A particle executes SHM according to equation `x=10(cm)cos[2pit+(pi)/(2)]`, where t is in seconds. The magnitude of the velocity of the particle at `t=(1)/(6)s` will be :-

Options

A. 24.7 cm/s
B. 20.5 cm/s
C. 28.3 cm/s
D. 31.4 cm/s

Answer

To find the magnitude of the velocity of the particle executing Simple Harmonic Motion (SHM) given by the equation \( x = 10 \, \text{cm} \cos(2\pi t + \frac{\pi}{2}) \) at \( t = \frac{1}{6} \, \text{s} \), we will follow these steps:

### Step 1: Differentiate the position function to find the velocity function
The velocity \( v \) of the particle is given by the derivative of the position \( x \) with respect to time \( t \):
\[ v = \frac{dx}{dt} \]
Given \( x = 10 \cos(2\pi t + \frac{\pi}{2}) \), we differentiate:
\[ v = \frac{d}{dt}[10 \cos(2\pi t + \frac{\pi}{2})] \]

### Step 2: Apply the chain rule
Using the chain rule, we differentiate:
\[ v = 10 \cdot (-\sin(2\pi t + \frac{\pi}{2})) \cdot \frac{d}{dt}(2\pi t + \frac{\pi}{2}) \]
The derivative of \( (2\pi t + \frac{\pi}{2}) \) with respect to \( t \) is \( 2\pi \):
\[ v = 10 \cdot (-\sin(2\pi t + \frac{\pi}{2})) \cdot 2\pi \]
\[ v = -20\pi \sin(2\pi t + \frac{\pi}{2}) \]

### Step 3: Simplify the sine term
Using the sine addition formula, we know:
\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]
Thus,
\[ \sin(2\pi t + \frac{\pi}{2}) = \sin(2\pi t)\cos(\frac{\pi}{2}) + \cos(2\pi t)\sin(\frac{\pi}{2}) = 0 + \cos(2\pi t) = \cos(2\pi t) \]
So,
\[ v = -20\pi \cos(2\pi t) \]

### Step 4: Substitute \( t = \frac{1}{6} \, \text{s} \)
Now, we substitute \( t = \frac{1}{6} \):
\[ v = -20\pi \cos(2\pi \cdot \frac{1}{6}) = -20\pi \cos(\frac{\pi}{3}) \]
Since \( \cos(\frac{\pi}{3}) = \frac{1}{2} \):
\[ v = -20\pi \cdot \frac{1}{2} = -10\pi \]

### Step 5: Calculate the magnitude of the velocity
The magnitude of the velocity is:
\[ |v| = 10\pi \, \text{cm/s} \]
Using \( \pi \approx 3.14 \):
\[ |v| \approx 10 \cdot 3.14 = 31.4 \, \text{cm/s} \]

### Final Answer
The magnitude of the velocity of the particle at \( t = \frac{1}{6} \, \text{s} \) is approximately \( 31.4 \, \text{cm/s} \).

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