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 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 inertiasuch as Earth, has a very large angular momentum. An object that has a large angular velocitysuch as a centrifuge, also has a rather large angular momentum. Making Connections Angular momentum is completely analogous to linear momentum, first presented in. 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. Discussion This number is large, demonstrating that Earth, as expected, has a tremendous angular momentum. The answer is approximate, because we have assumed a constant density for Earth in order to estimate its moment of inertia. When you push a merry-go-round, spin a bike wheel, or open a door, you exert a torque. If the torque you exert is greater than opposing torques, then the rotation accelerates, and angular momentum increases. The greater the net torque, the more rapid the increase in. The relationship between torque and angular momentum is Calculating the Torque Putting Angular Momentum Into a Lazy Susan shows a Lazy Susan food tray being rotated by a person in quest of sustenance. Suppose the person exerts a 2. A partygoer exerts a torque on a lazy Susan to make it rotate. The equation gives the relationship between torque and the angular momentum produced. Strategy We can find the angular momentum by solving forand using the given information to calculate the torque. The final angular momentum equals the change in angular momentum, because the lazy Susan starts from rest. To find the final velocity, we must calculate from the definition of dimensions of angular velocity. Solution for a Solving for gives Discussion Note that the imparted angular momentum does not depend on any property of the object but only on torque and time. The final angular velocity is equivalent to one revolution in 8. Calculating the Torque in a Kick The person whose leg is shown in kicks his leg by exerting a 2000-N force with his upper leg muscle. dimensions of angular velocity The effective perpendicular lever arm is 2. Given the moment of inertia of the lower leg isa find the angular acceleration of the leg. The muscle in the dimensions of angular velocity leg gives dimensions of angular velocity lower leg an angular acceleration and imparts rotational kinetic energy to it by exerting a torque about the knee. This example examines the situation. The moment of inertia is given and the torque can be found easily from the given force and perpendicular lever arm. Once dimensions of angular velocity angular acceleration is known, the final angular velocity and rotational kinetic energy can be calculated. The weight of the leg can be neglected in part a because it exerts no torque when the center of gravity of the lower leg is directly beneath the pivot in the knee. In part bthe force exerted by the upper leg is so large that its torque is much greater than that created by the weight of the lower leg as it rotates. The rotational kinetic energy given to the lower leg is enough that it could give a ball a significant dimensions of angular velocity by transferring some of this energy in a kick. Making Connections: Conservation Laws 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. Conservation of Angular Momentum 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. 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. What we have here is, in fact, another conservation law. If the net torque is zero, then angular momentum is constant or conserved. We can see this rigorously by considering for the situation in which the net torque is zero. In that case, 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 inin 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 a An ice skater is spinning on the tip of her skate with her arms extended. Her angular momentum is conserved because 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. Calculating the Angular Momentum of a Spinning Skater Suppose an ice skater, such as the one inis spinning at 0. She has a moment of inertia of with her arms extended and of with her arms close to her body. These moments of inertia are based on reasonable assumptions about a 60. To find this quantity, we use the conservation of angular momentum and note that the moments of inertia and initial angular velocity are given. To find the initial and final kinetic energies, we use the dimensions of angular velocity of rotational kinetic energy given by Discussion In both parts, there is an impressive increase. First, the final angular velocity is large, although most world-class skaters can achieve spin rates about this great. Second, the final kinetic energy is much greater than the initial kinetic energy. The increase in rotational kinetic energy comes from work done by the skater in pulling in her arms. There are several other examples of objects that increase their rate of spin because something reduced their moment of inertia. 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. 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. The Solar System coalesced from a cloud of gas and dust that was originally rotating. The orbital motions and spins of the planets are in the same direction as the original spin and conserve the angular momentum of the parent cloud. In case of human motion, one would not expect angular momentum to be conserved when a body interacts with the environment as its foot pushes off the ground. Astronauts floating in space aboard the International Space Station have no angular momentum relative to the inside of the ship if they are motionless. Their bodies will continue to have this zero value no matter how they twist about as long as they do not give themselves a push off the side of the vessel. 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. Suppose a child gets off a rotating merry-go-round. Does the angular velocity of the merry-go-round increase, decrease, or remain the same if: a He jumps off radially. Remember that the Moon keeps one side toward Earth at all times. Find the value of his moment of inertia dimensions of angular velocity his angular velocity decreases to 1. What average torque was exerted if this takes 15. Construct a problem in which you calculate the total angular momentum of the system including the spins of the Earth and the Moon on their axes and the orbital angular momentum of the Earth-Moon system in its nearly monthly rotation.
