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 cms^(-1)`
B. `5.3 cms^(-1)`
C. `6.3 cms^(-1)`
D. `7.3 cms^(-1)`

Explanation

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

### Step 1: Calculate the Circumference of the Circular Groove
The formula for the circumference \( C \) of a circle is given by:
\[ C = 2 \pi r \]
where \( r \) is the radius of the circle.

Given:
- Radius \( r = 12 \) cm

Substituting the value:
\[ C = 2 \pi \times 12 \text{ cm} = 24 \pi \text{ cm} \]

### Step 2: Calculate the Total Distance Traveled by the Insect
The insect completes 7 revolutions. Therefore, the total distance \( D \) traveled by the insect is:
\[ D = \text{Number of Revolutions} \times \text{Circumference} \]
Substituting the values:
\[ D = 7 \times 24 \pi \text{ cm} = 168 \pi \text{ cm} \]

### Step 3: Calculate the Time Taken
The total time \( t \) taken for the insect to complete these revolutions is given as:
\[ t = 100 \text{ seconds} \]

### Step 4: Calculate the Linear Speed
The linear speed \( v \) can be calculated using the formula:
\[ v = \frac{D}{t} \]
Substituting the values:
\[ v = \frac{168 \pi \text{ cm}}{100 \text{ s}} = \frac{168 \times 3.14}{100} \text{ cm/s} \]
Calculating this gives:
\[ v \approx \frac{526.32}{100} \text{ cm/s} \approx 5.26 \text{ cm/s} \]

### Final Answer
The linear speed of the insect is approximately:
\[ v \approx 5.26 \text{ cm/s} \]

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