A particle of mass 0.10 kg executes SHM with an amplitude 0.05 m and frequency 20 vib/s. Its energy of oscillation is

Options

A. 2 J
B. 4 J
C. 1 J
D. zero

Explanation

To find the energy of a particle executing simple harmonic motion (SHM), we can use the formula for the total energy of a simple harmonic oscillator:

\[ E = \frac{1}{2} m \omega^2 A^2 \]

where:
- \(E\) is the total energy,
- \(m\) is the mass of the particle,
- \(\omega\) is the angular frequency,
- \(A\) is the amplitude.

### Step-by-Step Solution:

**Step 1: Identify the given values.**
- Mass (\(m\)) = 0.10 kg
- Amplitude (\(A\)) = 0.05 m
- Frequency (\(f\)) = 20 vib/s

**Step 2: Calculate the angular frequency (\(\omega\)).**
The angular frequency is related to the frequency by the formula:
\[ \omega = 2\pi f \]
Substituting the given frequency:
\[ \omega = 2\pi \times 20 = 40\pi \text{ rad/s} \]

**Step 3: Substitute the values into the energy formula.**
Now we can substitute \(m\), \(\omega\), and \(A\) into the energy formula:
\[ E = \frac{1}{2} \times 0.10 \times (40\pi)^2 \times (0.05)^2 \]

**Step 4: Calculate \((40\pi)^2\) and \((0.05)^2\).**
First, calculate \((40\pi)^2\):
\[ (40\pi)^2 = 1600\pi^2 \]

Next, calculate \((0.05)^2\):
\[ (0.05)^2 = 0.0025 \]

**Step 5: Substitute these values back into the energy equation.**
Now we have:
\[ E = \frac{1}{2} \times 0.10 \times 1600\pi^2 \times 0.0025 \]

**Step 6: Simplify the expression.**
Calculating the constants:
\[ E = \frac{1}{2} \times 0.10 \times 1600 \times 0.0025 \times \pi^2 \]
\[ E = \frac{1}{2} \times 0.10 \times 4 = 0.20 \text{ (ignoring } \pi^2 \text{ for now)} \]

**Step 7: Calculate the numerical value.**
Now, we will consider \(\pi^2\) approximately as 10:
\[ E \approx 0.20 \times 10 = 2 \text{ Joules} \]

### Final Answer:
The total energy of the oscillation is approximately **2 Joules**.