Maximum K.E. of a mass of 1 kg executing SHM is18 J . Amplitude of motion is6 cm , its angular frequency is

Options

A. `25 rad s^(-1)`
B. ` 50 rad s^(-1)`
C. ` 75 rad s^(-1)`
D. ` 100 rad s^(-1)`

Answer

To solve the problem step by step, we will use the formula for the maximum kinetic energy (K.E.) in simple harmonic motion (SHM) and the given values.

### Step 1: Write down the formula for maximum kinetic energy in SHM.
The maximum kinetic energy (K.E.) of an object executing SHM is given by the formula:
\[ K.E._{max} = \frac{1}{2} m \omega^2 A^2 \]
where:
- \( K.E._{max} \) is the maximum kinetic energy,
- \( m \) is the mass of the object,
- \( \omega \) is the angular frequency,
- \( A \) is the amplitude of the motion.

### Step 2: Substitute the known values into the formula.
We know:
- \( K.E._{max} = 18 \, \text{J} \)
- \( m = 1 \, \text{kg} \)
- \( A = 6 \, \text{cm} = 0.06 \, \text{m} \) (convert cm to m)

Substituting these values into the formula:
\[ 18 = \frac{1}{2} \cdot 1 \cdot \omega^2 \cdot (0.06)^2 \]

### Step 3: Simplify the equation.
This simplifies to:
\[ 18 = \frac{1}{2} \cdot \omega^2 \cdot 0.0036 \]
\[ 18 = 0.0018 \cdot \omega^2 \]

### Step 4: Solve for \( \omega^2 \).
To isolate \( \omega^2 \), we can rearrange the equation:
\[ \omega^2 = \frac{18}{0.0018} \]
Calculating the right side:
\[ \omega^2 = 10000 \]

### Step 5: Find \( \omega \).
Taking the square root of both sides gives us:
\[ \omega = \sqrt{10000} = 100 \, \text{rad/s} \]

### Final Answer:
The angular frequency \( \omega \) is \( 100 \, \text{rad/s} \).

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