Answer to Question #181223 in Calculus for Aliyah

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 ustng 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

$F(x) = kx$

$10 = k(0.2)$

$k = 50\ N /m$

$W = ∫ ^b_ a F(x)\ dx$

$W = ∫ ^b _a kx\ dx$

$W = k∫ ^b_ a x\ dx$

$W = k [\dfrac{x²}{2} ] ^b_a = k [\dfrac{x²}{2} ] ^{0.5}_0$

$W = 50[\dfrac{0.5²}2-\dfrac{0²}{2}] = 50× \dfrac{0.25}{2} = 6.25\ J$

$\therefore 6.25 J$ of work should be done to stretch the spring to its maximum extent.