Question

By what fraction does the mass of a spring change when it is compressed by 1 cm? The mass of the spring is 200 g at its natural length and the spring constant is 500 N m⁻¹.

Answer 1

Given:
Compression in the string, x = 1 cm = 1 × 10⁻² m
Spring constant, k = 500 N/m
Mass of the spring, m = 200 g = 0.2 kg

Energy stored in the spring, E$=\frac{1}{2}k{x}^2$
$$\begin{gathered} \Rightarrow E=\frac{1}{2}\times 500\times {10}^{-4} \\ =0.025\mathrm{J} \end{gathered}$$

This energy can be converted into mass according to mass energy equivalence. Thus,

$$\begin{gathered} \mathrm{Increase}\mathrm{in}\mathrm{mass},∆m=\frac{E}{c^2} \\ =\frac{0.025}{c^2} \\ =\frac{0.025}{9\times {10}^{16}}\mathrm{kg} \end{gathered}$$

$$\begin{gathered} \mathrm{Fractional}\mathrm{change}\mathrm{of}\mathrm{mass},\frac{∆m}{m}=\frac{0.025}{9\times {10}^{16}}\times \frac{1}{0.2} \\ \Rightarrow \frac{∆m}{m}=0.01388\times {10}^{-16} \\ \Rightarrow \frac{∆m}{m}=1.4\times {10}^{-18} \end{gathered}$$