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 equation of simple harmonic motion (SHM), we will follow these steps:

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

### Step 2: Compare with the standard SHM equation
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.

From the given equation, we can identify:
- Amplitude \( a = 10 \) meters,
- Angular frequency \( \omega = 20 \) rad/s,
- Phase constant \( \phi = \frac{\pi}{3} \).

### Step 3: Calculate the time period
The time period \( T \) of a particle performing simple harmonic motion is related to the angular frequency \( \omega \) by the formula:
\[ T = \frac{2\pi}{\omega} \]

### Step 4: Substitute the value of \( \omega \)
Now, substituting the value of \( \omega \) into the formula:
\[ T = \frac{2\pi}{20} \]

### Step 5: Simplify the expression
Now simplify the expression:
\[ T = \frac{\pi}{10} \]

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

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