Question

The force exerted by a compression device is given by F(x)=kx (x−l) for 0≤x≤l, where l is the maximum possible compression, x is the compression and k is the constant. Work done to compress the device by a distance d will be maximum when?
A- \[d=\dfrac{1}{4}\]
B- \[d=\dfrac{1}{\sqrt{2}}\]
C- \[d=\dfrac{1}{2}\]
D- d= l

Answer 1

Hint: The force is given in the form of variables. We need to find the work done to get maximum extension. We can use differentiation here. We know work is given as a dot product of force and displacement. Also, for displacement we can integrate the product of force with it to get the total work done.

Complete step by step answer:
We know that in order for maximum work, the condition that needs to be satisfied is: \[\dfrac{dW}{dx}=0\]
But, \[\dfrac{dW}{dx}=F\]
\[\Rightarrow F=0\]
$\Rightarrow kx(x-l)=0 \\
\Rightarrow x-l=0 \\
\therefore x=l \\ $

So, the correct answer is “Option D”.

Additional Information:
Work is defined as the dot product of force and displacement. Since, dot product gives scalar quantity, work is a scalar quantity. On the other hand, both the force and the displacement are vector quantities. Since, work is given by the dot product, work can be positive, negative as well as zero. The SI unit of force is Newtons and that of displacement is metres.

Note:
External force and displacement are in the same direction
$\therefore$ Work will be positive continuously so it will be maximum when displacement is maximum. And the maximum displacement possible here is l, so this is one another way to solve this problem. Work is measured in units of Joules.