Ch. 08 - Conservation of Energy

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 08 - Conservation of Energy

Problem 73a

Chapter 8, Problem 73a

The graph of Fig. 8–43 shows the potential energy curve of a particle moving along the 𝓍 axis under the influence of a conservative force. Note that the total energy E > U(𝓍), so that the particle’s speed is never zero. In which interval(s) of 𝓍 is the force on the particle to the right?

Verified step by step guidance

Understand the relationship between force and potential energy: The force acting on a particle in a conservative field is related to the potential energy by the equation:

F = - \frac{d U}{d x}

. This means the force is the negative gradient (slope) of the potential energy curve.

Analyze the slope of the potential energy curve: Look at the graph of the potential energy

U (x)

. Identify the intervals where the slope of the curve is negative. A negative slope indicates that the force is positive (to the right).

Determine the intervals: For each interval of

x

, check whether the slope of the potential energy curve is negative. These intervals correspond to regions where the force is directed to the right.

Relate the total energy to the motion: Since the total energy

E

is greater than the potential energy

U (x)

, the particle is always in motion. This ensures that the particle can move through all regions where the force is to the right.

Summarize the intervals: Based on the analysis of the graph, list the specific intervals of

x

where the force is directed to the right. These intervals are determined by the regions where the slope of

U (x)

is negative.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Potential Energy and Conservative Forces

Potential energy is the energy stored in an object due to its position in a force field, such as gravitational or elastic forces. A conservative force is one where the work done is independent of the path taken and depends only on the initial and final positions. In this context, the potential energy curve indicates how the force acting on the particle changes with position along the x-axis.

Total Energy and Motion

The total energy of a system is the sum of its kinetic and potential energy. When the total energy E is greater than the potential energy U(x), the particle has kinetic energy, meaning it is in motion. This relationship implies that the particle cannot be at rest (speed is never zero) in the regions where E > U(x), which is crucial for determining the intervals where the force acts.

Force and Potential Energy Relationship

The force acting on a particle can be derived from the potential energy curve using the relationship F(x) = -dU/dx. This means that the force is related to the slope of the potential energy curve: if the slope is negative, the force is directed to the right, and if the slope is positive, the force is directed to the left. Analyzing the intervals of the potential energy curve helps identify where the force on the particle is directed.

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Textbook Question

The graph of Fig. 8–43 shows the potential energy curve of a particle moving along the 𝓍 axis under the influence of a conservative force. Note that the total energy E > U(𝓍), so that the particle’s speed is never zero. At what value(s) of 𝓍 is the magnitude of the force a minimum?

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