Question

A particle moves along X-axis from \[x = 0\] to $x = 5$ cm under the influence of force given by \[F = \left( {7 - 2x + 3{x^2}} \right)N\] . The work done in the process is
(a). 70 J
(b). 270 J
(c). 35 J
(d). 135 J

Answer 1

- Hint- In this question force varies with distance, here we have the force equation and variation limit of x , so in order to find out the work done in the process we will integrate force with respect to x.

Formula used- $W = \int\limits_{{x_i}}^{{x_f}} {F.dx} $

Complete step-by-step solution -

Given that the force acting on particle is \[F = \left( {7 - 2x + 3{x^2}} \right)N\] and the particle moves along the x axis from \[x = 0\] to $x = 5$
We know that work is said to be done when a force applied to an object moves that object.
Therefore the formula to calculate the work done by the force
$W = \int\limits_{{x_i}}^{{x_f}} {F.dx} $
Where F, is the force acting on the particle, ${x_f}$ is the final position of the particle and ${x_i}$ is the initial position of the particle.
Substitute the value of F, ${x_f}$ and ${x_i}$ in above formula, we have
$
  \because W = \int\limits_{{x_i}}^{{x_f}} {F.dx} \\
   \Rightarrow W = \int\limits_{x = 0}^{x = 5} {\left( {7 - 2x + 3{x^2}} \right)dx} \\
 $
Now let us solve the integration to find the value of work done.
\[
   \Rightarrow W = \left[ {7x - \dfrac{{2{x^2}}}{2} + \dfrac{{3{x^3}}}{3}} \right]_0^5 \\
   \Rightarrow W = \left[ {7\left( 5 \right) - \dfrac{{2{{\left( 5 \right)}^2}}}{2} + \dfrac{{3{{\left( 5 \right)}^3}}}{3} - 7\left( 0 \right) + \dfrac{{2{{\left( 0 \right)}^2}}}{2} - \dfrac{{3{{\left( 0 \right)}^3}}}{3}} \right] \\
   \Rightarrow W = \left[ {35 - 25 + 125 - 0 + 0 - 0} \right] \\
   \Rightarrow W = 135J \\
 \]
Hence work done by the force is 135J and the correct answer is option D.

Note- Work is said to be done when a force applied to an object moves that object. We can calculate work by multiplying the force by the movement of the object. The SI unit of work is the joule (J). A force is said to do positive work if the force has a component in the direction of the displacement of the point of application.