The figure below shows a ball with mass m = 0.299 kg attached to the end of a thin rod with length L = 0.388 m and negligible mass. The other end of the rod is pivoted so that the ball can move in a vertical circle. The rod is held horizontally as shown and then given enough of a downward push to cause the ball to swing down and around and just reach the vertically up position, with zero speed there.
How much work is done on the ball by the gravitational force from the initial point to the following points?
(a) the lowest point
J
(b) the highest point
J
(c) the point on the right level with the initial point
J
If the gravitational potential energy of the ball-Earth system is taken to be zero at the initial point, what is it when the ball reaches the following points?
(d) the lowest point
J
(e) the highest point
J
(f) the point on the right level with the initial point
J
To determine the work done on the ball by the gravitational force, we need to consider the change in gravitational potential energy as the ball moves between different points.
(a) The lowest point: At the lowest point, the ball is at its maximum height. The gravitational potential energy at this point is zero because it is taken to be zero at the initial point. Since the ball is moving downwards, the work done by gravity is negative. Therefore, the work done on the ball by the gravitational force at the lowest point is 0 J.
(b) The highest point: At the highest point, the ball is at its minimum height. The gravitational potential energy at this point is equal to the gravitational potential energy at the initial point because the ball-Earth system is taken to have zero potential energy at the initial point. Again, since the ball is moving upwards, the work done by gravity is negative. Therefore, the work done on the ball by the gravitational force at the highest point is 0 J.
(c) The point on the right level with the initial point: At this point, the ball is at the same height as the initial point. The gravitational potential energy at this point is equal to the gravitational potential energy at the initial point. Since the ball is not moving vertically, the work done by gravity is zero. Therefore, the work done on the ball by the gravitational force at this point is 0 J.
(d) The lowest point: The gravitational potential energy at the lowest point is zero because it was taken to be zero at the initial point.
(e) The highest point: The gravitational potential energy at the highest point is equal to the gravitational potential energy at the initial point because it was taken to have zero potential energy at the initial point.
(f) The point on the right level with the initial point: The gravitational potential energy at this point is equal to the gravitational potential energy at the initial point because it was taken to have zero potential energy at the initial point.
Therefore, the gravitational potential energy remains the same at all points - 0 J.