An insect trapped in circular groove of radius 12 cm moves along the groove steadily and completes 7 revolutions in 100s. The linear speed of the insect is

Options

A. 4.3 cm `s^(-1)`
B. 5.3 cm `s^(-1)`
C. 6.3 cm `s^(-1)`
D. 7.3 cm `s^(-1)`

Explanation

To find the linear speed of the insect moving in a circular groove, we can follow these steps:

### Step 1: Understand the given data
- Radius of the circular groove (r) = 12 cm
- Number of revolutions (N) = 7
- Time taken (t) = 100 s

### Step 2: Calculate the frequency (ν)
Frequency (ν) is defined as the number of revolutions per second. We can calculate it using the formula:
\[ \nu = \frac{N}{t} \]
Substituting the values:
\[ \nu = \frac{7 \text{ revolutions}}{100 \text{ seconds}} = 0.07 \text{ revolutions/second} \]

### Step 3: Convert frequency to angular velocity (ω)
The angular velocity (ω) in radians per second can be calculated using the formula:
\[ \omega = 2\pi\nu \]
Substituting the value of ν:
\[ \omega = 2\pi \times 0.07 \approx 0.4398 \text{ radians/second} \]

### Step 4: Calculate the linear speed (V)
The linear speed (V) can be calculated using the formula:
\[ V = r \cdot \omega \]
Substituting the values of r and ω:
\[ V = 12 \text{ cm} \times 0.4398 \text{ radians/second} \approx 5.2776 \text{ cm/second} \]

### Step 5: Round off the answer
Rounding off the linear speed to two decimal places, we get:
\[ V \approx 5.28 \text{ cm/second} \]

### Final Answer
The linear speed of the insect is approximately **5.28 cm/s**.

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