In Fig. 8-29, a single frictionless roller-coaster car of mass m=825 kg tops the first hill wit

First, we need to find the heights of points A, B, and C relative to the initial height h. We ca

In Fig. 8-29, a single frictionless roller-coaster car of...

In Fig. 8-29, a single frictionless roller-coaster car of mass $m=825 \mathrm{~kg}$ tops the first hill with speed $v_{0}=17.0 \mathrm{~m} / \mathrm{s}$ at height $h=42.0 \mathrm{~m}$. How much work does the gravitational force do on the car from that point to (a) point $A,(\mathrm{~b})$ point $B$, and $(\mathrm{c})$ point $C ?$ If the gravitational potential energy of the car-Earth system is taken to be zero at $C$, what is its value when the car is at $(\mathrm{d}) B$ and $(\mathrm{e}) A ?(\mathrm{f})$ If mass $m$ were doubled, would the change in the gravitational potential energy of the system between points $A$ and $B$ increase, decrease, or remain the same?

Key Concepts

Gravitational Potential Energy

Gravitational potential energy (GPE) quantifies the energy an object possesses due to its position in a gravitational field, usually expressed as mgh, where m is mass, g is the acceleration due to gravity, and h is the height above a chosen reference level. The concept includes the idea that potential energy differences are independent of the choice of the zero level, even though the absolute values depend on this reference. Changes in gravitational potential energy directly relate to the work done by gravity.

Conservative Forces

Gravitational force is a prime example of a conservative force, meaning that the work done by this force is path-independent and depends only on the initial and final positions of the object. For conservative forces, the work done over a closed path is zero, and there exists a potential energy function associated with the force. This principle is fundamental in energy conservation problems where energy transforms between kinetic and potential forms without losses.

Gravitational Force and Work

This concept addresses how the force of gravity performs work on objects as they move through a gravitational field. The work done by gravity is directly related to the displacement of the object and the vertical component of that displacement, and it is calculated as the product of the gravitational force and the vertical distance moved. In energy analysis, this work is typically negative when the object moves upward and positive when moving downward, reflecting the energy transfer between kinetic and gravitational potential energy.

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