Question

A force F = 2(5 - 3x) N acts on a particle. Find the minimum potential energy possessed by the particle.

Given: PE at origin is 8 J.

(a) 0 J

(b) 1/3 J

(c) -1/3 J

(d) 49/3 J

Solution:

F = 2(5 - 3x) N

int_(U_i)^(U_f) dU = - int_(x_i)^(x_f)F.dx

Given that x_i = 0 and U_i = 8J

=> U_i- U_f = [10x - 3x^2]_0^(x_f)

=> 8- U_f = [10x_f- 3(x_f)^2]

=> U_f = 8 - 10x_f+ 3(x_f)^2

For minimum potential energy, (dU)/(dx)= 0

=> (dU)/(dx) = 6x_f - 10 = 0

=> x_f = 5/3

U_(min)= 8 - 10(5/3)+ 3(5/3)^2

=>U_(min) = -1/3 J

Therefore, option (c) is correct.