Question

A mass \(m\) attached to free end of a spring executes SHM with a period of \(1 \,s\) If the mass is increased by \(3\, kg\) the period of oscillation increases by one second, the value of mass \(m\) is ______\(kg.\)

Hint:

The time period of SHM depends on mass and spring constant: \(T=2\pi\sqrt{\frac{m}{k}}\). For mass changes, compare squared ratios of T.

Answer 1

Correct Answer: 1

The time period of SHM is given by:

\[T=2\pi\sqrt{\frac{m}{k}}\]

Initially, \(T=1\) s:

\[1=2\pi\sqrt{\frac{m}{k}}\]

Squaring both sides:

\[\frac{m}{k}=\frac{1}{4\pi^{2}}\] ……………(1)

After increasing the mass by 3 kg, the period becomes 2 s:

\[2=2\pi\sqrt{\frac{m+3}{k}}\]

Squaring both sides:

\[\frac{m+3}{k}=\frac{4}{4\pi^{2}}=\frac{1}{\pi^{2}}\] ………………(2)

Dividing equation (2) by equation (1):

\[\frac{m+3}{m}=4\]

Simplify:

\[m+3=4m\]

\[3m=3 \Rightarrow m=1 \text{ kg}\]

Thus, the value of m is 1 kg.

Answer 2

The correct answer is 1.
T=2πkm​​=1 
T′=2πkm+3​​=2 
T′T​=m+3m​​=21​ 
⇒m+3m​=41​ 
m=1