Hooke's law states that a force is needed to stretch or compress a spring by a distance of x. The force is proportional to the distance x. It is written as `F = kx`
`F ` = force
`k` = proportionality constant or spring constant
`x` = length displacement from its natural length
Applying the given variable force: `F= 20` pounds to stretch a spring a total of `9` inches, we get:
Plug-in `k =20/9` on Hooke's law , we get:
`F = (20/9)x`
Work can be define with formula: `W = F*Deltax` where:
`F ` = force or ability to do work.
`Deltax` = displacement of the object’s position
With force function: `F(x)= (20/9)x ` and condition to stretch the spring by 1 foot (or 12 inches) from its natural position, we set-up the integral application for work as:
`W = int_a^b F(x) dx`
`W = int_0^12 (20/9)xdx`
Apply basic integration property: `int c*f(x)dx= c int f(x)dx` .
`W = (20/9)int_0^12 xdx`
Apply Power rule for integration: `int x^n(dx) = x^(n+1)/(n+1)` .
`W = (20/9) * x^(1+1)/(1+1)|_0^12`
`W = (20/9) * x^2/2|_0^12`
`W = (20x^2)/18|_0^12`
Apply definite integral formula: `F(x)|_a^b = F(b)-F(a)` .
`W = (10(12)^2)/9-(10(0)^2)/9`
`W =160 - 0`