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.

Short answer

Question: At the top of the circular path, the angular momentum of a ball attached to a string 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) equal to the angular momentum at the bottom of the circular path Answer: b) less than the angular momentum at the bottom of the circular path.

Step-by-step solution

Understand the formula for angular momentum

Angular momentum (L) is given by the formula: L = mvr, where m is the mass of the ball, v is the linear velocity, and r is the distance from the center of rotation to the ball.

Analyze the velocity at the top and bottom of the path

At the top of the circular path, the ball has to overcome gravitational force to move upward, thereby reducing its velocity. At the bottom of the circular path, the ball gains maximum velocity because the gravitational force acts in the same direction as the motion of the ball.

Compare the angular momentum at the top and bottom of the circular path

As we have established that the velocity of the ball is greater at the bottom than at the top, we can now compare the angular momentum. Since L = mvr, and the mass (m) and radius (r) remain constant in both positions, the higher velocity at the bottom of the circular path will result in greater angular momentum. Therefore, the correct answer is: b) less than the angular momentum at the bottom of the circular path.

Key concepts

Circular Motion

Circular motion refers to the movement of an object along the circumference of a circle or rotation along a circular path. It is a key concept in physics, particularly when analyzing the motions of celestial bodies, particles in accelerators, or even everyday objects like a ball on a string.

In the context of our exercise, the ball attached to a string swung in a vertical circle is a classic example of vertical circular motion. The ball experiences a centripetal force that keeps it moving in a circle, which in this case is provided by the tension of the string along with the gravitational force acting towards the center of the Earth.

Circular motion can be uniform, with a constant angular speed, or non-uniform, where the angular speed varies. The ball on the string exhibits non-uniform motion since gravity affects its velocity as it travels around the path. At the highest and lowest points in the circle, the ball's velocity changes due to gravity's influence, impacting several physics properties, including angular momentum.

Conservation of Angular Momentum

The principle of conservation of angular momentum is an important phenomenon in physics, asserting that if no external torque acts on a system, the total angular momentum of the system remains constant. This principle is a rotational analogue to the conservation of linear momentum and illustrates that the angular momentum is conserved in an isolated system.

In our exercise scenario, the ball and string system might seem isolated, but we must consider the gravitational force acting on it, especially since the ball is moving in a vertical circle. The crucial point is that while gravity does change the speed at which the ball travels, it acts directly towards or away from the center and therefore doesn't exert an external torque. Therefore, if the mass and radius remain constant and no outside forces interfere, the angular momentum would be conserved. However, due to the changing velocity—slower at the top and faster at the bottom—the ball’s angular momentum does change. It's important to note that the system’s angular momentum would only be conserved if the ball were to move in a perfectly horizontal circle or if no external forces, including gravitational forces, affected it.

Gravitational Force

Gravitational force is a natural phenomenon by which all things with mass or energy—including planets, stars, galaxies, and even light—are brought towards one another. On Earth, this force gives weight to physical objects and causes the ocean tides, among other effects.

In the case of the ball being swung in a circle on a string, gravitational force acts upon it throughout its motion. At the top of the circle, the ball moves against gravity, reducing its velocity because it is performing work against the Earth's gravitational pull, which requires energy. At the bottom of the circle, gravity accelerates the ball, increasing its velocity as the gravitational force acts in the same direction as the ball's motion.

Understanding gravitational force is essential to grasp not only the ball's velocity changes throughout its path but also how these velocity variations influence the ball's angular momentum at different points in the circular trajectory.