A body of mass 1 kg is executing simple harmonic motion. Its displacement y(cm) at t seconds is given by `y = 6 sin (100t + pi//4)` . Its maximum kinetic energy is

Options

A. 6 J
B. 18 J
C. 24 J
D. 36 J

Explanation

To find the maximum kinetic energy of a body executing simple harmonic motion (SHM), we can follow these steps:

### Step 1: Identify the parameters from the displacement equation
The displacement \( y \) is given by:
\[ y = 6 \sin(100t + \frac{\pi}{4}) \]
From this equation, we can identify:
- Amplitude \( a = 6 \) cm
- Angular frequency \( \omega = 100 \) rad/s

### Step 2: Convert amplitude to meters
Since the amplitude is given in centimeters, we need to convert it to meters:
\[ a = 6 \text{ cm} = 6 \times 10^{-2} \text{ m} = 0.06 \text{ m} \]

### Step 3: Write the formula for maximum kinetic energy
The maximum kinetic energy \( KE_{max} \) of a body in SHM is given by the formula:
\[ KE_{max} = \frac{1}{2} m \omega^2 a^2 \]
where:
- \( m \) is the mass of the body,
- \( \omega \) is the angular frequency,
- \( a \) is the amplitude.

### Step 4: Substitute the values into the formula
Given:
- Mass \( m = 1 \) kg,
- Angular frequency \( \omega = 100 \) rad/s,
- Amplitude \( a = 0.06 \) m.

Now substituting these values into the formula:
\[ KE_{max} = \frac{1}{2} \times 1 \times (100)^2 \times (0.06)^2 \]

### Step 5: Calculate \( KE_{max} \)
Calculating each part:
- \( (100)^2 = 10000 \)
- \( (0.06)^2 = 0.0036 \)

Now substituting these values:
\[ KE_{max} = \frac{1}{2} \times 1 \times 10000 \times 0.0036 \]
\[ KE_{max} = \frac{1}{2} \times 36 = 18 \text{ Joules} \]

### Final Answer
The maximum kinetic energy is:
\[ \boxed{18 \text{ Joules}} \]