Question

The resistance of a packing material to a sharp object penetrating it is a force proportional to the fourth power of the penetration depth x ; that is,F= -kx^4. The question asks for the work done and the answer is ıntegral 0 to x of -kx^4 which is kx^5/5. However, since W = Fx, why the answer is not -kx^4 * x = -kx^5?

Answer 1

Solution given below :-

Integral has to be used when force is not constant and is a function of x. Consider the Work Integral given by,

Here F(x) is the Force, let us assume that the force is constant and is equal to K, then we can write the Work done as,

But When the force is a variable and is a function of x, it can't be taken out of the integral, hence for the above question, the work done is,

Since force is a variable, therefore we can't say that the work done is simply -kx^4*x = -kx^5