Question

A force \[F = \alpha + bx\]  acts on a particle in the x-direction, where a and b are constants. Find the work done by this force during a displacement from x = 0 to x = d. 

 

 

Answer 1

Given that force is a function of displacement, i.e.

\[F = a + bx\] ,

where a and b are constants.
So, work done by this force during the displacement x = 0 to x = d,

\[W = \int\limits_0^d F \text{ dx } \]
\[W = \int\limits_0^d \left( a + bx \right) dx\]
\[W = \left[ ax + \frac{b x^2}{2} \right]_0^d \]
\[W = ad + \frac{b d^2}{2}\]
\[ \Rightarrow W = \left( a + \frac{bd}{2} \right)d\]