Question

A particle of mass m is in one dimension along the positive x-axis. It is acted on by a constant force directed toward the origin with magnitude 8 and an inverse-square law repulsive force with magnitude A.

1. Find the potential energy function.
2. Find the equilibrium positions.

Answer 1

To find the potential energy function, integrate the force function with respect to distance. The potential energy function is U(x) = -8x + A/x^2 + C. To find the equilibrium positions, set the net force equal to zero and solve for x: x = (A/8)^(1/2).

Step-by-step explanation:

In this scenario, the particle is subject to two forces: a constant force towards the origin and an inverse-square law repulsive force. To find the potential energy function, we need to integrate the force function with respect to distance. The potential energy function can be written as:

U(x) = -8x + A/x^2 + C

where C is the constant of integration. To find the equilibrium positions, we need to find the points where the force is zero. Set the net force equal to zero and solve for x:

8 - A/x^2 = 0

Solving for x:

x = (A/8)^(1/2)