The displacement of a simple harmonic oscillator is, x = 5 sin `(pit//3) m`. Then its velocity at t = 1 s, is

Options

A. `(pi)/(6)m//s`
B. `(5pi)/(6)m//s`
C. `(6pi)/(6)m//s`
D. `(pi)/(2)m//s`

Explanation

To find the velocity of a simple harmonic oscillator at a given time, we can follow these steps:

### Step 1: Write down the displacement equation
The displacement of the simple harmonic oscillator is given by:
\[ x(t) = 5 \sin\left(\frac{\pi t}{3}\right) \, \text{m} \]

### Step 2: Differentiate the displacement equation to find velocity
The velocity \( v(t) \) of a simple harmonic oscillator is the derivative of the displacement with respect to time:
\[ v(t) = \frac{dx}{dt} \]

Using the chain rule, we differentiate:
\[ v(t) = 5 \cdot \cos\left(\frac{\pi t}{3}\right) \cdot \frac{d}{dt}\left(\frac{\pi t}{3}\right) \]
\[ v(t) = 5 \cdot \cos\left(\frac{\pi t}{3}\right) \cdot \frac{\pi}{3} \]
\[ v(t) = \frac{5\pi}{3} \cos\left(\frac{\pi t}{3}\right) \]

### Step 3: Substitute \( t = 1 \) s into the velocity equation
Now we substitute \( t = 1 \) s into the velocity equation:
\[ v(1) = \frac{5\pi}{3} \cos\left(\frac{\pi \cdot 1}{3}\right) \]
\[ v(1) = \frac{5\pi}{3} \cos\left(\frac{\pi}{3}\right) \]

### Step 4: Calculate \( \cos\left(\frac{\pi}{3}\right) \)
We know that:
\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

### Step 5: Substitute the value of \( \cos\left(\frac{\pi}{3}\right) \) back into the equation
Now substituting back:
\[ v(1) = \frac{5\pi}{3} \cdot \frac{1}{2} \]
\[ v(1) = \frac{5\pi}{6} \]

### Final Answer
Thus, the velocity of the simple harmonic oscillator at \( t = 1 \) s is:
\[ v(1) = \frac{5\pi}{6} \, \text{m/s} \]

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