A transverse wave ` y = 0.05 sin ( 20 pi x - 50 pi t)` meters , is propagating along + ve X - axis on a string light insect starts crawling on the string with velocity of 5 cm /s at t= 0 along the +ve X - axis from point where x = 5 cm . After 5 s the difference in phase of its position is equal to

Options

A. `150 pi `
B. ` 250 pi `
C. `-245 pi`
D. `-5pi`

Answer

To solve the problem, we need to find the difference in phase of the insect's position after 5 seconds while it crawls along the string. Let's break down the steps:

### Step 1: Identify the wave equation and parameters
The given wave equation is:
\[ y = 0.05 \sin(20\pi x - 50\pi t) \]
From this equation, we can identify:
- The angular wave number \( k = 20\pi \) (which relates to the spatial component)
- The angular frequency \( \omega = 50\pi \) (which relates to the time component)

### Step 2: Calculate the initial phase at \( t = 0 \) and \( x = 5 \) cm
At \( t = 0 \) and \( x = 5 \) cm (which is 0.05 m):
\[ \text{Initial phase } \Phi_i = 20\pi(0.05) - 50\pi(0) \]
Calculating this gives:
\[ \Phi_i = 20\pi(0.05) = 1\pi = 100\pi \text{ radians} \]

### Step 3: Calculate the distance crawled by the insect after 5 seconds
The insect crawls at a velocity of 5 cm/s. The distance crawled in 5 seconds is:
\[ \text{Distance} = \text{velocity} \times \text{time} = 5 \, \text{cm/s} \times 5 \, \text{s} = 25 \, \text{cm} \]
Thus, the final position of the insect after 5 seconds is:
\[ x_f = 5 \, \text{cm} + 25 \, \text{cm} = 30 \, \text{cm} \]

### Step 4: Calculate the final phase at \( t = 5 \) seconds and \( x = 30 \) cm
Now, we need to find the final phase at \( t = 5 \) seconds and \( x = 30 \) cm (which is 0.30 m):
\[ \text{Final phase } \Phi_f = 20\pi(0.30) - 50\pi(5) \]
Calculating this gives:
\[ \Phi_f = 20\pi(0.30) - 50\pi(5) \]
\[ \Phi_f = 6\pi - 250\pi = -244\pi \text{ radians} \]

### Step 5: Calculate the phase difference
The phase difference \( \Delta \Phi \) is given by:
\[ \Delta \Phi = \Phi_f - \Phi_i \]
Substituting the values we found:
\[ \Delta \Phi = (-244\pi) - (100\pi) = -344\pi \text{ radians} \]

### Final Result
The difference in phase of the insect's position after 5 seconds is:
\[ \Delta \Phi = -344\pi \text{ radians} \]