Question

Supose that you need 2 Joules to expand a spring from its natural length of 30 cm to 42 cm: A)How much work its required to expand it from 35 to 40 cm? B)How long will expand from its natural length if a force of 30N is aplied to it?

I need the answer, using calculus.

Answer 1

A) `25/24` joules `quad` B) `10.8` cm

Explanation:

Work is force times distance:

`W = int F dx`

The spring obeys Hooke's Law:

`F = kx`

where the sign of `k` depends on whether we look at the work done or the force on the spring or by the spring; we won't worry about it.

`x` is the displacement from the spring's natural length, 30 here, so we'll just subtract 30 from the displacements given and divide by 100 to convert to meters, because I think joules and newtons are MKS.

Now we write that we needed two joules for the first .12 meters:

`2 = int _0^{.12} k x dx`

`= {kx^2}/2 |""_{x=0}^{x=.12} = {k (.12^2)}/2 = .0072 k`

`k = 2/.0072 = 2500/9`

Work from 35 to 40, which is .05 to .10 on our ruler,

`W = int _{.05}^{.10} k x dx = 2500/18( .10^2-.05^2)=25/24`joules

Force of 30 Newtons, means

`30 = k x`

`x = 30/k = 30/( 2500/9)=.1080` meters `= 10.8` cm