Problem 11 - University Physics with Modern Physics

Question

A ball attached to the end of a string is swung in a vertical circle. The angular momentum of the ball at the top of the circular path is a) greater than the angular momentum at the bottom of the circular path. b) less than the angular momentum at the bottom of the circular path. c) the same as the angular momentum at the bottom of the circular path.

Step-by-Step Solution

Answer

Question: Compare the angular momentum of a ball attached to a string at the top and bottom of its vertical circular path. Answer: b) less than the angular momentum at the bottom of the circular path.

Step 1: Understand the concept of Angular Momentum

Angular Momentum (L) is defined as the product of an object's moment of inertia (I) and its angular velocity (ω) (L = Iω). In this case, we can relate the linear momentum of the ball (p) to the angular momentum by multiplying the momentum by the radius of the circular path (r). Angular momentum will have the same magnitude if the ball's speed (v) and path radius (r) remain constant throughout the circle.

Step 2: Observe energy conservation

To understand the relationship between the ball's speed at the top and bottom of the circular path, we consider the conservation of mechanical energy. If we assume no energy losses due to friction or air resistance, the total mechanical energy (E) remains constant during the motion, consisting of the sum of potential energy (U) and kinetic energy (K). At the top of the circular path, the potential energy is at its maximum value (U = mgh) while kinetic energy is minimum (K = 1/2 mv^2). At the bottom of the circular path, potential energy is minimized (U = 0) and kinetic energy is at its maximum value.

Step 3: Determine the relationship between the ball's speed at the top and bottom

As we have just observed, due to the conservation of mechanical energy, the ball's speed will be greatest when its kinetic energy is maximized (at the bottom of the circular path). On the other hand, when the ball is at the top of the circular path and its potential energy is maximized, its speed will be minimum. Therefore, we can conclude that the speed of the ball at the bottom of the circular path is greater than its speed at the top.

Step 4: Compare the angular momentum at the top and bottom of the circular path

Since the angular momentum is the product of the linear momentum of the ball and the perpendicular distance from the axis, as we established in step 1, we can now compare the angular momentum at the top and bottom of the circular path. The speed of the ball is greater at the bottom of the circular path, as determined in step 3, so its linear momentum will also be greater. Since the radius remains constant during the entire path, the angular momentum will be greater at the bottom of the circular path as well. Therefore, the correct answer is: b) less than the angular momentum at the bottom of the circular path.