A particle performing SHM with angular frequency `omega=5000` radian/second and amplitude `A=2` cm and mass of 1 kg. Find the total energy of oscillation.

Options

A. 2 kJ
B. 5 kJ
C. 7 kJ
D. 15 kJ

Explanation

To find the total energy of a particle performing Simple Harmonic Motion (SHM), we can use the formula:

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

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

### Step 1: Identify the given values
- Angular frequency, \(\omega = 5000 \, \text{rad/s}\)
- Amplitude, \(A = 2 \, \text{cm} = 2 \times 10^{-2} \, \text{m}\) (convert cm to m)
- Mass, \(m = 1 \, \text{kg}\)

### Step 2: Substitute the values into the energy formula
Now, we can substitute the values into the energy formula:

\[ E = \frac{1}{2} \times 1 \, \text{kg} \times (5000 \, \text{rad/s})^2 \times (2 \times 10^{-2} \, \text{m})^2 \]

### Step 3: Calculate \(\omega^2\)
Calculate \(\omega^2\):

\[ \omega^2 = (5000)^2 = 25000000 \, \text{rad}^2/\text{s}^2 \]

### Step 4: Calculate \(A^2\)
Calculate \(A^2\):

\[ A^2 = (2 \times 10^{-2})^2 = 4 \times 10^{-4} \, \text{m}^2 \]

### Step 5: Substitute \(\omega^2\) and \(A^2\) back into the energy equation
Now substitute these values back into the energy equation:

\[ E = \frac{1}{2} \times 1 \times 25000000 \times 4 \times 10^{-4} \]

### Step 6: Simplify the equation
Calculate the total energy:

\[ E = \frac{1}{2} \times 25000000 \times 4 \times 10^{-4} = 12500000 \times 10^{-4} = 1250 \, \text{J} \]

### Step 7: Convert to kilojoules
Convert joules to kilojoules:

\[ E = 1.25 \, \text{kJ} \]

### Final Answer
The total energy of oscillation is:

\[ E = 1.25 \, \text{kJ} \]