The ratio of maximum velocity to the velocity of a particle performing S.H M at a point where potential energy is 25% of total energy is

Options

A. `2:sqrt3`
B. `sqrt3 : 2`
C. `1 : 2`
D. `1 : sqrt3`

Answer

To solve the problem, we need to find the ratio of the maximum velocity to the velocity of a particle performing Simple Harmonic Motion (SHM) at a point where the potential energy is 25% of the total energy.

### Step-by-Step Solution:

1. **Understanding the Energy Relations in SHM**:
- The potential energy (PE) in SHM is given by:
\[ PE = \frac{1}{2} m \omega^2 x^2 \]
- The kinetic energy (KE) is given by:
\[ KE = \frac{1}{2} m \omega^2 (A^2 - x^2) \]
- The total energy (E) in SHM is the sum of potential and kinetic energy:
\[ E = PE + KE = \frac{1}{2} m \omega^2 A^2 \]

2. **Setting Up the Given Condition**:
- According to the question, the potential energy is 25% of the total energy:
\[ PE = 0.25 E \]
- Substituting the expression for total energy:
\[ \frac{1}{2} m \omega^2 x^2 = 0.25 \left(\frac{1}{2} m \omega^2 A^2\right) \]
- Simplifying this gives:
\[ x^2 = 0.25 A^2 \]
- Therefore, we find:
\[ x = \frac{A}{2} \]

3. **Finding the Velocity at x = A/2**:
- The velocity (V) of the particle in SHM is given by:
\[ V = \omega \sqrt{A^2 - x^2} \]
- Substituting \(x = \frac{A}{2}\):
\[ V = \omega \sqrt{A^2 - \left(\frac{A}{2}\right)^2} = \omega \sqrt{A^2 - \frac{A^2}{4}} = \omega \sqrt{\frac{3A^2}{4}} = \frac{\omega A \sqrt{3}}{2} \]

4. **Finding the Maximum Velocity**:
- The maximum velocity (Vmax) in SHM is given by:
\[ V_{max} = \omega A \]

5. **Calculating the Ratio**:
- Now, we can find the ratio of maximum velocity to the velocity at the point where potential energy is 25% of total energy:
\[ \text{Ratio} = \frac{V_{max}}{V} = \frac{\omega A}{\frac{\omega A \sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]
- This can be expressed as:
\[ \text{Ratio} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \]

### Final Answer:
The ratio of maximum velocity to the velocity of the particle at the point where potential energy is 25% of total energy is:
\[ \frac{2}{\sqrt{3}} \quad \text{or} \quad 2 : \sqrt{3} \]