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, \[E= \frac{1}{2}k x^2\]

\[\Rightarrow E = \frac{1}{2} \times 500 \times {10}^{- 4} \]

= 0 . 025 J

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

Increase in mass, \[∆ m = \frac{E}{c^2}\]

\[ = \frac{0 . 025}{c^2}\]

\[ = \frac{0 . 025}{9 \times {10}^{16}} kg\]

Fractional change of mass, \[\frac{∆ m}{m} = \frac{0 . 025}{9 \times {10}^{16}} \times \frac{1}{0 . 2}\]

\[ \Rightarrow \frac{∆ m}{m} = 0 . 01388 \times {10}^{- 16} \]

\[ \Rightarrow \frac{∆ m}{m} = 1 . 4 \times {10}^{- 18} \]