Question

A particle moving on the $x-$axis experiences a force given by $F_x=q{x}^2$ , where $q$ is a constant. How much work is done on the particle as it moves from $x=0$ to $x=d$?

Short answer

Work done by particle$W_{0\to d}=\frac{q{d}^3}{3}$

Step-by-step solution

Step 1. Given information

We have,

Force$F_x=q{x}^3$

$q$is constant

Step 2. Find the expression for the work done in the interval

A force F that acted on an object with mass m is parallel to the direction of motion and causes displacement delta rightarrowx for that object

$W=\int F_{x}dx$

We are given the force as a function of x by Fx=qx2

$$\begin{gathered} W=\int F_{x}dx \\ ={\int }_0^{d}q{x}^{2}dx \\ =q{\int }_0^d{x}^{2}dx \\ =q\left[\frac{1}{3}{x}^3\right] \end{gathered}$$

Now,

$$\begin{gathered} W_{0\to d}=q\left[\frac{1}{3}{d}^3-\frac{1}{3}{\left(0\right)}^3\right] \\ =\frac{q{d}^3}{3} \end{gathered}$$