A particle is executing SHM along a straight line. Its velocities at distances `x_(1)` and `x_(2)` from the mean position are `v_(1)` and `v_(2)`, respectively. Its time period is

Options

A. `2pi sqrt((x_(1)^(2)+x_(2)^(2))/(v_(1)^(2)+v_(2)^(2)))`
B. `2pi sqrt((x_(2)^(2)-x_(1)^(2))/(v_(1)^(2)-v_(2)^(2)))`
C. `2pi sqrt((v_(1)^(2)+v_(2)^(2))/(x_(1)^(2)+x_(2)^(2)))`
D. `2pi sqrt((v_(1)^(2)-v_(2)^(2))/(x_(1)^(2)-x_(2)^(2)))`

Answer

To find the time period \( T \) of a particle executing simple harmonic motion (SHM) given its velocities at two different displacements from the mean position, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Velocity Equation in SHM**:
The velocity \( v \) of a particle in SHM at a displacement \( x \) from the mean position is given by:
\[ v = \sqrt{\omega^2 (A^2 - x^2)} \]
where \( \omega \) is the angular frequency and \( A \) is the amplitude.

2. **Set Up the Equations for Two Positions**:
For the two positions \( x_1 \) and \( x_2 \), we can write:
\[ v_1 = \sqrt{\omega^2 (A^2 - x_1^2)} \quad \text{(1)} \]
\[ v_2 = \sqrt{\omega^2 (A^2 - x_2^2)} \quad \text{(2)} \]

3. **Square Both Equations**:
Squaring both equations gives:
\[ v_1^2 = \omega^2 (A^2 - x_1^2) \quad \text{(3)} \]
\[ v_2^2 = \omega^2 (A^2 - x_2^2) \quad \text{(4)} \]

4. **Subtract the Two Equations**:
Subtract equation (4) from equation (3):
\[ v_1^2 - v_2^2 = \omega^2 (A^2 - x_1^2) - \omega^2 (A^2 - x_2^2) \]
This simplifies to:
\[ v_1^2 - v_2^2 = \omega^2 (x_2^2 - x_1^2) \quad \text{(5)} \]

5. **Express Angular Frequency in Terms of Time Period**:
We know that:
\[ \omega = \frac{2\pi}{T} \]
Substituting this into equation (5):
\[ v_1^2 - v_2^2 = \left(\frac{2\pi}{T}\right)^2 (x_2^2 - x_1^2) \]

6. **Rearranging for Time Period**:
Rearranging the equation to solve for \( T \):
\[ T^2 = \frac{(2\pi)^2 (x_2^2 - x_1^2)}{v_1^2 - v_2^2} \]
Taking the square root gives:
\[ T = 2\pi \sqrt{\frac{x_2^2 - x_1^2}{v_1^2 - v_2^2}} \]

### Final Answer:
The time period \( T \) of the particle is given by:
\[ T = 2\pi \sqrt{\frac{x_2^2 - x_1^2}{v_1^2 - v_2^2}} \]