A mass of 0.1 kg hangs at the end of a spring. When a mass of 0.01 kg more is added at the end of the spring, it stretches by 5 cm more. When the extra 0.01 kg mass is removed, determine the time period of vibration of the system.

Explanation

To solve the problem step by step, we will first determine the spring constant \( k \) using the information provided and then calculate the time period of the oscillation.

### Step 1: Understand the problem
We have a mass of \( 0.1 \, \text{kg} \) hanging from a spring. When an additional mass of \( 0.01 \, \text{kg} \) is added, the spring stretches by \( 5 \, \text{cm} \) (which is \( 0.05 \, \text{m} \)). We need to find the time period of the mass when the additional mass is removed.

### Step 2: Establish the equilibrium conditions
1. When the mass \( 0.1 \, \text{kg} \) is hanging, the force due to gravity is:
\[
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