A body of mass 1 kg executes SHM which is given by
` y=6.0 cos (100t+(pi)/(4)) cm`
What is the maximum kinetic energy ?

Options

A. 3J
B. 6J
C. 9J
D. 18J

Answer

To find the maximum kinetic energy of a body executing simple harmonic motion (SHM) given by the equation \( y = 6.0 \cos(100t + \frac{\pi}{4}) \) cm, we can follow these steps:

### Step 1: Identify the parameters from the SHM equation
From the equation \( y = 6.0 \cos(100t + \frac{\pi}{4}) \):
- The amplitude \( A \) is 6.0 cm.
- The angular frequency \( \omega \) is 100 rad/s.

### Step 2: Convert amplitude to meters
Since the amplitude is given in centimeters, we need to convert it to meters:
\[ A = 6.0 \text{ cm} = \frac{6.0}{100} \text{ m} = 0.06 \text{ m} \]

### Step 3: Use the formula for maximum kinetic energy
The maximum kinetic energy \( KE_{\text{max}} \) in SHM can be calculated using the formula:
\[ KE_{\text{max}} = \frac{1}{2} m \omega^2 A^2 \]
where:
- \( m \) is the mass of the body (1 kg),
- \( \omega \) is the angular frequency (100 rad/s),
- \( A \) is the amplitude (0.06 m).

### Step 4: Substitute the values into the formula
Substituting the values into the formula:
\[ KE_{\text{max}} = \frac{1}{2} \times 1 \text{ kg} \times (100 \text{ rad/s})^2 \times (0.06 \text{ m})^2 \]

### Step 5: Calculate \( KE_{\text{max}} \)
Calculating each part:
\[ (100 \text{ rad/s})^2 = 10000 \text{ rad}^2/\text{s}^2 \]
\[ (0.06 \text{ m})^2 = 0.0036 \text{ m}^2 \]
Now substituting these back:
\[ KE_{\text{max}} = \frac{1}{2} \times 1 \times 10000 \times 0.0036 \]
\[ KE_{\text{max}} = \frac{1}{2} \times 36 = 18 \text{ J} \]

### Final Answer
The maximum kinetic energy of the body is \( 18 \text{ J} \).

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