The time period and the amplitude of a simple pendulum are 4 seconds and 0.20 meter respectively. If the displacement is 0.1 meter at time t=0, the equation on its displacement is represented by :-

Options

A. `y=0.2sin(0.5pit)`
B. `y=0.1sin(0.5pit+(pi)/(6))`
C. `y=0.1sin(pit+(pi)/(6))`
D. `Y=0.2 sin(0.5pit+(pi)/(6))`

Answer

To find the equation of displacement for a simple pendulum, we can follow these steps:

### Step 1: Identify the parameters
We are given:
- Time period \( T = 4 \) seconds
- Amplitude \( A = 0.2 \) meters
- Displacement at \( t = 0 \), \( y(0) = 0.1 \) meters

### Step 2: Write the general equation of SHM
The general equation for the displacement in simple harmonic motion (SHM) can be expressed as:
\[ y(t) = A \sin(\omega t + \phi) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( \phi \) is the phase constant.

### Step 3: Calculate the angular frequency \( \omega \)
The angular frequency \( \omega \) is related to the time period \( T \) by the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting the given time period:
\[ \omega = \frac{2\pi}{4} = \frac{\pi}{2} \text{ radians/second} \]

### Step 4: Substitute known values into the equation
Now we can substitute \( A \) and \( \omega \) into the SHM equation:
\[ y(t) = 0.2 \sin\left(\frac{\pi}{2} t + \phi\right) \]

### Step 5: Find the phase constant \( \phi \)
At \( t = 0 \), the displacement is given as \( y(0) = 0.1 \):
\[ 0.1 = 0.2 \sin(\phi) \]
Dividing both sides by 0.2:
\[ \sin(\phi) = \frac{0.1}{0.2} = \frac{1}{2} \]
The angle \( \phi \) that satisfies this equation is:
\[ \phi = 30^\circ = \frac{\pi}{6} \text{ radians} \]

### Step 6: Write the final equation of displacement
Substituting \( \phi \) back into the equation:
\[ y(t) = 0.2 \sin\left(\frac{\pi}{2} t + \frac{\pi}{6}\right) \]

### Final Answer
The equation of displacement for the simple pendulum is:
\[ y(t) = 0.2 \sin\left(\frac{\pi}{2} t + \frac{\pi}{6}\right) \]