Question

In an experiment, one of the forces exerted on a proton is F⃗ =−αx2i^, where α=12N/m2. What is the potential-energy function for F⃗ ? Let U=0 when x=0.

Answer 1

Answer:

The potential energy is $-4x^3$

Explanation:

Given that,

Force $F=-\alpha x^2 i$

We need to calculate the potential energy

Using formula of work done

$\Delta U=F(x) dx$

Put the value of F

$\Delta U=-\alpha x^2\ dx$

On integration

$U=-\alpha \dfrac{x^3}{3}+C$

$U=-\dfrac{\alpha x^3}{3}+C$...(I)

U = 0, x = 0 so C = 0

Put the value of c and α in equation (I)

$U=-\dfrac{12x^3}{3}+0$

$U=-4x^3$

Hence, The potential energy is $-4x^3$