A spring is stretched a considerable distance. If the work required to stretch the spring one meter is W=2J, how much work is required to stretch the string from 1m to 2m

Question

anonymous · Answer

First note that work is the change in potential energy. The potential energy of a spring is $$U=\frac{kx^2}{2}$$ where x is the change in position. so the change in potential energy is $$U=\frac{k(1)^2}{2}=2$$ . because the x displacement is 1 m. Solving for k this means that k = 4.

Now the work to get between one and two meters is equal to the potentail energy at two meters minus the potential energy at 1 meter. This means that

$$W = U_2-U_1 = \frac{k(2)^2}{2}-\frac{k(1)^2}{2} $$ If we plug in our value for k and simplify this expression we get that the work is 6. 

So the work is 6 Joules