If the hill in Example 7–2 (Fig. 7–4) was not an even slope but rather an irregular curve as in Fig. 7–23, show that the same result would be obtained as in Example 7–2: namely, that the work done by gravity depends only on the height of the hill and not on its shape or the path taken.

검증된 단계별 안내

Understand the concept: The work done by gravity is path-independent and depends only on the vertical displacement (height) of the object. This is because gravity is a conservative force, meaning the work done by gravity depends only on the initial and final positions, not the path taken.

Express the work done by gravity mathematically: The work done by gravity is given by \( W = F \cdot d \cdot \cos(\theta) \), where \( F \) is the gravitational force, \( d \) is the displacement, and \( \theta \) is the angle between the force and displacement vectors. For vertical motion, \( \cos(\theta) = 1 \).

Relate gravitational force to weight: The gravitational force \( F \) is equal to \( mg \), where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity. Substituting \( F \) into the work formula gives \( W = mg \cdot h \), where \( h \) is the vertical height.

Explain why the path does not matter: Since the work done by gravity depends only on the change in height \( h \), the shape of the hill or the path taken does not affect the result. The work is determined solely by the difference in the initial and final vertical positions.

Conclude with the result: The work done by gravity is \( W = mg \cdot h \), which matches the result from Example 7–2. This demonstrates that the work done by gravity is independent of the irregular shape of the hill or the path taken.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Work Done by Gravity

The work done by gravity on an object is defined as the force of gravity acting on the object multiplied by the distance it moves in the direction of the force. This work is independent of the path taken, as it only depends on the vertical displacement of the object. In the context of hills, this means that regardless of the shape of the hill, the work done by gravity is determined solely by the change in height.

Conservative Forces

Gravity is a conservative force, meaning that the work done by gravity on an object moving between two points is the same regardless of the path taken. This property implies that the energy associated with gravitational force can be fully recovered, and the work done depends only on the initial and final positions. This concept is crucial for understanding why the shape of the hill does not affect the work done by gravity.

Potential Energy

Gravitational potential energy is the energy stored in an object due to its height above a reference point, typically the ground. It is calculated using the formula PE = mgh, where m is mass, g is the acceleration due to gravity, and h is height. The principle of conservation of energy states that the work done by gravity will convert potential energy into kinetic energy, reinforcing that the work done is dependent only on the height change, not the path taken.