Two simple harmonic are represented by the equation `y_(1)=0.1 sin (100pi+(pi)/3) and y_(2)=0.1 cos pit`.
The phase difference of the velocity of particle 1 with respect to the velocity of particle 2 is.

Options

A. (a) `(pi)/3`
B. (b) `(-pi)/6`
C. (c ) `(pi)/6`
D. (d) `(-pi)/3`

Explanation

To find the phase difference of the velocity of particle 1 with respect to the velocity of particle 2, we will follow these steps:

### Step 1: Identify the equations of motion
We have two simple harmonic motions given by:
- \( y_1 = 0.1 \sin(100\pi t + \frac{\pi}{3}) \)
- \( y_2 = 0.1 \cos(\pi t) \)

### Step 2: Differentiate to find the velocities
The velocity of a particle in simple harmonic motion can be found by differentiating the displacement with respect to time.

For particle 1:
\[ v_1 = \frac{dy_1}{dt} = 0.1 \cdot 100\pi \cos(100\pi t + \frac{\pi}{3}) = 10\pi \cos(100\pi t + \frac{\pi}{3}) \]

For particle 2:
\[ v_2 = \frac{dy_2}{dt} = -0.1 \cdot \pi \sin(\pi t) = -0.1\pi \sin(\pi t) \]

### Step 3: Rewrite the velocity of particle 2 in terms of cosine
We can express the sine function in terms of cosine:
\[ v_2 = -0.1\pi \sin(\pi t) = -0.1\pi \cos\left(\pi t - \frac{\pi}{2}\right) \]

### Step 4: Identify the phases of the velocities
Now we can identify the phases of both velocities:
- The phase of \( v_1 \) is \( \frac{\pi}{3} \).
- The phase of \( v_2 \) is \( -\frac{\pi}{2} \) (since \( \cos(x) \) has a phase shift of \( -\frac{\pi}{2} \) when rewritten from sine).

### Step 5: Calculate the phase difference
To find the phase difference of \( v_1 \) with respect to \( v_2 \), we calculate:
\[ \text{Phase difference} = \text{Phase of } v_1 - \text{Phase of } v_2 = \frac{\pi}{3} - \left(-\frac{\pi}{2}\right) \]
\[ = \frac{\pi}{3} + \frac{\pi}{2} \]

### Step 6: Find a common denominator and simplify
To combine these fractions, we find a common denominator:
\[ \frac{\pi}{3} = \frac{2\pi}{6}, \quad \frac{\pi}{2} = \frac{3\pi}{6} \]
Thus,
\[ \text{Phase difference} = \frac{2\pi}{6} + \frac{3\pi}{6} = \frac{5\pi}{6} \]

### Step 7: Final phase difference
The phase difference of the velocity of particle 1 with respect to the velocity of particle 2 is:
\[ \text{Phase difference} = \frac{5\pi}{6} \]