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∝ r3, if the distance from the P to the position of the particle is r. It has been determined that the constant of proportionality is 0.45 N/m3, i.e. the magnitude of the force on a particle can be written as 0.45r3, when the particle is at a distance r from the force center.

Answer 1

Answer:

       $PE = \dfrac{0.45 r^4}{4}$

Explanation:

given,

proportionality constant = 0.45

          F∝ r³

         F = k r³

using conservation of energy change in potential energy is equal to work done.

       d (PE) = F. dr

       d (PE) = k r³. dr

integrating both side

     $\int {d (PE)} = k \int {r^3} dr$

       $PE = \dfrac{k r^4}{4} + C$

given that at r = 0; PE = 0

       $0 = \dfrac{k 0^4}{4} + C$

                   C = 0

       $PE = \dfrac{k r^4}{4}$

       $PE = \dfrac{0.45 r^4}{4}$