Q24DQ - University Physics with Modern Physics

Question

if work W is required to stretch a spring distance x from its unstretched length, what work ( in terms of W) is required to stretch the spring an additional distance x?

Step-by-Step Solution

Answer

the total amount of work required to stretch the spring from a distance of 0 to 2x is $2{\mathrm{kx}}^2$

Step 1: About the work done by spring

The work done by spring is given by: $\mathbf{W}\mathbf{=}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{x}}\left[\mathrm{kx}\right]\mathbf{dx}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{kx}}^{\mathbf{2}}$

Here, k is the spring constant and x is the distance.

Step 2: Determination of work required to stretch the spring at an additional distance x

An additional distance would require work:

$$\begin{gathered} \mathrm{W}={\int }_{\mathrm{x}}^{2\mathrm{x}}\left[\mathrm{kx}\right]\mathrm{dx} \\ =\left[\frac{1}{2}\mathrm{k}{\left(2\mathrm{x}\right)}^2-\frac{1}{2}\mathrm{k}{\left(\mathrm{x}\right)}^2\right] \\ =\frac{1}{2}\mathrm{k}\left(4{\mathrm{x}}^2-{\mathrm{x}}^2\right) \\ =\frac{3}{2}{\mathrm{kx}}^2 \end{gathered}$$

Therefore, the total amount of work required to stretch the spring from a distance of 0 to 2x is given by-

$$\begin{gathered} \mathrm{W}=\frac{1}{2}{\mathrm{kx}}^2+\frac{3}{2}{\mathrm{kx}}^2 \\ =2{\mathrm{kx}}^2 \end{gathered}$$