Question

A mysterious force acts on all particles along a particular line and always points towards a particular point P on the line. The magnitude of the force on a particle increases as the cube of the distance from that point; that is F[infinity]r3, if the distance from P to the position of the particle is r. Let b be the proportionality constant, and write the magnitude of the force as F=br3. Find the potential energy of a particle subjected to this force when the particle is at a distance D from P, assuming the potential energy to be zero when the particle is at P.

Answer 1

To find the potential energy at a distance D from a point P given a force proportional to the cube of the distance (F=br³), integrate the force from 0 to D. The potential energy is - (b/4)D⁴, with zero potential energy at point P.

Step-by-step explanation:

The problem involves finding the potential energy of a particle experiencing a specific force. The force in question increases as the cube of the distance from a point P, with the relationship F = br3, where b is a proportionality constant and r is the distance from point P. To find the potential energy at a distance D from point P, we will need to integrate the force with respect to distance from point P to D.

The potential energy (U) at a distance D from point P is given by the negative of the work done by the force as the particle moves from P to D. Mathematically, we can express this as:

U = - ∫PD F dr = - ∫0D br3 dr

After integrating, we find the potential energy function:

U = -b⁄4 r4|0D = - (b/4)D4

This is the expression for the potential energy at a distance D from point P, with the given that the potential energy is zero at point P itself.