp \vec{\alpha}\) = 0):
\[\vec{r} \times \vec{a} = \vec{r} \times (\vec{\alpha} \times \vec{r}) = \vec{\alpha} (\vec{r}\; \cdotp \vec{r}) - \vec{r} (\vec{r}\; \cdotp \vec{\alpha}) = \vec{\alpha} (\vec{r}\; \cdotp \vec{r}) = \vec{\alpha} r^{2} \ldotp\]
We now form the cross product of Newton’s second law with the position vector \(\vec{r}\),
\[\sum (\vec{r} \times \vec{F}) = \vec{r} \times (m \vec{a}) = m \vec{r} \times \vec{a} = mr^{2} \vec{\alpha} \ldotp\]
Identifying the first term on the left as the sum of the torques, and mr 2 as the moment of inertia, we arrive at Newton’s second law of rotation:
\[\sum \vec\tau = I \vec\alpha \ldotp \label{10.26}\]
This equation is exactly Equation \ref{10.25}. An important point is that the torque vector is in the same direction as the angular acceleration.
Applying the Rotational Dynamics Equation
Before we apply the rotational dynamics equation to some everyday situations, let’s review a general problem-solving strategy for use with this category of problems.
- Examine the situation to determine that torque and mass are involved in the rotation. Draw a careful sketch of the situation.
- Determine the system of interest.
- Draw a free-body diagram. That is, draw and label all external forces acting on the system of interest.
- Identify the pivot point. If the object is in equilibrium, it must be in equilibrium for all possible pivot points––chose the one that simplifies your work the most.
- Apply \(\sum_{i} \tau_{i} = I \alpha\), the rotational equivalent of Newton’s second law, to solve the problem. Care must be taken to use the correct moment of inertia and to consider the torque about the point of rotation.
- As always, check the solution to see if it is reasonable.
Consider the father pushing a playground merry-go-round in Figure \(\PageIndex{2}\). He exerts a force of 250 N at the edge of the 200.0-kg merry-go-round, which has a 1.50-m radius. Calculate the angular acceleration produced (a) when no one is on the merry-go-round and (b) when an 18.0-kg child sits 1.25 m away from the center. Consider the merry-go-round itself to be a uniform disk with negligible friction.
- Strategy
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The net torque is given directly by the expression \(\sum_{i} \tau_{i} = I \alpha\), To solve for \(\alpha\), we must first calculate the net torque \(\tau\) (which is the same in both cases) and moment of inertia I (which is greater in the second case).
- Solutiony
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- The moment of inertia of a solid disk about this axis is given in Figure 10.5.4 to be $$\frac{1}{2} MR^{2} \ldotp$$We have M = 50.0 kg and R = 1.50 m, so $$I = (0.500)(50.0\; kg)(1.50\; m)^{2} = 56.25\; kg\; \cdotp m^{2} \ldotp$$To find the net torque, we note that the applied force is perpendicular to the radius and friction is negligible, so that $$\tau = rF \sin \theta = (1.50\; m)(250.0\; N) - 375.0\; N\; \cdotp m \ldotp$$Now, after we substitute the known values, we find the angular acceleration to be $$\alpha = \frac{\tau}{I} = \frac{375.0\; N\; \cdotp m}{56.25\; kg\; \cdotp m^{2}} = 6.67\; rad/s^{2} \ldotp$$
- We expect the angular acceleration for the system to be less in this part because the moment of inertia is greater when the child is on the merry-go-round. To find the total moment of inertia I, we first find the child’s moment of inertia I c by approximating the child as a point mass at a distance of 1.25 m from the axis. Then $$I_{c} = mR^{2} = (18.0\; kg)(1.25\; m)^{2} = 28.13\; kg\; \cdotp m^{2} \ldotp$$The total moment of inertia is the sum of the moments of inertia of the merry-go-round and the child (about the same axis): $$I = (28.13\; kg\; \cdotp m^{2}) + (56.25\; kg\; \cdotp m^{2}) = 84.38\; kg\; \cdotp m^{2} \ldotp$$Substituting known values into the equation for \(\alpha\) gives $$\alpha =\frac{\tau}{I} = \frac{375.0\; N\; \cdotp m}{84.38\; kg\; \cdotp m^{2}} = 4.44\; rad/s \ldotp$$
Significance
The angular acceleration is less when the child is on the merry-go-round than when the merry-go-round is empty, as expected. The angular accelerations found are quite large, partly due to the fact that friction was considered to be negligible. If, for example, the father kept pushing perpendicularly for 2.00 s, he would give the merry-goround an angular velocity of 13.3 rad/s when it is empty but only 8.89 rad/s when the child is on it. In terms of revolutions per second, these angular velocities are 2.12 rev/s and 1.41 rev/s, respectively. The father would end up running at about 50 km/h in the first case.
A toy rocket is attached to a string on a horizontal frictionless table, as shown in Figure \(\PageIndex{3}\). The rocket has a mass \(m\) and produces a constant force of thrust with a magnitude \(\vec{F}\) that accelerates the rocket along a circle of radius \(R\) (the length of the string). If the rocket starts at rest, what distance along the circumference of the circle will the rocket have traveled after a time, \(t\) ?
- Solution
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We can model the rocket as a point particle of mass \(m\) with the following forces exerted on it:
- \(\vec F\) , the thrust of the rocket, always acting tangent to the circle.
- \(\vec T\) , the force of tension in the string, always acting towards the center of the circle.
- \(\vec F_g\) , the rocket’s weight, acting into the page, with magnitude \(mg\) .
- \(\vec F_N\) , a normal force exerted by the table, out of the page, with magnitude \(mg\) .
Because the normal force and the weight are equal in magnitude and opposite in direction, the net force will be the sum of the force of thrust and the force of tension, which are always perpendicular to each other. Thinking about this with Newton’s Second Law, we could model the force of thrust as increasing the speed of the particle, while the force of tension keeps the rocket moving in a circle (it cannot increase the speed, since it is always perpendicular to the motion).
We introduce a coordinate system whose origin coincides with the center of the circle, as shown in Figure \(\PageIndex{4}\), so that \(\vec r\) corresponds to the position of the rocket relative to the origin. The force of thrust and the tension are also shown in the diagram. We choose the direction of the \(x\) axis such that the rocket was located at the intersection of the \(x\) axis and the circle at time, \(t=0\) .The net torque on the rocket about the point of rotation is given by the cross-product between the thrust force, \(\vec F\) , and the position vector, \(\vec r\) :
\[\begin{aligned} \vec\tau^{net} = \vec r\times\vec F\end{aligned}\]
and will point in the positive \(z\) direction (as given by the right hand rule). \(\vec r\) and \(\vec F\) are perpendicular, so the magnitude of the net torque is given by:
\[\begin{aligned} \tau^{net} = rF \sin(90^{\circ}) = RF\end{aligned}\]
where \(R\) is the magnitude of \(\vec r\) . The net torque vector is thus:
\[\begin{aligned} \vec\tau^{net} = RF \hat z\end{aligned}\]
Applying the rotational version of Newton’s Second Law allows us to determine the angular acceleration:
\[\begin{aligned} \vec \tau ^{net} &= mr^2\vec\alpha\\ RF \hat z&= mR^2\vec\alpha\\ \therefore \vec \alpha &= \frac{F}{mR}\hat z\end{aligned}\]
The angular acceleration vector points in the positive \(z\) direction (as does the net torque), and indicates that the rocket is accelerating in the counter-clockwise direction about the \(z\) axis.
After a period of time \(t\) , the rocket will have covered an angular displacement, \(\Delta \theta\) , given by:
\[\begin{aligned} \Delta \theta &= \theta(t)-\theta_0 = \omega_0t + \frac{1}{2}\alpha t^2\\ &=\frac{1}{2}\frac{F}{mR} t^2\end{aligned}\]
The linear displacement, \(\Delta s\) , that corresponds to this angular displacement is:
\[\begin{aligned} \Delta s = R \Delta\theta = \frac{1}{2}\frac{F}{m} t^2\end{aligned}\]
- Discussion
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The formula that we found for the total linear displacement is the same that we would have found if the particle were moving in a straight line with a net force \(F\) applied to it (as the particle would have a constant acceleration given by \(F/m\) ).
The fan blades on a jet engine have a moment of inertia 30.0 kg • m 2 . In 10 s, they rotate counterclockwise from rest up to a rotation rate of 20 rev/s. (a) What torque must be applied to the blades to achieve this angular acceleration? (b) What is the torque required to bring the fan blades rotating at 20 rev/s to a rest in 20 s?


6: Graphs
In this chapter, we study two representations of graphs and basic algorithms that use these representations.
Mathematically, a (directed) graph is a pair \(G=(V,E)\) where \(V\) is a set of vertices and \(E\) is a set of ordered pairs of vertices called edges . An edge \(\mathtt{(i,j)}\) is directed from \(\mathtt{i}\) to \(\mathtt{j}\); \(\mathtt{i}\) is called the source of the edge and \(\mathtt{j}\) is called the target . A path in \(G\) is a sequence of vertices \(v_0,\ldots,v_k\) such that, for every \(i\in\{1,\ldots,k\}\), the edge \((v_{i-1},v_{i})\) is in \(E\). A path \(v_0,\ldots,v_k\) is a cycle if, additionally, the edge \((v_k,v_0)\) is in \(E\). A path (or cycle) is simple if all of its vertices are unique. If there is a path from some vertex \(v_i\) to some vertex \(v_j\) then we say that \(v_j\) is reachable from \(v_i\). An example of a graph is shown in Figure \(\PageIndex{1}\).
Due to their ability to model so many phenomena, graphs have an enormous number of applications. There are many obvious examples. Computer networks can be modelled as graphs, with vertices corresponding to computers and edges corresponding to (directed) communication links between those computers. City streets can be modelled as graphs, with vertices representing intersections and edges representing streets joining consecutive intersections.
Less obvious examples occur as soon as we realize that graphs can model any pairwise relationships within a set. For example, in a university setting we might have a timetable conflict graph whose vertices represent courses offered in the university and in which the edge \(\mathtt{(i,j)}\) is present if and only if there is at least one student that is taking both class \(\mathtt{i}\) and class \(\mathtt{j}\). Thus, an edge indicates that the exam for class \(\mathtt{i}\) should not be scheduled at the same time as the exam for class \(\mathtt{j}\).
Throughout this section, we will use \(\mathtt{n}\) to denote the number of vertices of \(G\) and \(\mathtt{m}\) to denote the number of edges of \(G\). That is, \(\mathtt{n}=\vert V\vert\) and \(\mathtt{m}=\vert E\vert\). Furthermore, we will assume that \(V=\{0,\ldots,\mathtt{n}-1\}\). Any other data that we would like to associate with the elements of \(V\) can be stored in an array of length \(\mathtt{n}\).
Some typical operations performed on graphs are:
- \(\mathtt{addEdge(i,j)}\): Add the edge \((\mathtt{i},\mathtt{j})\) to \(E\).
- \(\mathtt{removeEdge(i,j)}\): Remove the edge \((\mathtt{i},\mathtt{j})\) from \(E\).
- \(\mathtt{hasEdge(i,j)}\): Check if the edge \((\mathtt{i},\mathtt{j})\in E\)
- \(\mathtt{outEdges(i)}\): Return a List of all integers \(\mathtt{j}\) such that \((\mathtt{i},\mathtt{j})\in E\)
- \(\mathtt{inEdges(i)}\): Return a List of all integers \(\mathtt{j}\) such that \((\mathtt{j},\mathtt{i})\in E\)
Note that these operations are not terribly difficult to implement efficiently. For example, the first three operations can be implemented directly by using a USet , so they can be implemented in constant expected time using the hash tables discussed in Chapter 7 . The last two operations can be implemented in constant time by storing, for each vertex, a list of its adjacent vertices.
However, different applications of graphs have different performance requirements for these operations and, ideally, we can use the simplest implementation that satisfies all the application's requirements. For this reason, we discuss two broad categories of graph representations.


9.1: Introduction
Module 9
Geologic Time
Introduction
Geologic time is immense. The Earth has existed for approximately 4.6 billion years, and many geologic processes occur over timescales of millions of years. Given that it is unusual for a person to live over 100 years, grasping the immensity of geologic time is challenging. In this module, you will learn about ways in which relative time can be determined, and about techniques that can be used to estimate numerical ages of Earth materials. You will also learn about the units that are used to measure geologic time.
The amount of time that is involved in the carving of the landscape, the formation of rocks, or the movement of the continents is an important scientific question. Different hypotheses about the age of the Earth can essentially change our perspective of the workings of geologic events that molded the Earth. If the geologic time is relatively short then catastrophic events would be required to form the features we see on the surface of the Earth, whereas a vast amount of time allows the slow and steady pace that we can easily observe around us today.
Geologists have used many methods to reconstruct geologic time and mapping the major events in Earth’s history as well as their duration. Scientists studying rocks were able to piece together a progression of rocks through time to construct the Geologic Time Scale below. This time scale was constructed by lining up in order rocks that had particular features such as rock types, environmental indicators, or fossils. Scientists looked at clues within the rocks and determined the age of these rocks in a comparative sense. This process is called relative dating, which is the process of determining the comparative age of two objects or events. For example, you are younger than your parents. It doesn’t matter your age or your parents’ age, as long as you can establish that one is older than the other. As time progressed, scientists discovered and developed techniques to date certain rocks, as well as the Earth itself. They discovered the Earth was billions of years old (4.54 billion years old) and put a time frame to the geologic time scale. This process is called absolute dating, which is the process of determining the exact amount of time that has passed since an object was formed or an event occurred.
Both absolute and relative dating have advantages and are still frequently used by geologists. Dating rocks using relative dating allows a geologist to reconstruct a series of events cheaply, often very quickly, and can be used out in the field on a rocky outcrop. Relative dating also can be used on many different types of rocks, where absolute dating is restricted to certain minerals or materials. However, absolute dating is the only method that allows scientists to place an exact age to a particular rock.
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Module Objectives
At the completion of this module you will be able to:
- Apply basic geological principles to the determination of the relative ages of rocks.
- Explain the difference between relative and absolute age-dating techniques.
- Summarize the history of the geological time scale and the relationships between eons, eras, periods, and epochs.
- Understand the importance and significance of unconformities.
- Describe the applications and limitations of using isotopes, tree rings, and magnetic data for geological dating.
Activities Overview
See the Schedule of Work for dates of availability and due dates.
Be sure to read through the directions for all of this module’s activities before getting started so that you can plan your time accordingly. You are expected to work on this course throughout the week.
Read
Physical Geology by Steven Earle
- Chapter 8 (Measuring Geologic Time)
Module 9 Assignment: Relative Dating and Cross Cutting Relationships
10 points
After you complete the reading, you can start working on Module 9 Assignment – Relative Dating and Cross Cutting Relationships
Module 9 Quiz
10 points
Module 9 Quiz has 10 multiple-choice questions and is based on the content of the Module 9 readings and Assignment 9.
The quiz is worth a total of 10 points (1 points per question). You will have only 10 minutes to complete the quiz, and you may take this quiz only once. Note : that is not enough time to look up the answers!
Make sure that you fully understand all of the concepts presented and study for this quiz as though it were going to be proctored in a classroom, or you will likely find yourself running out of time.
Keep track of the time, and be sure to look over your full quiz results after you have submitted it for a grade.
Your Questions and Concerns…
Please contact me if you have any questions or concerns.
General course questions : If your question is of a general nature such that other students would benefit from the answer, then go to the discussions area and post it as a question thread in the “General course questions” discussion area.
Personal questions : If your question is personal, (e.g. regarding my comments to you specifically), then send me an email from within this course.


17.8: Partition Functions can be Decomposed into Partition Functions of Each Degree of Freedom
From the previous sections, the partition function for a system of \(N\) indistinguishable and independent molecules is:
\[ Q(N,V,\beta) = \dfrac{\sum_i{e^{-\beta E_i}}}{N!} \label{ID1} \]
And the average energy of the system is:
\[ \langle E \rangle = kT^2 \left(\dfrac{\partial \ln{Q}}{\partial T}\right) \label{ID2} \]
We can combine these two equations to obtain:
\[ \begin{split} \langle E \rangle &= kT^2 \left(\dfrac{\partial \ln{Q}}{\partial T}\right)_{N,V} \\ &= NkT^2 \left(\dfrac{\partial \ln{q}}{\partial T}\right)_V \\ &= N\sum_i{\epsilon_i \dfrac{e^{-\epsilon_i/kT}}{q(V,T)}} \end{split} \label{ID3} \]
The average energy is equal to:
\[ \langle E \rangle = N \langle \epsilon \rangle \label{aveE} \]
where \(\langle \epsilon \rangle\) is the average energy of a single particle. If we compare Equation \(\ref{ID2}\) with Equation \(\ref{ID2}\), we can see:
\[ \langle \epsilon \rangle = \sum_i{\epsilon_i \dfrac{e^{-\epsilon_i/kT}}{q(V,T)}} \nonumber \]
The probability that a particle is in state \(i\), \(\pi_i\), is given by:
\[ \langle \epsilon \rangle = \dfrac{e^{-\epsilon_i/kT}}{q(V,T)} = \dfrac{e^{-\epsilon_i/kT}}{\sum_i{e^{-\epsilon_i/kT}}} \nonumber \]
The energy of a particle is a sum of the energy of each degree of freedom for that particle. In the case of a molecule, the energy is:
\[ \epsilon = \epsilon_\text{trans} + \epsilon_\text{rot} + \epsilon_\text{vib} + \epsilon_\text{elec} \nonumber \]
The molecular partition function is the product of the degree of freedom partition functions:
\[ q(V,T) = q_\text{trans} q_\text{rot} q_\text{vib} q_\text{elec} \nonumber \]
The partition function for each degree of freedom follows the same is related to the Boltzmann distribution. For example, the vibrational partition function is:
\[ q_\text{vib} = \sum_i{e^{-\epsilon_i/kT}} \nonumber \]
The average energy of each degree of freedom follows the same pattern as before. For example, the average vibrational energy is:
\[ \langle \epsilon_\text{vib} \rangle = kT^2\dfrac{\partial \ln{q_\text{vib}}}{\partial t} = -\dfrac{\partial \ln{q_\text{vib}}}{\partial \beta} \nonumber \]


1.1.5: Taking Charge of your health - Making Healthy Changes
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Increase Your Opportunity for Successful Change
Behavior change is hard for many reasons. Here are some strategies to help increase your chances of successfully achieving your health and wellness goals.
Set approach goals instead of avoidance goals [10]
Approach goals are goals where you are taking actions to meet a desired outcome, whereas avoidance goals are focused on not taking actions. Research has shown that we are more successful when we set approach goals rather than avoidance goals. For example, rather than setting an avoidance goal such as “I will not drink soda” the focus could be “I will drink water instead of soda.”
Set mastery (learning) goals instead of performance goals [11]
A performance goal might be to lose 20 pounds, whereas a mastery (learning) goal would focus on the actions you are taking to lose weight, such as focusing on your nutrition or exercise. A mastery skill involves increasing existing abilities or learning new skills. Performance goals may be less effective than mastery goals. It is ok to set a performance goal, but it should always be accompanied by a mastery goal.
Find Your Why
Take a moment to think about why you want to change. How will this change make your life better? How will this change help you feel better, have better relationships with yourself and others, or help your family or friends? You might like to refer to this as finding or discovering your why.
Your Daily Action Plan
If you think about your SMART goal as your destination, how will you get there and what are the directions or what route will you take? Developing an action plan is a helpful strategy to increase your motivation. An action plan specifies the what (always a behavior), how much, how often, and usually when (which days and times), in other words it is the steps by which the SMART goal will be achieved. Your action plan provides you with your goal for each day allowing you to focus on your day-to-day actions rather than just on your long term goal.
Boost Your Self Efficacy
Implementing a carefully and thoughtfully crafted action plan can help to improve health and build your self efficacy [12] . Self-Efficacy is the confidence in one’s ability to achieve a specific goal, it is the belief in oneself that you can do it. Implementing and meeting the goals of your action plan is a great strategy for building your self-efficacy, which in turn builds your motivation to change.
A Positive Mindset
Having a positive mindset can also help to increase your self-efficacy. This might include focusing on positive self-talk whereby purposefully telling yourself that you can complete your goals. A positive mindset may also take the form of finding enjoyment in your changes. When the thought of your change brings you enjoyment, you are more likely to engage in the behavior change and stick to it [13] .