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

Step 1: Understanding the Concept:
The time period \( T \) of a spring-mass system is directly proportional to the square root of the mass \( M \).
: Key Formula or Approach:
\[ T = 2\pi\sqrt{\frac{M}{k}} \implies T \propto \sqrt{M} \]
Step 2: Detailed Explanation:
Initial state: \( T_1 = 2 \text{ s} \) and mass \( = M \).
Final state: \( T_2 = 2 + 1 = 3 \text{ s} \) and mass \( = M + 4 \).
Using the proportionality:
\[ \frac{T_1}{T_2} = \sqrt{\frac{M}{M + 4}} \]
\[ \frac{2}{3} = \sqrt{\frac{M}{M + 4}} \]
Squaring both sides:
\[ \frac{4}{9} = \frac{M}{M + 4} \]
\[ 4(M + 4) = 9M \]
\[ 4M + 16 = 9M \]
\[ 5M = 16 \]
\[ M = \frac{16}{5} = 3.2 \text{ kg} \]
Step 3: Final Answer:
The initial mass M was 3.2 kg.