The phase difference between two SHM `y_(1) = 10 sin (10 pi t + (pi)/(3))` and `y_(2) = 12 sin (8 pi t + (pi)/(4))` at `t = 0.5s` is

Options

A. `(11 pi)/(12)`
B. `(13 pi)/(12)`
C. `pi`
D. `(17 pi)/(12)`

Explanation

To find the phase difference between the two simple harmonic motions (SHM) given by the equations \( y_1 = 10 \sin(10 \pi t + \frac{\pi}{3}) \) and \( y_2 = 12 \sin(8 \pi t + \frac{\pi}{4}) \) at \( t = 0.5 \) seconds, we will follow these steps:

### Step 1: Identify the Phase of Each SHM
The phase of the first SHM \( y_1 \) is given by:
\[ \phi_1 = 10 \pi t + \frac{\pi}{3} \]
The phase of the second SHM \( y_2 \) is given by:
\[ \phi_2 = 8 \pi t + \frac{\pi}{4} \]

### Step 2: Substitute the Given Time \( t = 0.5 \) seconds
Now, we will substitute \( t = 0.5 \) into both phases.

For \( y_1 \):
\[ \phi_1 = 10 \pi (0.5) + \frac{\pi}{3} = 5\pi + \frac{\pi}{3} \]

For \( y_2 \):
\[ \phi_2 = 8 \pi (0.5) + \frac{\pi}{4} = 4\pi + \frac{\pi}{4} \]

### Step 3: Simplify the Phases
Now we will simplify both expressions.

For \( \phi_1 \):
\[ \phi_1 = 5\pi + \frac{\pi}{3} = \frac{15\pi}{3} + \frac{\pi}{3} = \frac{16\pi}{3} \]

For \( \phi_2 \):
\[ \phi_2 = 4\pi + \frac{\pi}{4} = \frac{16\pi}{4} + \frac{\pi}{4} = \frac{17\pi}{4} \]

### Step 4: Calculate the Phase Difference
The phase difference \( \Delta \phi \) is given by:
\[ \Delta \phi = \phi_1 - \phi_2 \]
Substituting the values we calculated:
\[ \Delta \phi = \frac{16\pi}{3} - \frac{17\pi}{4} \]

### Step 5: Find a Common Denominator and Simplify
To subtract these fractions, we need a common denominator. The least common multiple of 3 and 4 is 12.

Convert \( \frac{16\pi}{3} \) and \( \frac{17\pi}{4} \) to have a denominator of 12:
\[ \frac{16\pi}{3} = \frac{64\pi}{12} \]
\[ \frac{17\pi}{4} = \frac{51\pi}{12} \]

Now, substituting back into the phase difference:
\[ \Delta \phi = \frac{64\pi}{12} - \frac{51\pi}{12} = \frac{13\pi}{12} \]

### Final Answer
Thus, the phase difference between the two SHMs at \( t = 0.5 \) seconds is:
\[ \Delta \phi = \frac{13\pi}{12} \]

---