21.5: Stomach
By the end of the section, you will be able to:
- Label on a diagram the four main regions of the stomach, its curvatures, and its sphincter
- Identify the four main types of secreting cells in gastric glands, and their important products
- Explain why the stomach does not digest itself
- Describe the mechanical and chemical digestion of food entering the stomach
Although a minimal amount of carbohydrate digestion occurs in the mouth, chemical digestion really gets underway in the stomach. An expansion of the alimentary canal that lies immediately inferior to the esophagus, the stomach links the esophagus to the first part of the small intestine (the duodenum) and is relatively fixed in place at its esophageal and duodenal ends. In between, however, it can be a highly active structure, contracting and continually changing position and size. These contractions provide mechanical assistance to digestion. The empty stomach is only about the size of your fist, but can stretch to hold as much as 4 liters of food and fluid, or more than 75 times its empty volume, and then return to its resting size when empty. Although you might think that the size of a person’s stomach is related to how much food that individual consumes, body weight does not correlate with stomach size. Rather, when you eat greater quantities of food—such as at holiday dinner—you stretch the stomach more than when you eat less.
Popular culture tends to refer to the stomach as the location where all digestion takes place. Of course, as you have already begun to learn, this is not true; the bulk of digestion occurs in the small intestine. An important function of the stomach is to serve as a temporary holding chamber. You can ingest a meal far more quickly than it can be digested and absorbed by the small intestine. Thus, the stomach holds food and parses only small amounts into the small intestine at a time. Foods are not processed in the order they are eaten; rather, they are mixed together with digestive juices in the stomach until they are converted into chyme, which is released into the small intestine.
As you will see in the sections that follow, the stomach plays several important roles in chemical digestion, including the continued digestion of carbohydrates and the initial digestion of proteins and triglycerides. Little,if any, nutrient absorption occurs in the stomach, with the exception of the negligible amount of nutrients in alcohol.
Structure
There are four main regions in the stomach : the cardia, fundus, body, and pylorus (Figure 21.5.1). It is unique with three layers of the muscularis externa: outer longitudinal layer, middle circular layer, and inner oblique layer. The cardia (or cardiac region) is the point where the esophagus connects to the stomach and through which food passes into the stomach. Located inferior to the diaphragm, above and to the left of the cardia, is the dome-shaped fundus . Below the fundus is the body , the main part of the stomach. The funnel-shaped pylorus connects the stomach to the duodenum. The wider end of the funnel, the pyloric antrum , connects to the body of the stomach. The narrower end is called the pyloric canal , which connects to the duodenum. The smooth muscle pyloric sphincter is located at this latter point of connection and controls stomach emptying. In the absence of food, the stomach deflates inward, and its mucosa and submucosa fall into large folds called rugae .
The convex lateral surface of the stomach is called the greater curvature; the concave medial border is the lesser curvature. The stomach is held in place by the lesser omentum, which extends from the liver to the lesser curvature, and the greater omentum, which runs from the greater curvature, down the anterior abdominal cavity, forming a large, insulating fold and then connecting to the transverse colon.
Histology
The wall of the stomach is made of the same four layers as most of the rest of the alimentary canal, but with adaptations to the mucosa and muscularis for the unique functions of this organ. In addition to the typical circular and longitudinal smooth muscle layers, the muscularis has an inner oblique smooth muscle layer (Figure 21.5.2). As a result, in addition to moving food through the canal, the stomach can vigorously churn food, mechanically breaking it down into smaller particles.
Figure 21.5.2 also shows the different types of gastric gland in the epithelial lining. The stomach mucosa’s simple columnar epithelial lining consists only of surface mucous cells , which secrete a protective coat of alkaline mucus. A vast number of gastric pits dot the surface of the epithelium, giving it the appearance of a well-used pincushion, and mark the entry to deeper exocrine tubules called gastric glands , which secrete a complex digestive fluid referred to as gastric juice.
Although the walls of the gastric pits are made up primarily of mucous cells, the gastric glands are made up of different types of cells. The glands of the cardia and pylorus are composed primarily of mucus-secreting cells. Cells that make up the pyloric antrum secrete mucus and a number of hormones, including the majority of the stimulatory hormone, gastrin . The much larger glands of the fundus and body of the stomach, the site of most chemical digestion, produce most of the gastric secretions. These glands are made up of a variety of secretory cells. These include parietal cells, chief cells, mucous neck cells, and enteroendocrine cells.
Parietal cells —Located primarily in the middle region of the gastric glands are parietal cells , which are among the most highly differentiated of the body’s epithelial cells. These relatively large cells produce both hydrochloric acid (HCl) and intrinsic factor . HCl is responsible for the high acidity (pH 1.5 to 3.5) of the stomach contents and is needed to activate the protein-digesting enzyme, pepsin. The acidity also kills much of the bacteria you ingest with food and helps to denature proteins, making them more available for enzymatic digestion. Intrinsic factor is a glycoprotein necessary for the absorption of vitamin B 12 in the small intestine.
Chief cells —Located primarily in the basal regions of gastric glands are chief cells , which secrete pepsinogen , the inactive proenzyme form of pepsin. HCl is necessary for the conversion of pepsinogen to pepsin, an enzyme to digest proteins.
Mucous neck cells —Gastric glands in the upper part of the stomach contain mucous neck cells that secrete thin, acidic mucus that is much different from the mucus secreted by the goblet cells of the surface epithelium. The role of this mucus is not currently known.
Enteroendocrine cells —Finally, enteroendocrine cells found in the gastric glands secrete various hormones into the interstitial fluid of the lamina propria. These include gastrin, which is released mainly by enteroendocrine G cells .
The mucosa of the stomach is exposed to the highly corrosive acidity of gastric juice. Gastric enzymes that can digest protein can also digest the stomach itself. The stomach is protected from self-digestion by the mucosal barrier . This barrier has several components. First, the stomach wall is covered by a thick coating of bicarbonate-rich mucus. This mucus forms a physical barrier, and its bicarbonate ions neutralize acid. Second, the epithelial cells of the stomach's mucosa meet at tight junctions, which block gastric juice from penetrating the underlying tissue layers. Finally, stem cells located where gastric glands join the gastric pits quickly replace damaged epithelial mucosal cells, when the epithelial cells are shed. In fact, the surface epithelium of the stomach is completely replaced every 3 to 6 days.
| Hormone | Production site | Production stimulus | Target organ | Action |
|---|---|---|---|---|
| Gastrin | Stomach mucosa, mainly G cells of the pyloric antrum | Presence of peptides and amino acids in stomach | Stomach | Increases secretion by gastric glands; promotes gastric emptying |
| Gastrin | Stomach mucosa, mainly G cells of the pyloric antrum | Presence of peptides and amino acids in stomach | Small intestine | Promotes intestinal muscle contraction |
| Gastrin | Stomach mucosa, mainly G cells of the pyloric antrum | Presence of peptides and amino acids in stomach | Ileocecal valve | Relaxes valve |
| Gastrin | Stomach mucosa, mainly G cells of the pyloric antrum | Presence of peptides and amino acids in stomach | Large intestine | Triggers mass movements |
| Ghrelin | Stomach mucosa, mainly fundus | Fasting state (levels increase just prior to meals) | Hypothalamus | Regulates food intake, primarily by stimulating hunger and satiety |
| Histamine | Stomach mucosa | Presence of food in the stomach | Stomach | Stimulates parietal cells to release HCl |
| Serotonin | Stomach mucosa | Presence of food in the stomach | Stomach | Contracts stomach muscle |
| Somatostatin | Mucosa of stomach, especially pyloric antrum; also duodenum | Presence of food in the stomach; sympathetic axon stimulation | Stomach | Restricts all gastric secretions, gastric motility, and emptying |
| Somatostatin | Mucosa of stomach, especially pyloric antrum; also duodenum | Presence of food in the stomach; sympathetic axon stimulation | Pancreas | Restricts pancreatic secretions |
| Somatostatin | Mucosa of stomach, especially pyloric antrum; also duodenum | Presence of food in the stomach; sympathetic axon stimulation | Small intestine | Reduces intestinal absorption by reducing blood flow |
DISORDER OF THE...
Digestive System: Ulcers in Mucosal Barrier
As effective as the mucosal barrier is, it is not a “fail-safe” mechanism. Sometimes, gastric juice eats away at the superficial lining of the stomach mucosa, creating erosions, which mostly heal on their own. Deeper and larger erosions are called ulcers.
Why does the mucosal barrier break down? A number of factors can interfere with its ability to protect the stomach lining. The majority of all ulcers are caused by either excessive intake of non-steroidal anti-inflammatory drugs (NSAIDs), including aspirin, or Helicobacter pylori infection.
Antacids help relieve symptoms of ulcers such as “burning” pain and indigestion. When ulcers are caused by NSAID use, switching to other classes of pain relievers allows healing. When caused by H. pylori infection, antibiotics are effective.
A potential complication of ulcers is perforation: Perforated ulcers create a hole in the stomach wall, resulting in peritonitis (inflammation of the peritoneum). These ulcers must be repaired surgically.
Digestive Functions of the Stomach
The stomach participates in virtually all the digestive activities with the exception of ingestion and defecation. Although almost all absorption takes place in the small intestine, the stomach does absorb some nonpolar substances, such as alcohol and aspirin.
Mechanical Digestion
Within a few moments after food after enters your stomach, mixing waves begin to occur at intervals of approximately 20 seconds. A mixing wave is a unique type of peristalsis that mixes and softens the food with gastric juices to create chyme. The initial mixing waves are relatively gentle, but these are followed by more intense waves, starting at the body of the stomach and increasing in force as they reach the pylorus. It is fair to say that long before your sushi exits through the pyloric sphincter, it bears little resemblance to the sushi you ate.
The pylorus, which holds around 30 mL (1 fluid ounce) of chyme, acts as a filter, permitting only liquids and small food particles to pass through the mostly, but not fully, closed pyloric sphincter. In a process called gastric emptying , rhythmic mixing waves force about 3 mL of chyme at a time through the pyloric sphincter and into the duodenum. Release of a greater amount of chyme at one time would overwhelm the capacity of the small intestine to handle it. The rest of the chyme is pushed back into the body of the stomach, where it continues mixing. This process is repeated when the next mixing waves force more chyme into the duodenum.
Gastric emptying is regulated by both the stomach and the duodenum. The presence of chyme in the duodenum activates receptors that inhibit gastric secretion. This prevents additional chyme from being released by the stomach before the duodenum is ready to process it.
Chemical Digestion
The fundus plays an important role, because it stores both undigested food and gases that are released during the process of chemical digestion. Food may sit in the fundus of the stomach for a while before being mixed with the chyme. While the food is in the fundus, the digestive activities of salivary amylase continue until the food begins mixing with the acidic chyme. Ultimately, mixing waves incorporate this food with the chyme, the acidity of which inactivates salivary amylase and activates lingual lipase. Lingual lipase then begins chemical digestion of lipids, breaking down triglycerides into free fatty acids, and mono- and diglycerides.
The secretion of gastric juice is controlled by both nerves and hormones. Stimuli in the brain, stomach, and small intestine activate or inhibit gastric juice production. This is why the three phases of gastric secretion are called the cephalic, gastric, and intestinal phases. However, once gastric secretion begins, all three phases can occur simultaneously.
The breakdown of protein begins in the stomach through the actions of HCl and the enzyme pepsin. During infancy, gastric glands also produce rennin, an enzyme that helps digest milk protein.
Its numerous digestive functions notwithstanding, there is only one stomach function necessary to life: the production of intrinsic factor. The intestinal absorption of vitamin B 12 , which is necessary for both the production of mature red blood cells and normal neurological functioning, cannot occur without intrinsic factor. People who undergo total gastrectomy (stomach removal)—for life-threatening stomach cancer, for example—can survive with minimal digestive dysfunction if they receive vitamin B 12 injections.
The contents of the stomach are completely emptied into the duodenum within 2 to 4 hours after you eat a meal. Different types of food take different amounts of time to process. Foods heavy in carbohydrates empty fastest, followed by high-protein foods. Meals with a high triglyceride content remain in the stomach the longest. Since enzymes in the small intestine digest fats slowly, food can stay in the stomach for 6 hours or longer when the duodenum is processing fatty chyme. However, note that this is still a fraction of the 24 to 72 hours that full digestion typically takes from start to finish.
Concept Review
The stomach is a curved organ that connects the esophagus to the small intestine. It has a greater curvature and lesser curvature and can expand because of rugae. There are four main regions in the stomach : the cardia, fundus, body, and pylorus. The mucosa has gastric pits that lead to gastric glands that secrete gastric juices, comprised of mucus, acid, and different proteins to aid in digestion. The muscularis externa has an additional layer, the oblique smooth muscle layer to aid in churning of chyme. The stomach participates in all digestive activities except ingestion and defecation. The stomach begins the digestion of protein and continues the digestion of carbohydrates and fats. It stores food as chyme, and releases it gradually into the small intestine through the pyloric sphincter. It absorbs certain drugs, including aspirin and some alcohol.
Review Questions
Q. Which is not related to the stomach?
A. lesser omentum
B. greater curvature
C. stratified squamous cells
D. enteroendocrine cells
- Answer
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Answer: C
Q. Which is/are unique about the stomach?
A. Muscularis has 3 layers of muscles.
B. Rugae allow the stomach to expand.
C. It has pits leading to glands.
D. All of the above.
- Answer
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Answer: D
Q. During gastric emptying, chyme is released into the duodenum through the ________.
A. esophageal hiatus
B. pyloric antrum
C. pyloric canal
D. pyloric sphincter
- Answer
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Answer: D
Q. Parietal cells are located in the ________.
A. gastric pits
B. gastric glands
C. lamina propria
D. muscularis mucosa
- Answer
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Answer: B
Critical Thinking Questions
Q. Explain how the stomach is protected from self-digestion and why this is necessary.
- Answer
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A. The mucosal barrier protects the stomach from self-digestion. It includes a thick coating of bicarbonate-rich mucus; the mucus is physically protective, and bicarbonate neutralizes gastric acid. Epithelial cells meet at tight junctions, which block gastric juice from penetrating the underlying tissue layers, and stem cells quickly replace sloughed off epithelial mucosal cells.
Q. Describe unique anatomical features that enable the stomach to perform digestive functions.
- Answer
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A. The stomach has an additional inner oblique smooth muscle layer that helps the muscularis churn and mix food. The epithelium includes gastric glands that secrete gastric fluid. The gastric fluid consists mainly of mucous, HCl, and the enzyme pepsin released as pepsinogen.
Glossary
- body
- mid-portion of the stomach
- cardia
- (also, cardiac region) part of the stomach surrounding the cardiac orifice (esophageal hiatus)
- cephalic phase
- (also, reflex phase) initial phase of gastric secretion that occurs before food enters the stomach
- chief cell
- gastric gland cell that secretes pepsinogen
- enteroendocrine cell
- gastric gland cell that releases hormones
- fundus
- dome-shaped region of the stomach above and to the left of the cardia
- G cell
- gastrin-secreting enteroendocrine cell
- gastric emptying
- process by which mixing waves gradually cause the release of chyme into the duodenum
- gastric gland
- gland in the stomach mucosal epithelium that produces gastric juice
- gastric phase
- phase of gastric secretion that begins when food enters the stomach
- gastric pit
- narrow channel formed by the epithelial lining of the stomach mucosa
- gastrin
- peptide hormone that stimulates secretion of hydrochloric acid and gut motility
- hydrochloric acid (HCl)
- digestive acid secreted by parietal cells in the stomach
- intrinsic factor
- glycoprotein required for vitamin B 12 absorption in the small intestine
- intestinal phase
- phase of gastric secretion that begins when chyme enters the intestine
- mixing wave
- unique type of peristalsis that occurs in the stomach
- mucosal barrier
- protective barrier that prevents gastric juice from destroying the stomach itself
- mucous neck cell
- gastric gland cell that secretes a uniquely acidic mucus
- parietal cell
- gastric gland cell that secretes hydrochloric acid and intrinsic factor
- pepsinogen
- inactive form of pepsin
- pyloric antrum
- wider, more superior part of the pylorus
- pyloric canal
- narrow, more inferior part of the pylorus
- pyloric sphincter
- sphincter that controls stomach emptying
- pylorus
- lower, funnel-shaped part of the stomach that is continuous with the duodenum
- ruga
- fold of alimentary canal mucosa and submucosa in the empty stomach and other organs
- stomach
- alimentary canal organ that contributes to chemical and mechanical digestion of food from the esophagus before releasing it, as chyme, to the small intestine
Contributors and Attributions
-
OpenStax Anatomy & Physiology (CC BY 4.0). Access for free at https://openstax.org/books/anatomy-and-physiology


Describe the use of shared power in U.S. foreign policymaking
Explain why presidents lead more in foreign policy than in domestic policy
Discuss why individual House and Senate members rarely venture into foreign policy
List the actors who engage in foreign policy
Institutional relationships in foreign policy constitute a paradox. On the one hand, there are aspects of foreign policymaking that necessarily engage multiple branches of government and a multiplicity of actors. Indeed, there is a complexity to foreign policy that is bewildering, in terms of both substance and process. On the other hand, foreign policymaking can sometimes call for nothing more than for the president to make a formal decision, quickly endorsed by the legislative branch. This section will explore the institutional relationships present in U.S. foreign policymaking.
Foreign Policy and Shared Power
While presidents are more empowered by the Constitution in foreign than in domestic policy, they nonetheless must seek approval from Congress on a variety of matters; chief among these is the basic budgetary authority needed to run foreign policy programs. Indeed, most if not all of the foreign policy instruments described earlier in this chapter require interbranch approval to go into effect. Such approval may sometimes be a formality, but it is still important. Even a sole executive agreement often requires subsequent funding from Congress in order to be carried out, and funding calls for majority support from the House and Senate. Presidents lead, to be sure, but they must consult with and engage the Congress on many matters of foreign policy. Presidents must also delegate a great deal in foreign policy to the bureaucratic experts in the foreign policy agencies. Not every operation can be run from the West Wing of the White House.
At bottom, the United States is a separation-of-powers political system with authority divided among executive and legislative branches, including in the foreign policy realm. Table 17.1 shows the formal roles of the president and Congress in conducting foreign policy.
Table 17.1 Roles of the President and Congress in Conducting Foreign Policy
Policy Output
Presidential Role
Congressional Role
Public laws
Proposes, signs into law
Proposes, approves for passage
Agency reauthorizations
Proposes, signs into law
Approves for passage
Foreign policy budget
Proposes, signs into law
Authorizes/appropriates for passage
Treaties
Negotiates, ratifies
Senate consents to treaty (two-thirds)
Sole executive agreements
Negotiates, approves
None (unless funding is required)
Congressional-executive agreements
Negotiates
Approves by majority vote
Declaration of war
Proposes
Approves by majority vote
Military use of force
Carries out operations at will (sixty days)
Approves for operations beyond sixty days
Presidential appointments
Nominates candidates
Senate approves by majority vote
Shared power creates an incentive for the branches to cooperate. Even in the midst of a crisis, such as the Cuban Missile Crisis in 1962, it is common for the president or senior staff to brief congressional leaders in order to keep them up to speed and ensure the country can stand unified on international matters. That said, there are areas of foreign policy where the president has more discretion, such as the operation of intelligence programs, the holding of foreign policy summits, and the mobilization of troops or agents in times of crisis. Moreover, presidents have more power and influence in foreign policymaking than they do in domestic policymaking. It is to that power that we now turn.
The Two Presidencies Thesis
When the media cover a domestic controversy, such as social unrest or police brutality, reporters consult officials at different levels and in branches of government, as well as think tanks and advocacy groups. In contrast, when an international event occurs, such as a terrorist bombing in Paris or Brussels, the media flock predominately to one actor—the president of the United States—to get the official U.S. position.
In the realm of foreign policy and international relations, the president occupies a leadership spot that is much clearer than in the realm of domestic policy. This dual domestic and international role has been described by the two presidencies thesis. This theory originated with University of California–Berkeley professor Aaron Wildavsky and suggests that there are two distinct presidencies, one for foreign policy and one for domestic policy, and that presidents are more successful in foreign than domestic policy. Let’s look at the reasoning behind this thesis.
The Constitution names the president as the commander-in-chief of the military, the nominating authority for executive officials and ambassadors, and the initial negotiator of foreign agreements and treaties. The president is the agenda-setter for foreign policy and may move unilaterally in some instances. Beyond the Constitution, presidents were also gradually given more authority to enter into international agreements without Senate consent by using the executive agreement. We saw above that the passage of the War Powers Resolution in 1973, though intended as a statute to rein in executive power and reassert Congress as a check on the president, effectively gave presidents two months to wage war however they wish. Given all these powers, we have good reason to expect presidents to have more influence and be more successful in foreign than in domestic policy.
A second reason for the stronger foreign policy presidency has to do with the informal aspects of power. In some eras, Congress will be more willing to allow the president to be a clear leader and speak for the country. For instance, the Cold War between the Eastern bloc countries (led by the Soviet Union) and the West (led by the United States and Western European allies) prompted many to want a single actor to speak for the United States. A willing Congress allowed the president to take the lead because of urgent circumstances (Figure 17.12). Much of the Cold War also took place when the parties in Congress included more moderates on both sides of the aisle and the environment was less partisan than today. A phrase often heard at that time was, “Partisanship stops at the water’s edge.” This means that foreign policy matters should not be subject to the bitter disagreements seen in party politics.
Does the thesis’s expectation of a more successful foreign policy presidency apply today? While the president still has stronger foreign policy powers than domestic powers, the governing context has changed in two key ways. First, the Cold War ended in 1989 with the demolition of the Berlin Wall, the subsequent disintegration of the Soviet Union, and the eventual opening up of Eastern European territories to independence and democracy. These dramatic changes removed the competitive superpower aspect of the Cold War, in which the United States and the USSR were dueling rivals on the world stage. The absence of the Cold War has led to less of a rally-behind-the-president effect in the area of foreign policy.
Second, beginning in the 1980s and escalating in the 1990s, the Democratic and Republican parties began to become polarized in Congress. The moderate members in each party all but disappeared, while more ideologically motivated candidates began to win election to the House and later the Senate. Hence, the Democrats in Congress became more liberal on average, the Republicans became more conservative, and the moderates from each party, who had been able to work together, were edged out. It became increasingly likely that the party opposite the president in Congress might be more willing to challenge his initiatives, whereas in the past it was rare for the opposition party to publicly stand against the president in foreign policy.
Finally, several analysts have tried applying the two presidencies thesis to contemporary presidential-congressional relationships in foreign policy. Is the two presidencies framework still valid in the more partisan post–Cold War era? The answer is mixed. On the one hand, presidents are more successful on foreign policy votes in the House and Senate, on average, than on domestic policy votes. However, the gap has narrowed. Moreover, analysis has also shown that presidents are opposed more often in Congress, even on the foreign policy votes they win.
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Democratic leaders regularly challenged Republican George W. Bush on the Iraq War and it became common to see the most senior foreign relations committee members of the Republican Party opposing the foreign policy positions of Democratic president Barack Obama. Such challenging of the president by the opposition party simply didn’t happen during the Cold War.
In the Trump administration, there was a distinct shift in foreign policy style. While for some regions, like South America, Trump was content to let the foreign policy bureaucracies proceed as they always had, in certain areas, the president was pivotal in changing the direction of American foreign policy. For example, he stepped away from two key international agreements—the Iran-Nuclear Deal and the Paris climate change accords. Moreover, his actions in Syria were quite unilateral, employing bombing raids unilaterally on two occasions. This approach reflected more of a neoconservative foreign policy approach, similar to Obama's widespread use of drone strikes.
Therefore, it seems presidents no longer enjoy unanimous foreign policy support as they did in the early 1960s. They have to work harder to get a consensus and are more likely to face opposition. Still, because of their formal powers in foreign policy, presidents are overall more successful on foreign policy than on domestic policy.
The Perspective of House and Senate Members
Congress is a bicameral legislative institution with 100 senators serving in the Senate and 435 representatives serving in the House. How interested in foreign policy are typical House and Senate members?
While key White House, executive, and legislative leaders monitor and regularly weigh in on foreign policy matters, the fact is that individual representatives and senators do so much less often. Unless there is a foreign policy crisis, legislators in Congress tend to focus on domestic matters, mainly because there is not much to be gained with their constituents by pursuing foreign policy matters.
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Domestic policy matters resonate more strongly with the voters at home. A sluggish economy, increasing health care costs, and crime matter more to them than U.S. policy toward North Korea, for example. In an open-ended Gallup poll question from early 2021 about the “most important problem” in the United States, fewer than 10 percent of respondents named a foreign policy topic (and most of those respondents mentioned immigration). These results suggest that foreign policy is not at the top of many voters’ minds. In the end, legislators must be responsive to constituents in order to be good representatives and to achieve reelection.
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However, some House and Senate members do wade into foreign policy matters. First, congressional party leaders in the majority and minority parties speak on behalf of their institution and their party on all types of issues, including foreign policy. Some House and Senate members ask to serve on the foreign policy committees, such as the Senate Committee on Foreign Relations, the House Foreign Affairs Committee, and the two defense committees (Figure 17.13). These members might have military bases within their districts or states and hence have a constituency reason for being interested in foreign policy. Legislators might also simply have a personal interest in foreign policy matters that drives their engagement in the issue. Finally, they may have ambitions to move into an executive branch position that deals with foreign policy matters, such as secretary of state or defense, CIA director, or even president.
Get Connected!
Let People Know What You Think!
Most House and Senate members do not engage in foreign policy because there is no electoral benefit to doing so. Thus, when citizens become involved, House members and senators will take notice. Research by John Kingdon on roll-call voting and by Richard Hall on committee participation found that when constituents are activated, their interest becomes salient to a legislator and the legislator will respond.
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One way you can become active in the foreign policy realm is by writing a letter or e-mail to your House member and/or your two U.S. senators about what you believe the U.S. foreign policy approach in a particular area ought to be. Perhaps you want the United States to work with other countries to protect dolphins from being accidentally trapped in tuna nets. You can also state your position in a letter to the editor of your local newspaper, or post an opinion on the newspaper’s website where a related article or op-ed piece appears. You can share links to news coverage on Facebook or Twitter and consider joining a foreign policy interest group such as Greenpeace.
When you engaged in foreign policy discussion as suggested above, what type of response did you receive?
A variety of actors carry out the various and complex activities of U.S. foreign policy: White House staff, executive branch staff, and congressional leaders.
The White House staff members engaged in foreign policy are likely to have very regular contact with the president about their work. The national security advisor heads the president’s National Security Council, a group of senior-level staff from multiple foreign policy agencies, and is generally the president’s top foreign policy advisor. Also reporting to the president in the White House is the director of the Central Intelligence Agency (CIA). Even more important on intelligence than the CIA director is the director of national intelligence, a position created in the government reorganizations after 9/11, who oversees the entire intelligence community in the U.S. government. The Joint Chiefs of Staff consist of six members, one each from the Army, Navy, Air Force, and Marines, plus a chair and vice chair. The chair of the Joint Chiefs of Staff is the president’s top uniformed military officer. In contrast, the secretary of defense is head of the entire Department of Defense but is a nonmilitary civilian. The U.S. trade representative develops and directs the country’s international trade agenda. Finally, within the Executive Office of the President, another important foreign policy official is the director of the president’s Office of Management and Budget (OMB). The OMB director develops the president’s yearly budget proposal, including funding for the foreign policy agencies and foreign aid.
In addition to those who work directly in the White House or Executive Office of the President, several important officials work in the broader executive branch and report to the president in the area of foreign policy. Chief among these is the secretary of state. The secretary of state is the nation’s chief diplomat, serves in the president’s cabinet, and oversees the Foreign Service. The secretary of defense, who is the civilian (nonmilitary) head of the armed services housed in the Department of Defense, is also a key cabinet member for foreign policy (as mentioned above). A third cabinet secretary, the secretary of homeland security, is critically important in foreign policy, overseeing the massive Department of Homeland Security (Figure 17.14).
Insider Perspective
Former Secretary of Defense Robert Gates
Former secretary of defense Robert Gates served under both Republican and Democratic presidents. First Gates rose through the ranks of the CIA to become the director during the George H. W. Bush administration. He then left government to serve as an academic administrator at Texas A&M University in College Station, Texas, where he rose to the position of university president. He was able to win over reluctant faculty and advance the university’s position, including increasing the faculty at a time when budgets were in decline in Texas. Then, when Secretary of Defense Donald Rumsfeld resigned, President George W. Bush invited Gates to return to government service as Rumsfeld’s replacement. Gates agreed, serving in that capacity for the remainder of the Bush years and then for several years in the Obama administration before retiring from government service a second time (Figure 17.15). He has generally been seen as thorough, systematic, and fair.
In his memoir,
Duty: Memoirs of a Secretary at War
,
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Secretary Gates takes issue with the actions of both the presidents for whom he worked, but ultimately he praises them for their service and for upholding the right principles in protecting the United States and U.S. military troops. In this and earlier books, Gates discusses the need to have an overarching plan but says plans cannot be too tight or they will fail when things change in the external environment. After leaving politics, Gates served as president of the Boy Scouts of America, where he presided over the change in policy that allowed openly gay scouts and leaders, an issue with which he had had experience as secretary of defense under President Obama. In that role Gates oversaw the end of the military’s “Don’t ask, don’t tell” policy.
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What do you think about a cabinet secretary serving presidents from two different political parties? Is this is a good idea? Why or why not?
The final group of official key actors in foreign policy are in the U.S. Congress. The Speaker of the House, the House minority leader, and the Senate majority and minority leaders are often given updates on foreign policy matters by the president or the president’s staff. They are also consulted when the president needs foreign policy support or funding. However, the experts in Congress who are most often called on for their views are the committee chairs and the highest-ranking minority members of the relevant House and Senate committees. In the House, that means the Foreign Affairs Committee and the Committee on Armed Services. In the Senate, the relevant committees are the Committee on Foreign Relations and the Armed Services Committee. These committees hold regular hearings on key foreign policy topics, consider budget authorizations, and debate the future of U.S. foreign policy.


11.4E: Exercises
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Practice Makes Perfect
Verify Solutions to an Inequality in Two Variables
In the following exercises, determine whether each ordered pair is a solution to the given inequality.
Exercise \(\PageIndex{1}\)
Determine whether each ordered pair is a solution to the inequality \(y>x−1\):
- \((0,1)\)
- \((−4,−1)\)
- \((4,2)\)
- \((3,0)\)
- \((−2,−3)\)
Exercise \(\PageIndex{2}\)
Determine whether each ordered pair is a solution to the inequality \(y>x−3\):
- \((0,0)\)
- \((2,1)\)
- \((−1,−5)\)
- \((−6,−3)\)
- \((1,0)\)
- Answer
-
- yes
- no
- no
- yes
- no
Exercise \(\PageIndex{3}\)
Determine whether each ordered pair is a solution to the inequality \(y<x+2\):
- \((0,3)\)
- \((−3,−2)\)
- \((−2,0)\)
- \((0,0)\)
- \((−1,4)\)
Exercise \(\PageIndex{4}\)
Determine whether each ordered pair is a solution to the inequality \(y<x+5\):
- \((−3,0)\)
- \((1,6)\)
- \((−6,−2)\)
- \((0,1)\)
- \((5,−4)\)
- Answer
-
- yes
- no
- no
- yes
- yes
Exercise \(\PageIndex{5}\)
Determine whether each ordered pair is a solution to the inequality \(x+y>4\):
- \((5,1)\)
- \((−2,6)\)
- \((3,2)\)
- \((10,−5)\)
- \((0,0)\)
Exercise \(\PageIndex{6}\)
Determine whether each ordered pair is a solution to the inequality \(x+y>2\):
- \((1,1)\)
- \((4,−3)\)
- \((0,0)\)
- \((−8,12)\)
- \((3,0)\)
- Answer
-
- no
- no
- no
- yes
- yes
Recognize the Relation Between the Solutions of an Inequality and its Graph
In the following exercises, write the inequality shown by the shaded region.
Exercise \(\PageIndex{7}\)
Write the inequality shown by the graph with the boundary line \(y=3x−4\).
Exercise \(\PageIndex{8}\)
Write the inequality shown by the graph with the boundary line \(y=2x−4\).
- Answer
-
\(y<2 x-4\)
Exercise \(\PageIndex{9}\)
Write the inequality shown by the graph with the boundary line \(y=\frac{1}{2} x+1\)
Exercise \(\PageIndex{10}\)
Write the inequality shown by the graph with the boundary line \(y=-\frac{1}{3} x-2\)
- Answer
-
\(y \leq-\frac{1}{3} x-2\)
Exercise \(\PageIndex{11}\)
Write the inequality shown by the shaded region in the graph with the boundary line \(x+y=5\).
Exercise \(\PageIndex{12}\)
Write the inequality shown by the shaded region in the graph with the boundary line \(x+y=3\).
- Answer
-
\(x+y \geq 3\)
Exercise \(\PageIndex{13}\)
Write the inequality shown by the shaded region in the graph with the boundary line \(2x+y=−4\).
Exercise \(\PageIndex{14}\)
Write the inequality shown by the shaded region in the graph with the boundary line \(x+2y=−2\).
- Answer
-
\(x+2 y \geq-2\)
Exercise \(\PageIndex{15}\)
Write the inequality shown by the shaded region in the graph with the boundary line \(3x−y=6\).
Exercise \(\PageIndex{16}\)
Write the inequality shown by the shaded region in the graph with the boundary line \(2x−y=4\).
- Answer
-
\(2 x-y<4\)
Exercise \(\PageIndex{17}\)
Write the inequality shown by the shaded region in the graph with the boundary line \(2x−5y=10\).
Exercise \(\PageIndex{18}\)
Write the inequality shown by the shaded region in the graph with the boundary line \(4x−3y=12\).
- Answer
-
\(4 x-3 y>12\)
Graph Linear Inequalities
In the following exercises, graph each linear inequality.
Exercise \(\PageIndex{19}\)
Graph the linear inequality \(y>\frac{2}{3} x-1\)
Exercise \(\PageIndex{20}\)
Graph the linear inequality \(y<\frac{3}{5} x+2\)
- Answer
-
Exercise \(\PageIndex{21}\)
Graph the linear inequality \(y \leq-\frac{1}{2} x+4\)
Exercise \(\PageIndex{22}\)
Graph the linear inequality \(y \geq-\frac{1}{3} x-2\)
- Answer
-
Exercise \(\PageIndex{23}\)
Graph the linear inequality \(x-y \leq 3\)
Exercise \(\PageIndex{24}\)
Graph the linear inequality \(x-y \geq -2\)
- Answer
-
Exercise \(\PageIndex{25}\)
Graph the linear inequality \(4x+y>-4\)
Exercise \(\PageIndex{26}\)
Graph the linear inequality \(x+5y<-5\)
- Answer
-
Exercise \(\PageIndex{27}\)
Graph the linear inequality \(3 x+2 y \geq-6\)
Exercise \(\PageIndex{28}\)
Graph the linear inequality \(4 x+2 y \geq-8\)
- Answer
-
Exercise \(\PageIndex{29}\)
Graph the linear inequality \(y>4x\)
Exercise \(\PageIndex{30}\)
Graph the linear inequality \(y>x\)
- Answer
-
Exercise \(\PageIndex{31}\)
Graph the linear inequality \(y \leq-x\)
Exercise \(\PageIndex{32}\)
Graph the linear inequality \(y \leq-3 x\)
- Answer
-
Exercise \(\PageIndex{33}\)
Graph the linear inequality \(y \geq-2\)
Exercise \(\PageIndex{34}\)
Graph the linear inequality \(y<-1\)
- Answer
-
Exercise \(\PageIndex{35}\)
Graph the linear inequality \(y<4\)
Exercise \(\PageIndex{36}\)
Graph the linear inequality \(y \geq 2\)
- Answer
-
Exercise \(\PageIndex{37}\)
Graph the linear inequality \(x \leq 5\)
Exercise \(\PageIndex{38}\)
Graph the linear inequality \(x>-2\)
- Answer
-
Exercise \(\PageIndex{39}\)
Graph the linear inequality \(x>-3\)
Exercise \(\PageIndex{40}\)
Graph the linear inequality \(x \leq 4\)
- Answer
-
Exercise \(\PageIndex{41}\)
Graph the linear inequality \(x-y<4\)
Exercise \(\PageIndex{42}\)
Graph the linear inequality \(x-y<-3\)
- Answer
-
Exercise \(\PageIndex{43}\)
Graph the linear inequality \(y \geq \frac{3}{2} x\)
Exercise \(\PageIndex{44}\)
Graph the linear inequality \(y \leq \frac{5}{4} x\)
- Answer
-
Exercise \(\PageIndex{45}\)
Graph the linear inequality \(y>-2 x+1\)
Exercise \(\PageIndex{46}\)
Graph the linear inequality \(y<-3 x-4\)
- Answer
-
Exercise \(\PageIndex{47}\)
Graph the linear inequality \(x \leq-1\)
Exercise \(\PageIndex{48}\)
Graph the linear inequality \(x \geq 0\)
- Answer
-
Everyday Math
Exercise \(\PageIndex{49}\)
Money. Gerry wants to have a maximum of $100 cash at the ticket booth when his church carnival opens. He will have $1 bills and $5 bills. If \(x\) is the number of $1 bills and \(y\) is the number of $5 bills, the inequality \(x+5y \leq 100\) models the situation.
- Graph the inequality.
- List three solutions to the inequality \(x+5y \leq 100\) where both \(x\) and \(y\) are integers.
Exercise \(\PageIndex{50}\)
Shopping. Tula has $20 to spend at the used book sale. Hardcover books cost $2 each and paperback books cost $0.50 each. If \(x\) is the number of hardcover books Tula can buy and \(y\) is the number of paperback books she can buy, the inequality \(2x+\frac{1}{2} y \leq 20\) models the situation.
- Graph the inequality.
- List three solutions to the inequality \(2x+\frac{1}{2} y \leq 20\) where both \(x\) and \(y\) are whole numbers.
- Answer
-
1.
2. Answers will vary.
Writing Exercises
Exercise \(\PageIndex{51}\)
Lester thinks that the solution of any inequality with a \(>\) sign is the region above the line and the solution of any inequality with a \(<\) sign is the region below the line. Is Lester correct? Explain why or why not.
Exercise \(\PageIndex{52}\)
Explain why in some graphs of linear inequalities the boundary line is solid but in other graphs it is dashed.
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. What does this checklist tell you about your mastery of this section? What steps will you take to improve?


9: African History to 1500
-
- 9.2: Introduction
- Africa is the second-largest continent in the world. Today, it has over 50 independent countries. You can also find just about every imaginable environment, from savannahs, rainforests, and deserts, to glaciers and snow-capped mountains in Africa. Its over 1,000 languages (or about one-third of the world’s languages) demonstrate the continent’s diversity.1 Africa is home to more than a billion people, who are living, working, and raising their families.
-
- 9.3: Aksum and Ethiopia
- From the 4th through 6th Centuries, Aksum was located in what are today Ethiopia, Eritrea, and parts of Sudan. At its high point, Aksum extended its influence beyond Africa into parts of the southern Arabian Peninsula. If you are familiar with accounts of the Queen of Sheba, you know pieces of the story that Ethiopians use to explain the origins of the Ethiopian Solomonic Dynasty and their possession of the Ark of the Covenant. The capital of the kingdom was the city of Aksum and its most import
-
- 9.4: The Western Sudanic States
- Straddling regions with different climates, the people of the Western Sudan developed productive agriculture, trade networks, and an urban culture. The architecture states stands out for its use of mud (adobe) to construct monumental buildings, such as the Great Mosque in Djenne (Figure 9.4.1, and Aksia the Great’s tomb in Gao (Figure 9.4.2). From roughly 800 to 1600 CE, the people of this region organized and supported—sometimes under duress—the large states that dominated the Western Sudan.
-
- 9.5: The Spread of Agriculture and Great Zimbabwe
- During European colonization 150 years ago, anthropologists grouped Africans into “tribes” based on presumed physical, cultural, and linguistic similarities. As a result of this study, they found that people in most of Sub-Saharan Africa spoke languages that used the root –ntu to refer to person, with the prefix ba- added in the plural. Combining the root and the plural prefix, 19th Century colonial anthropologists referred to people in these communities as Bantu and later traced Bantu languages
-
- 9.7: Conclusion
- Africa was not isolated. Instead, oceans and deserts were “highways” in these periods. Aksum, the Western Sudanic states, Great Zimbabwe, and the Swahili coast were all commercially linked to Europe, the Mediterranean, the Arabian Peninsula, the Persian Gulf, Southeast Asia, and even China. Although there were local differences, the ruling classes in each of the states collected tribute from outlying areas and participated in long-distance trade. The wealth of these states supported labor specia
Thumbnail: The Aksum Obelisk) in Aksum (Tigray Region, Ethiopia). Image used wtih permission (CC BY 2.5 Generic; Ondřej Žváček via Wikipedia).


6.4: Rotational Kinetic Energy
Learning Objectives
- Guess the equation for rotational kinetic energy by analogy.
- Calculate rotational kinetic energy.
In this module, we will learn about work and energy associated with rotational motion. Figure \(\PageIndex{1}\) shows a worker using an electric grindstone propelled by a motor. Sparks are flying, and noise and vibration are created as layers of steel are pared from the pole. The stone continues to turn even after the motor is turned off, but it is eventually brought to a stop by friction. Clearly, the motor had to work to get the stone spinning. This work went into heat, light, sound, vibration, and considerable rotational kinetic energy .
Instead of deriving the formula for rotational kinetic energy by calculating the work done by torque on a rotating object, we can simply guess the correct formula by analogy. We know the kinetic energy in linear, or translational motion, \(\mathrm{KE}=\frac{1}{2} m v^{2}\). We can find the rotational version of kinetic energy by replacing mass \(m\) with rotational version of mass, rotational inertia \(I\), and by replacing speed \(v\) with rotational speed \(\omega\). Rotational speed is measured in units of radians per second. With these replacements, we get the formula for rotational kinetic energy,
\[\mathrm{KE}_{\mathrm{rot}}=\frac{1}{2} I \omega^{2}. \nonumber \]
Rotational kinetic energy has important effects. Flywheels, for example, can be used to store large amounts of rotational kinetic energy in a vehicle, as seen in Figure \(\PageIndex{2}\).
Helicopter pilots are quite familiar with rotational kinetic energy. They know, for example, that a point of no return will be reached if they allow their blades to slow below a critical angular velocity during flight. The blades lose lift, and it is impossible to immediately get the blades spinning fast enough to regain it. Rotational kinetic energy must be supplied to the blades to get them to rotate faster, and enough energy cannot be supplied in time to avoid a crash. Because of weight limitations, helicopter engines are too small to supply both the energy needed for lift and to replenish the rotational kinetic energy of the blades once they have slowed down. The rotational kinetic energy is put into them before takeoff and must not be allowed to drop below this crucial level. One possible way to avoid a crash is to use the gravitational potential energy of the helicopter to replenish the rotational kinetic energy of the blades by losing altitude and aligning the blades so that the helicopter is spun up in the descent. Of course, if the helicopter’s altitude is too low, then there is insufficient time for the blade to regain lift before reaching the ground.
Example \(\PageIndex{1}\): Calculating Helicopter Energies
A typical small rescue helicopter, similar to the one in Figure \(\PageIndex{3}\), has four blades, each is 4.00 m long and has a mass of 50.0 kg. The blades can be approximated as thin rods that rotate about one end of an axis perpendicular to their length. The helicopter has a total loaded mass of 1000 kg. (a) Calculate the rotational kinetic energy in the blades when they rotate at 300 rpm. (b) Calculate the translational kinetic energy of the helicopter when it flies at 20.0 m/s, and compare it with the rotational energy in the blades. (c) To what height could the helicopter be raised if all of the rotational kinetic energy could be used to lift it?
Strategy
Rotational and translational kinetic energies can be calculated from their definitions. The last part of the problem relates to the idea that energy can change form, in this case from rotational kinetic energy to gravitational potential energy.
Solution for (a)
The rotational kinetic energy is
\[\mathrm{KE}_{\mathrm{rot}}=\frac{1}{2} I \omega^{2}. \nonumber\]
We must convert the angular velocity to radians per second and calculate the moment of inertia before we can find \(\mathrm{KE}_{\mathrm{rot}}\). The angular velocity \(\omega\) is
\[\omega=\frac{300 \mathrm{rev}}{1.00 \mathrm{~min}} \cdot \frac{2 \pi \mathrm{rad}}{1 \mathrm{rev}} \cdot \frac{1.00 \mathrm{~min}}{60.0 \mathrm{~s}}=31.4 \frac{\mathrm{rad}}{\mathrm{s}}. \nonumber\]
The moment of inertia of one blade will be that of a thin rod rotated about its end, \(I_{\mathrm{rod}}=\frac{1}{3} M \ell^{2}\). The total \(I\) is four times this moment of inertia, because there are four blades. Thus,
\[I=4 \frac{M \ell^{2}}{3}=4 \times \frac{(50.0 \mathrm{~kg})(4.00 \mathrm{~m})^{2}}{3}=1067 \mathrm{~kg} \cdot \mathrm{m}^{2}. \nonumber\]
Entering \(\omega\) and \(I\) into the expression for rotational kinetic energy gives
\[\begin{aligned}
\mathrm{KE}_{\mathrm{rot}} &=0.5\left(1067 \mathrm{~kg} \cdot \mathrm{m}^{2}\right)(31.4 \mathrm{rad} / \mathrm{s})^{2} \\
&=5.26 \times 10^{5} \mathrm{~J}
\end{aligned} \nonumber\]
Solution for (b)
Entering the given values of mass and velocity into formula for translational kinetic energy, we obtain
\[\mathrm{KE}_{\text {trans }}=\frac{1}{2} m v^{2}=(0.5)(1000 \mathrm{~kg})(20.0 \mathrm{~m} / \mathrm{s})^{2}=2.00 \times 10^{5} \mathrm{~J}. \nonumber\]
To compare kinetic energies, we take the ratio of translational kinetic energy to rotational kinetic energy. This ratio is
\[\frac{2.00 \times 10^{5} \mathrm{~J}}{5.26 \times 10^{5} \mathrm{~J}}=0.380. \nonumber\]
Solution for (c)
At the maximum height, all rotational kinetic energy will have been converted to gravitational energy. To find this height, we equate those two energies:
\[\mathrm{KE}_{\text {rot }}=\mathrm{PE}_{\text {grav }} \nonumber\]
or
\[\frac{1}{2} I \omega^{2}=m g h. \nonumber\]
We now solve for \(h\) and substitute known values into the resulting equation
\[h=\frac{\frac{1}{2} I \omega^{2}}{m g}=\frac{5.26 \times 10^{5} \mathrm{~J}}{(1000 \mathrm{~kg})\left(9.80 \mathrm{~m} / \mathrm{s}^{2}\right)}=53.7 \mathrm{~m}. \nonumber\]
Discussion
The ratio of translational energy to rotational kinetic energy is only 0.380. This ratio tells us that most of the kinetic energy of the helicopter is in its spinning blades—something you probably would not suspect. The 53.7 m height to which the helicopter could be raised with the rotational kinetic energy is also impressive, again emphasizing the amount of rotational kinetic energy in the blades.
How Thick Is the Soup? Or Why Don’t All Objects Roll Downhill at the Same Rate?
One of the quality controls in a tomato soup factory consists of rolling filled cans down a ramp. If they roll too fast, the soup is too thin. Why should cans of identical size and mass roll down an incline at different rates? And why should the thickest soup roll the slowest?
The easiest way to answer these questions is to consider energy. Suppose each can starts down the ramp from rest. Each can starting from rest means each starts with the same gravitational potential energy \(\mathrm{PE}_{\text {grav }}\), which is converted entirely to KE, provided each rolls without slipping. KE, however, can take the form of \(\mathrm{KE}_{\text {trans }}\) or \(\mathrm{KE}_{\text {rot }}\), and total KE is the sum of the two. If a can rolls down a ramp, it puts part of its energy into rotation, leaving less for translation. Thus, the can goes slower than it would if it slid down. Furthermore, the thin soup does not rotate, whereas the thick soup does, because it sticks to the can. The thick soup thus puts more of the can’s original gravitational potential energy into rotation than the thin soup, and the can rolls more slowly, as seen in Figure \(\PageIndex{4}\).
Assuming no losses due to friction, there is only one force doing work—gravity. Therefore the total work done is the change in kinetic energy. As the cans start moving, the potential energy is changing into kinetic energy. Conservation of energy gives
\[\mathrm{PE}_{\mathrm{i}}=\mathrm{KE}_{\mathrm{f}}. \nonumber \]
More specifically,
\[\mathrm{PE}_{\text {grav }}=\mathrm{KE}_{\text {trans }}+\mathrm{KE}_{\mathrm{rot}} \nonumber \]
or
\[m g h=\frac{1}{2} m v^{2}+\frac{1}{2} I \omega^{2}. \nonumber \]
So, the initial \(mgh\) is divided between translational kinetic energy and rotational kinetic energy; and the greater \(I\) is, the less energy goes into translation. If the can slides down without friction, then \(ω=0\) and all the energy goes into translation; thus, the can goes faster.
TAKE-HOME EXPERIMENT
Locate several cans each containing a different type of food (for example, canned soup with different kinds of soup). First, predict which can will win the race down an inclined plane and explain why. See if your prediction is correct. A good contrast can be obtained between a thick soup concentrate (for example, condensed cream of mushroom) and a thin soup (for example, chicken noodle soup). If you feel adventurous, try other foods, such as canned corn or peas; try predicting how these would compare to, for example, condensed cream of mushroom can in the race down an inclined plane.
Example \(\PageIndex{2}\): Calculating the Speed of a Cylinder Rolling Down an Incline
Calculate the final speed of a solid cylinder that rolls down a 2.00-m-high incline. The cylinder starts from rest, has a mass of 0.750 kg, and has a radius of 4.00 cm.
Strategy
We can solve for the final velocity using conservation of energy, but we must first express rotational quantities in terms of translational quantities to end up with v as the only unknown.
Solution
Conservation of energy for this situation is written as described above:
\[m g h=\frac{1}{2} m v^{2}+\frac{1}{2} I \omega^{2}. \nonumber\]
Before we can solve for \(v\) , we must look up an expression for \(I\) for a solid disk: \(I_{\text {disk }}=\frac{1}{2} M \ell^{2}\). Because \(v\) and \(\omega\) are related (note here that the cylinder is rolling without slipping), we must also substitute the relationship \(\omega=v / R\) into the expression. These substitutions yield
\[m g h=\frac{1}{2} m v^{2}+\frac{1}{2}\left(\frac{1}{2} m R^{2}\right)\left(\frac{v^{2}}{R^{2}}\right). \nonumber\]
Interestingly, the cylinder’s radius \(R\) and mass \(m\) cancel, yielding
\[g h=\frac{1}{2} v^{2}+\frac{1}{4} v^{2}=\frac{3}{4} v^{2}. \nonumber\]
Solving algebraically, the equation for the final velocity \(v\) gives
\[v=\left(\frac{4 g h}{3}\right)^{1 / 2}. \nonumber\]
Substituting known values into the resulting expression yields
\[v=\left[\frac{4\left(9.80 \mathrm{~m} / \mathrm{s}^{2}\right)(2.00 \mathrm{~m})}{3}\right]^{1 / 2}=5.11 \mathrm{~m} / \mathrm{s}. \nonumber\]
Discussion
Because \(m\) and \(R\) cancel, the result \(v=\left(\frac{4}{3} g h\right)^{1 / 2}\) is valid for any solid cylinder, implying that all solid cylinders will roll down an incline at the same rate independent of their masses and sizes. (Rolling cylinders down inclines is what Galileo actually did to show that objects fall at the same rate independent of mass.) Note that if the cylinder slid without friction down the incline without rolling, then the entire gravitational potential energy would go into translational kinetic energy. Thus, \(\frac{1}{2} m v^{2}=m g h\) and \(v=(2 g h)^{1 / 2}\), which is 22% greater than \((4 g h / 3)^{1 / 2}\). That is, the cylinder would go faster at the bottom.
Exercise \(\PageIndex{1}\)
Analogy of Rotational and Translational Kinetic Energy
Is rotational kinetic energy completely analogous to translational kinetic energy? What, if any, are their differences? Give an example of each type of kinetic energy.
- Answer
-
Yes, rotational and translational kinetic energy are exact analogs. They both are the energy of motion involved with the coordinated (non-random) movement of mass relative to some reference frame. The only difference between rotational and translational kinetic energy is that translational is straight line motion while rotational is not. An example of both kinetic and translational kinetic energy is found in a bike tire while being ridden down a bike path. The rotational motion of the tire means it has rotational kinetic energy while the movement of the bike along the path means the tire also has translational kinetic energy. If you were to lift the front wheel of the bike and spin it while the bike is stationary, then the wheel would have only rotational kinetic energy relative to the Earth.
Section Summary
-
The rotational kinetic energy \(\mathrm{KE}_{\text {rot }}\) for an object with a moment of inertia \(I\) and an angular velocity \(\omega\) is given by
\[\mathrm{KE}_{\mathrm{rot}}=\frac{1}{2} I \omega^{2}. \nonumber\]
- Helicopters store large amounts of rotational kinetic energy in their blades. This energy must be put into the blades before takeoff and maintained until the end of the flight. The engines do not have enough power to simultaneously provide lift and put significant rotational energy into the blades.
- Work and energy in rotational motion are completely analogous to work and energy in translational motion.
Glossary
- rotational kinetic energy
- the kinetic energy due to the rotation of an object. This is part of its total kinetic energy


1.0: Introduction
Think of a drawer in your kitchen used to store flatware. This drawer likely holds forks, spoons, and knives, and possibly other items such as a meat thermometer and a can opener. The drawer in this case represents a tool used to group a collection of objects. The members of the group are the individual items in the drawer, such as a fork or a spoon.
The members of a set can be anything, such as people, numbers, or letters of the alphabet. In statistical studies, a set is a well-defined collection of objects used to identify an entire population of interest. For example, in a research study examining the effects of a new medication, there can be two sets of people: one set that is given the medication and a different set that is given a placebo (control group). In this chapter, we will discuss sets and Venn diagrams, which are graphical ways to show relationships between different groups.