Answer to Question #181728 in Calculus for Pornesian

Question

A manufacturer wants to build a spring that takes a force 10 N (in negative direction) to compress it 0.2 m from the equilibrium position. The spring should be able to stretch 0.5 m from the equilibrium position.

As a mechanical engineer, you were asked to present how much work should be done to stretch the spring.The presentation shall include important components of the problem, complete and correct computations and a logical and organized explanation.

Hints:

First, create a force function F(x) by finding the spring constant k in F(x) = kx where F(x) is the force and x is the position from the equilibrium.

Then, calculate work using the concept of the definite integral.

If a variable force F(x) moves an object in a positive direction along the x-axis from point a to point b, then the work done on the object is W = ∫ ^b a F(x) dx.

Answer 1

The work done in stretching a spring a distance x from its rest position is:

$W=kx^2/2$

$k=F/x$

$F$ is the force to stretch a spring

Then:

$k=10/0.2=50\ H/m$

The work done to stretch the spring 0.5 m from the equilibrium position:

$W=50\cdot0.5^2/2=6.25\ J$