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 L as.
As we would expect, an object that has a large moment of inertia I , such as Earth, has a very large angular momentum. First, according to Figure 1, the formula for the moment of inertia of a sphere is. 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.
Figure 2 shows a Lazy Susan food tray being rotated by a person in quest of sustenance. Suppose the person exerts a 2. Figure 2. A partygoer exerts a torque on a lazy Susan to make it rotate. The final angular momentum equals the change in angular momentum, because the lazy Susan starts from rest. 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. The person whose leg is shown in Figure 3 kicks his leg by exerting a N force with his upper leg muscle.
The effective perpendicular lever arm is 2. Given the moment of inertia of the lower leg is 1. Figure 3. The muscle in the upper leg gives the lower leg an angular acceleration and imparts rotational kinetic energy to it by exerting a torque about the knee. F is a vector that is perpendicular to r. This example examines the situation. The moment of inertia I is given and the torque can be found easily from the given force and perpendicular lever arm. Because the force and the perpendicular lever arm are given and the leg is vertical so that its weight does not create a torque, the net torque is thus.
The kinetic energy is then. These values are reasonable for a person kicking his leg starting from the position shown. 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 b , the 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 velocity by transferring some of this energy in a kick. We can now understand why Earth keeps on spinning. 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 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. 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 in Figure 4, 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. Key Takeaways Key Points When an object is spinning in a closed system and no external torques are applied to it, it will have no change in angular momentum.
The conservation of angular momentum explains the angular acceleration of an ice skater as she brings her arms and legs close to the vertical axis of rotation.
If the net torque is zero, then angular momentum is constant or conserved. Key Terms quantum mechanics : The branch of physics that studies matter and energy at the level of atoms and other elementary particles; it substitutes probabilistic mechanisms for classical Newtonian ones.
Rotational Collisions In a closed system, angular momentum is conserved in a similar fashion as linear momentum. Learning Objectives Evaluate the difference in equation variables in rotational versus angular momentum. Which is the moment of inertia times the angular velocity, or the radius of the object crossed with the linear momentum. In a closed system, angular momentum is conserved in all directions after a collision.
Since momentum is conserved, part of the momentum in a collision may become angular momentum as an object starts to spin after a collision.
Key Terms momentum : of a body in motion the product of its mass and velocity. Licenses and Attributions. CC licensed content, Shared previously. Another popular example of the conservation of angular momentum is that of a person holding a spinning bicycle wheel on a rotating chair.
The person then turns over the bicycle wheel, causing it to rotate in an opposite direction, as shown below. In b , the direction of spin is reversed, causing the person to spin on the chair to conserve angular momentum. Initially, the wheel has an angular momentum in the upward direction. When the person turns over the wheel, the angular momentum of the wheel reverses direction. Because the person-wheel-chair system is an isolated system, total angular momentum must be conserved, and the person begins to rotate in an opposite direction as the wheel.
The vector sum of angular momentum in a and b is the same, and momentum is conserved. This example is quite counterintuitive. It seems odd that simply moving a bicycle wheel would cause one to rotate.
The answer is that her angular momentum is constant, so that. It is interesting to see how the rotational kinetic energy of the skater changes when she pulls her arms in. Her initial rotational energy is. The source of this additional rotational kinetic energy is the work required to pull her arms inward.
This work causes an increase in the rotational kinetic energy, while her angular momentum remains constant. Since she is in a frictionless environment, no energy escapes the system. Thus, if she were to extend her arms to their original positions, she would rotate at her original angular velocity and her kinetic energy would return to its original value. The solar system is another example of how conservation of angular momentum works in our universe.
Our solar system was born from a huge cloud of gas and dust that initially had rotational energy. Gravitational forces caused the cloud to contract, and the rotation rate increased as a result of conservation of angular momentum Figure We continue our discussion with an example that has applications to engineering.
Before contact, only one flywheel is rotating. Therefore, the ratio of the final kinetic energy to the initial kinetic energy is. A merry-go-round at a playground is rotating at 4.
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