Concept of potential energy

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.