Angular Momentum and Its Conservation Quiz

PDF page 460: Why does Earth keep on spinning? What started it spinning to begin with? And how does an ice skater manage to spin faster and faster simply by pulling her arms in? Why does she not have to exert a torque to spin faster? Questions like these have answers based in angular momentum, the rotational analog to linear momentum. By now the pattern is clear—every rotational phenomenon has a direct translational analog. It seems quite reasonable, then, to define angular momentum as 10.90 This equation is an analog to the definition of linear momentum as . Units for linear momentum are while units for angular momentum are . As we would expect, an object that has a large moment of inertia , such as Earth, has a very large angular momentum. An object that has a large angular velocity , such as a centrifuge, also has a rather large angular momentum.

PDF page 460: Why does Earth keep on spinning? What started it spinning to begin with? And how does an ice skater manage to spin faster and faster simply by pulling her arms in? Why does she not have to exert a torque to spin faster? Questions like these have answers based in angular momentum, the rotational analog to linear momentum. By now the pattern is clear—every rotational phenomenon has a direct translational analog. It seems quite reasonable, then, to define angular momentum as 10.90 This equation is an analog to the definition of linear momentum as . Units for linear momentum are while units for angular momentum are . As we would expect, an object that has a large moment of inertia , such as Earth, has a very large angular momentum. An object that has a large angular velocity , such as a centrifuge, also has a rather large angular momentum.

PDF page 460: Angular momentum is completely analogous to linear momentum, first presented in Uniform Circular Motion and Gravitation. It has the same implications in terms of carrying rotation forward, and it is conserved when the net external torque is zero. Angular momentum, like linear momentum, is also a property of the atoms and subatomic particles.

PDF page 463: Angular momentum, like energy and linear momentum, is conserved. This universally applicable law is another sign of underlying unity in physical laws. Angular momentum is conserved when net external torque is zero, just as linear momentum is conserved when the net external force is zero.

PDF page 461: This expression is exactly analogous to the relationship between force and linear momentum, . The equation is very fundamental and broadly applicable. It is, in fact, the rotational form of Newton’s second law.

PDF page 464: If the change in angular momentum is zero, then the angular momentum is constant; thus, 10.110 or 10.111 These expressions are the law of conservation of angular momentum. Conservation laws are as scarce as they are important. An example of conservation of angular momentum is seen in Figure 10.21, in which an ice skater is executing a spin. The net torque on her is very close to zero, because there is relatively little friction between her skates and the ice and because the friction is exerted very close to the pivot point. (Both and are small, and so is negligibly small.) Consequently, she can spin for quite some time. She can do something else, too. She can increase her rate of spin by pulling her arms and legs in. Why does pulling her arms and legs in increase her rate of spin? The answer is that her angular momentum is constant, so that 10.112 Expressing this equation in terms of the moment of inertia, 10.113 where the primed quantities refer to conditions after she has pulled in her arms and reduced her moment of inertia.

PDF page 464: If the change in angular momentum is zero, then the angular momentum is constant; thus, 10.110 or 10.111 These expressions are the law of conservation of angular momentum. Conservation laws are as scarce as they are important. An example of conservation of angular momentum is seen in Figure 10.21, in which an ice skater is executing a spin. The net torque on her is very close to zero, because there is relatively little friction between her skates and the ice and because the friction is exerted very close to the pivot point. (Both and are small, and so is negligibly small.) Consequently, she can spin for quite some time. She can do something else, too. She can increase her rate of spin by pulling her arms and legs in. Why does pulling her arms and legs in increase her rate of spin? The answer is that her angular momentum is constant, so that 10.112 Expressing this equation in terms of the moment of inertia, 10.113 where the primed quantities refer to conditions after she has pulled in her arms and reduced her moment of inertia.

PDF page 464: Because is smaller, the angular velocity must increase to keep the angular momentum constant. The change

PDF page 464: the net torque on her is negligibly small. In the next image, her rate of spin increases greatly when she pulls in her arms, decreasing her moment of inertia. The work she does to pull in her arms results in an increase in rotational kinetic energy.

PDF page 464: Because is smaller, the angular velocity must increase to keep the angular momentum constant. The change

PDF page 463: We can now understand why Earth keeps on spinning. As we saw in the previous example, . This equation means that, to change angular momentum, a torque must act over some period of time. Because Earth has a large angular momentum, a large torque acting over a long time is needed to change its rate of spin. So what external torques are there? Tidal friction exerts torque that is slowing Earth’s rotation, but tens of millions of years must pass before the change is very significant. Recent research indicates the length of the day was 18 h some 900 million years ago. Only the tides exert significant retarding torques on Earth, and so it will continue to spin, although ever more slowly, for many billions of years.

PDF page 463: We can now understand why Earth keeps on spinning. As we saw in the previous example, . This equation means that, to change angular momentum, a torque must act over some period of time. Because Earth has a large angular momentum, a large torque acting over a long time is needed to change its rate of spin. So what external torques are there? Tidal friction exerts torque that is slowing Earth’s rotation, but tens of millions of years must pass before the change is very significant. Recent research indicates the length of the day was 18 h some 900 million years ago. Only the tides exert significant retarding torques on Earth, and so it will continue to spin, although ever more slowly, for many billions of years.

PDF page 463: Angular momentum, like energy and linear momentum, is conserved. This universally applicable law is another sign of underlying unity in physical laws. Angular momentum is conserved when net external torque is zero, just as linear momentum is conserved when the net external force is zero.

PDF page 465: There are several other examples of objects that increase their rate of spin because something reduced their moment of inertia. Tornadoes are one example. Storm systems that create tornadoes are slowly rotating. When the radius of rotation narrows, even in a local region, angular velocity increases, sometimes to the furious level of a tornado. Earth is another example. Our planet was born from a huge cloud of gas and dust, the rotation of which came from turbulence in an even larger cloud. Gravitational forces caused the cloud to contract, and the rotation rate increased as a result. (See Figure 10.22.)

PDF page 465: There are several other examples of objects that increase their rate of spin because something reduced their moment of inertia. Tornadoes are one example. Storm systems that create tornadoes are slowly rotating. When the radius of rotation narrows, even in a local region, angular velocity increases, sometimes to the furious level of a tornado. Earth is another example. Our planet was born from a huge cloud of gas and dust, the rotation of which came from turbulence in an even larger cloud. Gravitational forces caused the cloud to contract, and the rotation rate increased as a result. (See Figure 10.22.)