A particle that can move along the x-axis is part of a system with potential energy U(x) = A/x2 − B/x where A and B are positive constants. Where are the particle's equilibrium positions?

Ch 10: Interactions and Potential Energy

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 10: Interactions and Potential Energy

Problem 61a

Chapter 10, Problem 61a

A particle that can move along the x-axis is part of a system with potential energy U(x) = A/x² − B/x where A and B are positive constants. Where are the particle's equilibrium positions?

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Step 1: Recall that equilibrium positions occur where the force acting on the particle is zero. The force is related to the potential energy by the equation F(x) = -dU(x)/dx. Begin by calculating the derivative of the given potential energy function U(x) = A/x^2 - B/x with respect to x.

Step 2: Differentiate U(x) with respect to x. Using the power rule for derivatives, the derivative of A/x^2 is -2A/x^3, and the derivative of -B/x is B/x^2. Combine these results to find dU(x)/dx = -2A/x^3 + B/x^2.

Step 3: Set the force equal to zero to find the equilibrium positions. This means solving the equation -dU(x)/dx = 0, or equivalently, -2A/x^3 + B/x^2 = 0.

Step 4: Factorize the equation to simplify. Rewrite the equation as (B/x^2) - (2A/x^3) = 0. Factor out 1/x^2, resulting in (1/x^2)(B - 2A/x) = 0. Since 1/x^2 cannot be zero, solve the remaining equation B - 2A/x = 0.

Step 5: Solve for x to find the equilibrium positions. Rearrange the equation B - 2A/x = 0 to isolate x, giving x = 2A/B. This is the position where the particle is in equilibrium.

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Key Concepts

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

Potential Energy

Potential energy is the energy stored in a system due to its position or configuration. In this case, the potential energy U(x) is defined as a function of the particle's position along the x-axis. Understanding how potential energy varies with position is crucial for analyzing the forces acting on the particle and determining its equilibrium positions.

Equilibrium Position

An equilibrium position occurs when the net force acting on a particle is zero, meaning the particle is in a state of balance. For a particle moving along the x-axis, this can be found by setting the derivative of the potential energy function U(x) to zero. At these points, the particle experiences no net force and can remain at rest or move with constant velocity.

Force from Potential Energy

The force acting on a particle can be derived from the potential energy function using the relationship F(x) = -dU/dx. This means that the force is equal to the negative gradient of the potential energy with respect to position. By calculating this derivative, one can identify the points where the force is zero, which correspond to the equilibrium positions of the particle.

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Textbook Question

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