Question

The force exerted by a compression device is given by $F\left(x\right)=kx(x-l)$, where $l$ is the maximum possible compression, $x$ is the compression and $k$ is a constant. The work required to compress the device by a distance $d$ will be maximum when

• A. $d=\frac{l}{4}$
• B. $d=\frac{l}{\sqrt{2}}$
• C. $d=\frac{l}{2}$
• D. $d=l$

Answer 1

The correct option is D $d=l$

For workdone $\left(W\right)$ to be maximum; $\frac{dW}{dx}=0=F\left(x\right);i.e.F\left(x\right)=0=kx(x-l)\Rightarrow x=l,x=0$
Clearly for $d=l$, the work done is maximum.

Alternate solution:
External force and displacement are in the same direction
$\therefore$ Work will be positive continuously so it will be maximum when displacement is maximum (i.e. at $d=l$).