Two masses 8 kg 4 kg are suspended together by a massless spring of spring constant `1000 Nm^(-1)` . When the masses are in equilibrium 8 kg is removed without disturbing the system. The amplitude of oscillation is

Options

A. 0.5 m
B. 0.08 m
C. 0.4 m
D. 0.04 m

Explanation

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the System
We have two masses, 8 kg and 4 kg, suspended together by a massless spring with a spring constant \( k = 1000 \, \text{N/m} \). When both masses are attached, they will stretch the spring to a certain equilibrium position.

### Step 2: Calculate the Total Mass
The total mass when both weights are attached is:
\[ m_{\text{total}} = 8 \, \text{kg} + 4 \, \text{kg} = 12 \, \text{kg} \]

### Step 3: Find the Equilibrium Position
Using Hooke's Law, the extension \( x \) of the spring at equilibrium can be calculated using the formula:
\[ kx = mg \]
where \( m \) is the total mass and \( g \) is the acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \)).

Substituting the values:
\[ 1000 \, x = 12 \, \text{kg} \times 10 \, \text{m/s}^2 \]
\[ 1000 \, x = 120 \, \text{N} \]
\[ x = \frac{120}{1000} = 0.12 \, \text{m} \]

### Step 4: Remove the 8 kg Mass
Now, we remove the 8 kg mass from the system. The remaining mass is 4 kg.

### Step 5: Calculate the New Equilibrium Position
Now we need to find the new equilibrium position with only the 4 kg mass:
\[ kx_1 = mg \]
Substituting the values:
\[ 1000 \, x_1 = 4 \, \text{kg} \times 10 \, \text{m/s}^2 \]
\[ 1000 \, x_1 = 40 \, \text{N} \]
\[ x_1 = \frac{40}{1000} = 0.04 \, \text{m} \]

### Step 6: Calculate the Amplitude of Oscillation
The amplitude of oscillation is the difference between the initial maximum displacement (with both masses) and the new maximum displacement (with only the 4 kg mass):
\[ \text{Amplitude} = x - x_1 = 0.12 \, \text{m} - 0.04 \, \text{m} = 0.08 \, \text{m} \]

### Final Answer
The amplitude of oscillation after removing the 8 kg mass is:
\[ \text{Amplitude} = 0.08 \, \text{m} \]