The amplitude of a body executing shm is 2 cm. Its mass is 20 g and the frequency of vibration is 20 Hz. Find its (i) maximum velocity (ii) energy ?

Explanation

To solve the given problem step by step, we will find the maximum velocity and the energy of a body executing simple harmonic motion (SHM) using the provided parameters.

### Given:
- Amplitude (A) = 2 cm = 0.02 m
- Mass (m) = 20 g = 0.02 kg
- Frequency (f) = 20 Hz

### Step 1: Calculate Angular Frequency (ω)
The angular frequency (ω) can be calculated using the formula:
\[ \omega = 2\pi f \]
Substituting the value of frequency:
\[ \omega = 2\pi \times 20 = 40\pi \, \text{rad/s} \]

### Step 2: Calculate Maximum Velocity (V_max)
The maximum velocity (V_max) in SHM is given by the formula:
\[ V_{\text{max}} = A \cdot \omega \]
Substituting the values of amplitude and angular frequency:
\[ V_{\text{max}} = 0.02 \, \text{m} \times 40\pi \, \text{rad/s} \]
Calculating this:
\[ V_{\text{max}} = 0.02 \times 40\pi = 0.8\pi \, \text{m/s} \]
To convert this to cm/s:
\[ V_{\text{max}} = 80\pi \, \text{cm/s} \]

### Step 3: Calculate Energy (E)
The total mechanical energy (E) in SHM is given by the formula:
\[ E = \frac{1}{2} m V_{\text{max}}^2 \]
Substituting the values:
\[ E = \frac{1}{2} \times 0.02 \, \text{kg} \times (80\pi \, \text{cm/s})^2 \]
First, convert cm/s to m/s:
\[ E = \frac{1}{2} \times 0.02 \times \left(0.8\pi\right)^2 \]
Calculating \( (80\pi)^2 \):
\[ (80\pi)^2 = 6400\pi^2 \]
Now substituting:
\[ E = \frac{1}{2} \times 0.02 \times 6400\pi^2 \]
Calculating this:
\[ E = 0.01 \times 6400\pi^2 = 64\pi^2 \, \text{J} \]
Now, calculating the numerical value:
\[ E \approx 6.3 \times 10^{-2} \, \text{J} \]

### Final Answers:
(i) Maximum Velocity \( V_{\text{max}} = 80\pi \, \text{cm/s} \)
(ii) Energy \( E \approx 6.3 \times 10^{-2} \, \text{J} \)