An object of mass 0.2 kg executes simple harmonic oscillation along the x-axis with a frequency `(25)/pi`. At the position x = 0.04m, the object has kinetic energy 0.5J and potential energy 0.4J. amplitude of oscillation is (potential energy is zero mean position).

Options

A. 6cm
B. 4cm
C. 8 cm
D. 2cm

Answer

To find the amplitude of oscillation for the object executing simple harmonic motion, we can follow these steps:

### Step 1: Calculate Total Mechanical Energy
The total mechanical energy (E) in simple harmonic motion is the sum of kinetic energy (KE) and potential energy (PE). Given:
- Kinetic Energy (KE) = 0.5 J
- Potential Energy (PE) = 0.4 J

Total Mechanical Energy (E) can be calculated as:
\[ E = KE + PE \]
\[ E = 0.5 \, \text{J} + 0.4 \, \text{J} = 0.9 \, \text{J} \]

### Step 2: Relate Total Mechanical Energy to Amplitude
In simple harmonic motion, the total mechanical energy is also related to the amplitude (A) by the formula:
\[ E = \frac{1}{2} k A^2 \]
Where \( k \) is the spring constant.

### Step 3: Express Spring Constant in Terms of Mass and Frequency
The spring constant \( k \) can be expressed in terms of mass (m) and angular frequency (ω):
\[ k = m \omega^2 \]
Where \( \omega = 2\pi f \) and \( f \) is the frequency.

Given:
- Mass (m) = 0.2 kg
- Frequency (f) = \( \frac{25}{\pi} \)

Calculate \( \omega \):
\[ \omega = 2\pi f = 2\pi \left(\frac{25}{\pi}\right) = 50 \, \text{rad/s} \]

Now substitute \( \omega \) back to find \( k \):
\[ k = m \omega^2 = 0.2 \times (50)^2 = 0.2 \times 2500 = 500 \, \text{N/m} \]

### Step 4: Substitute Values into Energy Equation
Now substitute \( k \) back into the energy equation:
\[ E = \frac{1}{2} k A^2 \]
Substituting the known values:
\[ 0.9 = \frac{1}{2} \times 500 \times A^2 \]

### Step 5: Solve for Amplitude (A)
Rearranging the equation to solve for \( A^2 \):
\[ 0.9 = 250 A^2 \]
\[ A^2 = \frac{0.9}{250} \]
\[ A^2 = \frac{9}{2500} \]

Taking the square root to find \( A \):
\[ A = \sqrt{\frac{9}{2500}} = \frac{3}{50} \, \text{m} \]

### Step 6: Convert to Centimeters
To convert meters to centimeters:
\[ A = \frac{3}{50} \times 100 = 6 \, \text{cm} \]

### Final Answer
The amplitude of oscillation is **6 cm**.

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