The displacement of a particle in SHM is indicated by equation y=10` "sin"(20t+pi//3`) where, y is in metres. The value of time period of vibration will be (in second)

Options

A. `10//pi`
B. `pi//10`
C. `2pi//10`
D. `10//2pi`

Explanation

To find the time period of the vibration for the given displacement equation of a particle in simple harmonic motion (SHM), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given equation**: The displacement of the particle in SHM is given by the equation:
\[ y = 10 \sin(20t + \frac{\pi}{3}) \]
Here, \(y\) is in meters.

2. **Compare with the standard form**: The standard form of the SHM equation is:
\[ y = A \sin(\omega t + \phi) \]
where:
- \(A\) is the amplitude,
- \(\omega\) is the angular frequency,
- \(\phi\) is the phase constant.

3. **Extract the values from the equation**: From the given equation, we can identify:
- Amplitude \(A = 10\) meters,
- Angular frequency \(\omega = 20\) radians/second.

4. **Relate angular frequency to time period**: The relationship between angular frequency \(\omega\) and time period \(T\) is given by:
\[ \omega = \frac{2\pi}{T} \]
Rearranging this gives:
\[ T = \frac{2\pi}{\omega} \]

5. **Substitute the value of \(\omega\)**: Now, substitute \(\omega = 20\) radians/second into the equation for \(T\):
\[ T = \frac{2\pi}{20} \]

6. **Calculate the time period**: Simplifying this gives:
\[ T = \frac{\pi}{10} \text{ seconds} \]

### Final Answer:
The time period of the vibration is:
\[ T = \frac{\pi}{10} \text{ seconds} \]