Calculate the energy possessed by a body of mass 20g executing SHM of amplitude 1cm and time peeriod 4s.

Answer

To calculate the energy possessed by a body of mass 20g executing Simple Harmonic Motion (SHM) with an amplitude of 1cm and a time period of 4s, we can follow these steps:

### Step 1: Understand the formula for total energy in SHM
The total energy (E) of a body in SHM is given by the formula:
\[ E = \frac{1}{2} m \omega^2 A^2 \]
where:
- \( m \) is the mass of the body,
- \( \omega \) is the angular frequency,
- \( A \) is the amplitude.

### Step 2: Convert units
Convert the mass from grams to kilograms and the amplitude from centimeters to meters:
- Mass: \( 20 \, \text{g} = 0.02 \, \text{kg} \)
- Amplitude: \( 1 \, \text{cm} = 0.01 \, \text{m} \)

### Step 3: Calculate angular frequency (\( \omega \))
The angular frequency \( \omega \) can be calculated using the formula:
\[ \omega = \frac{2\pi}{T} \]
where \( T \) is the time period. Given \( T = 4 \, \text{s} \):
\[ \omega = \frac{2\pi}{4} = \frac{\pi}{2} \, \text{rad/s} \]

### Step 4: Calculate \( \omega^2 \)
Now, calculate \( \omega^2 \):
\[ \omega^2 = \left(\frac{\pi}{2}\right)^2 = \frac{\pi^2}{4} \]

### Step 5: Substitute values into the energy formula
Now substitute the values of \( m \), \( \omega^2 \), and \( A \) into the energy formula:
\[ E = \frac{1}{2} \times 0.02 \times \frac{\pi^2}{4} \times (0.01)^2 \]
\[ E = \frac{1}{2} \times 0.02 \times \frac{\pi^2}{4} \times 0.0001 \]
\[ E = \frac{0.02 \times \pi^2 \times 0.0001}{8} \]
\[ E = \frac{0.000002 \times \pi^2}{8} \]

### Step 6: Calculate the numerical value
Using \( \pi^2 \approx 9.87 \):
\[ E \approx \frac{0.000002 \times 9.87}{8} \]
\[ E \approx \frac{0.00001974}{8} \]
\[ E \approx 0.0000024675 \, \text{J} \]

### Step 7: Convert to appropriate units
To convert joules to ergs (1 J = 10^7 ergs):
\[ E \approx 0.0000024675 \times 10^7 \, \text{ergs} \]
\[ E \approx 24.675 \, \text{ergs} \]

### Final Answer
The energy possessed by the body is approximately **24.675 ergs**.

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