A particle is moving along a circle such that it completes one revolution in 40 seconds. In 2 minuts 20 seconds, the ratio `("|displacement|")/("distance")` is distance

Options

A. Zero
B. `(1)/(7)`
C. `(2)/(7)`
D. `(1)/(11)`

Answer

To solve the problem, we need to find the ratio of the magnitude of displacement to the distance traveled by a particle moving in a circle. Let's break it down step by step.

### Step 1: Determine the time taken for one complete revolution
The particle completes one revolution in 40 seconds.

### Step 2: Convert the total time into seconds
The total time given is 2 minutes and 20 seconds. We need to convert this into seconds:
\[ 2 \text{ minutes} = 2 \times 60 = 120 \text{ seconds} \]
So,
\[ 2 \text{ minutes and } 20 \text{ seconds} = 120 + 20 = 140 \text{ seconds} \]

### Step 3: Calculate the number of complete revolutions in 140 seconds
Since one revolution takes 40 seconds, we can find the number of complete revolutions in 140 seconds:
\[ \text{Number of revolutions} = \frac{140 \text{ seconds}}{40 \text{ seconds/revolution}} = 3.5 \text{ revolutions} \]

### Step 4: Calculate the distance traveled
The distance traveled in one complete revolution is the circumference of the circle, which is given by:
\[ \text{Distance for one revolution} = 2\pi r \]
For 3.5 revolutions, the distance traveled is:
\[ \text{Total distance} = 3.5 \times 2\pi r = 7\pi r \]

### Step 5: Calculate the displacement
Displacement is the shortest distance from the initial position to the final position. After 3.5 revolutions, the particle will be at the opposite side of the circle (half a revolution) from its starting point. Therefore, the displacement is:
\[ \text{Displacement} = 2r \]

### Step 6: Calculate the ratio of displacement to distance
Now we can find the ratio of the magnitude of displacement to the distance traveled:
\[ \text{Ratio} = \frac{|\text{Displacement}|}{\text{Distance}} = \frac{2r}{7\pi r} \]
The \( r \) cancels out:
\[ \text{Ratio} = \frac{2}{7\pi} \]

### Step 7: Simplify the ratio
Using the approximation \( \pi \approx \frac{22}{7} \):
\[ \text{Ratio} = \frac{2}{7 \times \frac{22}{7}} = \frac{2}{22} = \frac{1}{11} \]

### Conclusion
Thus, the ratio of the magnitude of displacement to the distance traveled is:
\[ \frac{1}{11} \]