Question

A mass \(M\) is attached to a string, oscillates with a period of 2 s. If the mass is increased by 4 kg, the time period increases by 1 s. Assuming Hooke's law is obeyed, the initial mass \(M\) was:

• A. 3.2 kg
• B. 1 kg
• C. 2 kg
• D. 8 kg

Hint:

In SHM, use ratio method instead of full formula — saves time in exams.

Answer 1

The Correct Option is A

Concept: \[ T = 2\pi \sqrt{\frac{m}{k}} \;\Rightarrow\; T \propto \sqrt{m} \]

Step 1: Use ratio \[ \frac{T_2}{T_1} = \sqrt{\frac{M+4}{M}} \]

Step 2: Substitute values \[ T_1 = 2\,s,\quad T_2 = 3\,s \] \[ \frac{3}{2} = \sqrt{\frac{M+4}{M}} \]

Step 3: Solve \[ \left(\frac{3}{2}\right)^2 = \frac{M+4}{M} \] \[ \frac{9}{4} = \frac{M+4}{M} \] \[ 9M = 4M + 16 \Rightarrow 5M = 16 \Rightarrow M = \frac{16}{5} = 3.2 \, \text{kg} \] Final: 3.2 kg