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Chapter 50: Problem 1

A ball tends to roll down a hill. a) Which is the lower energy state: the ball at the top of the hill or the ball at the bottom of the hill? b) Is the change in the potential energy for this process, \(V_{\mathrm{f}}-V_{\mathrm{i}}\), positive, negative, or zero? c) Why doesn't the ball roll up the hill?

Short Answer

Expert verified
a) The ball at the bottom of the hill is at a lower energy state. b) The change in potential energy is negative. c) The ball doesn't roll up the hill because of the force of gravity pulling it down and the nature of the ball to be in the state of lower potential energy.

Step by step solution

01

Define Lower Energy State

The potential energy of an object is given by the equation \(PE=mgh\), where \(m\) is the mass of the object, \(g\) is the acceleration due to gravity and \(h\) is the height of the object from a reference position, typically the ground level. The lower energy state is achieved when the potential energy is least. Since the height of the ball is least at the bottom of the hill, the potential energy of the ball at the bottom is smaller than at the top of the hill. Thus, the lower energy state is when the ball is at the bottom of the hill.
02

Determine the Sign of the Energy Change

The change in potential energy can be calculated as \(V_{\mathrm{f}}-V_{\mathrm{i}}\), where \(V_{\mathrm{f}}\) is the final potential energy and \(V_{\mathrm{i}}\) is the initial potential energy. As the ball rolls down from the top (initial position) to the bottom (final position) of the hill, the height and thus potential energy decreases. Hence, \(V_{\mathrm{f}}-V_{\mathrm{i}}\) is negative.
03

Understand Gravity's Role

The ball doesn't roll up the hill because of the force of gravity. Gravity is pulling the ball down, which opposes the ball's ability to roll up the hill without an external force or energy input. Plus, the bottom of the hill is a lower energy state compared to the top, so the ball naturally wants to move to the state with the least potential energy, which is down the hill.

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