If displacement time equation of an S.H.M. is`x= sin . (pit)/(6)` , then the moment of times at which peak value of acceleration is attained are

Options

A. 2,4,6,…....s
B. 1,3,5 ….....s
C. 3,9,15 ….....s
D. 1,2,3…......s

Answer

To solve the problem, we need to find the moments of time at which the peak value of acceleration is attained given the displacement-time equation of simple harmonic motion (S.H.M.):

**Step 1: Understand the given equation**
The displacement-time equation is given as:
\[ x = \sin\left(\frac{\pi t}{6}\right) \]

**Step 2: Find the velocity**
To find the velocity, we differentiate the displacement with respect to time \( t \):
\[ v = \frac{dx}{dt} = \frac{d}{dt}\left(\sin\left(\frac{\pi t}{6}\right)\right) \]
Using the chain rule:
\[ v = \cos\left(\frac{\pi t}{6}\right) \cdot \frac{\pi}{6} \]
So, the velocity equation is:
\[ v = \frac{\pi}{6} \cos\left(\frac{\pi t}{6}\right) \]

**Step 3: Find the acceleration**
Next, we differentiate the velocity to find the acceleration:
\[ a = \frac{dv}{dt} = \frac{d}{dt}\left(\frac{\pi}{6} \cos\left(\frac{\pi t}{6}\right)\right) \]
Using the chain rule again:
\[ a = -\frac{\pi}{6} \cdot \sin\left(\frac{\pi t}{6}\right) \cdot \frac{\pi}{6} \]
Thus, the acceleration equation is:
\[ a = -\frac{\pi^2}{36} \sin\left(\frac{\pi t}{6}\right) \]

**Step 4: Determine when acceleration is at its peak value**
The peak value of acceleration occurs when the sine function is at its maximum or minimum, which means:
\[ \sin\left(\frac{\pi t}{6}\right) = \pm 1 \]
This occurs at:
\[ \frac{\pi t}{6} = n\pi + \frac{\pi}{2} \quad \text{for } n \in \mathbb{Z} \]
This can be simplified to:
\[ t = 6\left(n + \frac{1}{2}\right) = 6n + 3 \]

**Step 5: Find specific moments of time**
Now, substituting different integer values for \( n \):
- For \( n = 0 \): \( t = 3 \) seconds
- For \( n = 1 \): \( t = 9 \) seconds
- For \( n = 2 \): \( t = 15 \) seconds
- And so on...

Thus, the moments of time at which the peak value of acceleration is attained are:
\[ t = 3, 9, 15, \ldots \text{ seconds} \]

**Final Answer:**
The moments of time at which the peak value of acceleration is attained are \( t = 3 \, \text{s}, 9 \, \text{s}, 15 \, \text{s}, \ldots \)

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