A body of mass 8 kg performs S.H.M. of amplitude 60 cm. The restoring force is 120 N, when the displacement is 60 cm. The time period is

Options

A. `0.628 s`
B. `1.256 s`
C. `1.884 s`
D. ` 2.5 12s`

Answer

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

### Step 1: Identify the given values
- Mass (m) = 8 kg
- Amplitude (A) = 60 cm = 0.6 m (convert to meters)
- Restoring force (F) = 120 N
- Displacement (x) = 60 cm = 0.6 m (convert to meters)

### Step 2: Use the formula for restoring force in SHM
The restoring force in Simple Harmonic Motion (SHM) is given by:
\[ F = -Kx \]
Where:
- \( K \) is the force constant (spring constant)
- \( x \) is the displacement

### Step 3: Calculate the force constant (K)
From the equation \( F = Kx \), we can rearrange it to find \( K \):
\[ K = \frac{F}{x} \]
Substituting the values:
\[ K = \frac{120 \, \text{N}}{0.6 \, \text{m}} = 200 \, \text{N/m} \]

### Step 4: Use the formula for the time period (T) of SHM
The time period \( T \) of SHM is given by:
\[ T = 2\pi \sqrt{\frac{m}{K}} \]

### Step 5: Substitute the values into the time period formula
Substituting the values of mass and force constant:
\[ T = 2\pi \sqrt{\frac{8 \, \text{kg}}{200 \, \text{N/m}}} \]

### Step 6: Simplify the expression
Calculating the fraction:
\[ \frac{8}{200} = 0.04 \]
Now, take the square root:
\[ \sqrt{0.04} = 0.2 \]

### Step 7: Calculate the time period
Now substitute back into the time period formula:
\[ T = 2\pi \times 0.2 \]
\[ T = 0.4\pi \]

### Step 8: Calculate the numerical value
Using \( \pi \approx 3.14 \):
\[ T \approx 0.4 \times 3.14 \approx 1.256 \, \text{s} \]

### Final Answer
The time period \( T \) is approximately **1.256 seconds**.

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