An insect trapped in a circular groove of radius, 12 cm moves along the groove steadily and completes 7 revolutions in 100s. What is the angular speed and the linear speed of the motion ?

Answer

To solve the problem step by step, we will calculate both the angular speed and the linear speed of the insect moving in the circular groove.

### Step 1: Calculate the Angular Speed
The angular speed (ω) can be calculated using the formula:
\[ \omega = \frac{\Delta \theta}{\Delta t} \]
where \(\Delta \theta\) is the change in angle in radians and \(\Delta t\) is the time in seconds.

Given that the insect completes 7 revolutions in 100 seconds, we first need to convert revolutions to radians. Since one revolution is \(2\pi\) radians, we have:
\[ \Delta \theta = 7 \text{ revolutions} \times 2\pi \text{ radians/revolution} = 14\pi \text{ radians} \]

Now, substituting the values into the formula:
\[ \omega = \frac{14\pi \text{ radians}}{100 \text{ seconds}} = \frac{14\pi}{100} \text{ radians/second} = 0.14\pi \text{ radians/second} \]

### Step 2: Calculate the Linear Speed
The linear speed (v) can be calculated using the formula:
\[ v = \omega \times r \]
where \(r\) is the radius of the circular path.

Given that the radius \(r = 12 \text{ cm}\), we need to convert this into meters:
\[ r = 12 \text{ cm} = 12 \times 10^{-2} \text{ m} = 0.12 \text{ m} \]

Now substituting the values of \(\omega\) and \(r\) into the formula:
\[ v = (0.14\pi \text{ radians/second}) \times (0.12 \text{ m}) \]
Calculating this gives:
\[ v = 0.0168\pi \text{ m/s} \]

### Final Answers
- Angular Speed (ω): \(0.14\pi \text{ radians/second}\)
- Linear Speed (v): \(0.0168\pi \text{ m/s}\)