The equation of a particle executing simple harmonic motion is given by `x = sin pi (t +1/3)` m At t = 1 s , the speed of particle will be `(" Given" pi = 3.14)` .

Options

A. `0 " cm s"^(-1)`
B. `157 " cm s"^(-1)`
C. `272 " cm s"^(-1)`
D. `314 " cms"^(-1)`

Explanation

To find the speed of the particle at \( t = 1 \) second, we start with the given equation of motion:

\[ x = \sin(\pi(t + \frac{1}{3})) \]

### Step 1: Differentiate the displacement equation to find the velocity
The velocity \( v \) is the time derivative of the displacement \( x \). Thus, we differentiate \( x \) with respect to \( t \):

\[ v = \frac{dx}{dt} = \frac{d}{dt} \left( \sin\left(\pi\left(t + \frac{1}{3}\right)\right)\right) \]

Using the chain rule, we get:

\[ v = \cos\left(\pi\left(t + \frac{1}{3}\right)) \cdot \frac{d}{dt}\left(\pi\left(t + \frac{1}{3}\right)\right) \]

The derivative of \( \pi(t + \frac{1}{3}) \) with respect to \( t \) is \( \pi \):

\[ v = \pi \cos\left(\pi\left(t + \frac{1}{3}\right)\right) \]

### Step 2: Substitute \( t = 1 \) second into the velocity equation
Now we substitute \( t = 1 \) into the velocity equation:

\[ v = \pi \cos\left(\pi\left(1 + \frac{1}{3}\right)\right) = \pi \cos\left(\pi \cdot \frac{4}{3}\right) \]

### Step 3: Calculate the cosine value
Next, we need to calculate \( \cos\left(\frac{4\pi}{3}\right) \). The angle \( \frac{4\pi}{3} \) is in the third quadrant where cosine is negative:

\[ \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2} \]

### Step 4: Substitute back to find the velocity
Now substituting this back into the velocity equation:

\[ v = \pi \left(-\frac{1}{2}\right) = -\frac{\pi}{2} \]

### Step 5: Find the magnitude of the speed
Since speed is the magnitude of velocity, we take the absolute value:

\[ |v| = \frac{\pi}{2} \]

### Step 6: Substitute the value of \( \pi \)
Using the given value of \( \pi = 3.14 \):

\[ |v| = \frac{3.14}{2} = 1.57 \text{ m/s} \]

### Step 7: Convert to centimeters per second
To convert meters per second to centimeters per second, we multiply by 100:

\[ |v| = 1.57 \times 100 = 157 \text{ cm/s} \]

Thus, the speed of the particle at \( t = 1 \) second is:

\[ \boxed{157 \text{ cm/s}} \]