The instantaneous displacement of a simple pendulum oscillator is given by `x= A cos (omegat+ pi / 4 )`Its speed will be maximum at time

Options

A. `(2pi)/(omega)`
B. `(omega)/(2pi)`
C. `(omega)/(pi)`
D. `(pi)/(4omega)`

Answer

To find the time at which the speed of the simple pendulum oscillator is maximum, we start with the given displacement equation:

\[ x = A \cos(\omega t + \frac{\pi}{4}) \]

### Step 1: Differentiate the displacement to find the velocity
The velocity \( v \) is the time derivative of displacement \( x \):

\[ v = \frac{dx}{dt} = \frac{d}{dt}[A \cos(\omega t + \frac{\pi}{4})] \]

Using the chain rule, we differentiate:

\[ v = -A \sin(\omega t + \frac{\pi}{4}) \cdot \frac{d}{dt}(\omega t + \frac{\pi}{4}) = -A \omega \sin(\omega t + \frac{\pi}{4}) \]

### Step 2: Determine when the speed is maximum
The speed will be maximum when the sine function reaches its maximum value. The maximum value of \( \sin(\theta) \) is 1. Therefore, we set:

\[ \sin(\omega t + \frac{\pi}{4}) = 1 \]

### Step 3: Solve for \( \omega t + \frac{\pi}{4} \)
The sine function equals 1 at:

\[ \omega t + \frac{\pi}{4} = \frac{\pi}{2} + 2n\pi \quad (n \in \mathbb{Z}) \]

For the first occurrence (n=0):

\[ \omega t + \frac{\pi}{4} = \frac{\pi}{2} \]

### Step 4: Isolate \( \omega t \)
Now, we isolate \( \omega t \):

\[ \omega t = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4} \]

### Step 5: Solve for \( t \)
Now, we solve for \( t \):

\[ t = \frac{\pi}{4\omega} \]

### Conclusion
Thus, the time at which the speed of the simple pendulum oscillator is maximum is:

\[ t = \frac{\pi}{4\omega} \]