A bob of simple pendulum of mass 1g is oscillating with a frequency 5 vibrations per second and its amplitude is 3cm. Find the kinetic energy and kinetic energy at that instant.

Answer

To find the kinetic energy of a simple pendulum bob at its lowest point, we can follow these steps:

### Step 1: Understand the Given Data
- Mass of the bob, \( m = 1 \, \text{g} = 0.001 \, \text{kg} \) (convert grams to kilograms)
- Frequency of oscillation, \( f = 5 \, \text{Hz} \)
- Amplitude of oscillation, \( A = 3 \, \text{cm} = 0.03 \, \text{m} \) (convert centimeters to meters)

### Step 2: Calculate Angular Frequency
The angular frequency \( \omega \) can be calculated using the formula:
\[ \omega = 2\pi f \]
Substituting the value of frequency:
\[ \omega = 2\pi \times 5 \approx 31.42 \, \text{rad/s} \]

### Step 3: Calculate Total Energy
The total mechanical energy \( E \) in a simple harmonic motion is given by:
\[ E = \frac{1}{2} m \omega^2 A^2 \]
Substituting the known values:
\[ E = \frac{1}{2} \times 0.001 \times (31.42)^2 \times (0.03)^2 \]

Calculating \( (31.42)^2 \) and \( (0.03)^2 \):
\[ (31.42)^2 \approx 987.76, \quad (0.03)^2 = 0.0009 \]
Now substituting these values:
\[ E = \frac{1}{2} \times 0.001 \times 987.76 \times 0.0009 \]
\[ E \approx \frac{1}{2} \times 0.001 \times 0.888984 \approx 0.000444492 \, \text{J} \]

### Step 4: Convert Energy to Erg
Since \( 1 \, \text{J} = 10^7 \, \text{erg} \):
\[ E \approx 0.000444492 \times 10^7 \approx 4444.92 \, \text{erg} \]

### Step 5: Conclusion
The kinetic energy of the pendulum bob at its lowest point is approximately:
\[ \text{Kinetic Energy} \approx 4444.92 \, \text{erg} \]