If the length of second`s hand in a clock is 4 cm,the angular velocity and linear velocity of the tip is

Options

A. `pi/30 rad s^-1,pi/750m/s`
B. `pi/6 rad s^-1,pi/1500m/s`
C. `pi/15 rad s^-1,pi/375m/s`
D. `pi/12 rads^-1,pi/350m/s`

Answer

To solve the problem of finding the angular velocity and linear velocity of the tip of the second's hand of a clock, we can follow these steps:

### Step 1: Understand the problem
We are given the length of the second's hand of a clock, which is 4 cm. We need to find both the angular velocity and the linear velocity of the tip of the second's hand.

### Step 2: Determine the angular velocity
The second's hand completes one full rotation (360 degrees) in 60 seconds.

- **Calculate the angle traversed in one second:**
\[ \text{Angle traversed in 1 second} = \frac{360 \text{ degrees}}{60 \text{ seconds}} = 6 \text{ degrees} \]

- **Convert degrees to radians:**
We know that \(180 \text{ degrees} = \pi \text{ radians}\). Therefore, to convert 6 degrees to radians:
\[ \text{Angle in radians} = 6 \times \frac{\pi}{180} = \frac{\pi}{30} \text{ radians} \]

- **Calculate angular velocity (\(\omega\)):**
Angular velocity is defined as the angle traversed per unit time:
\[ \omega = \frac{\theta}{t} = \frac{\frac{\pi}{30} \text{ radians}}{1 \text{ second}} = \frac{\pi}{30} \text{ radians/second} \]

### Step 3: Determine the linear velocity
The linear velocity (\(v\)) of the tip of the second's hand can be calculated using the formula:
\[ v = r \cdot \omega \]
where \(r\) is the radius (length of the second's hand).

- **Convert the length of the second's hand to meters:**
\[ r = 4 \text{ cm} = \frac{4}{100} \text{ meters} = 0.04 \text{ meters} \]

- **Substituting the values into the linear velocity formula:**
\[ v = 0.04 \text{ m} \cdot \frac{\pi}{30} \text{ radians/second} \]
\[ v = \frac{0.04\pi}{30} \text{ m/s} \]
\[ v = \frac{\pi}{750} \text{ m/s} \]

### Final Results
- **Angular Velocity (\(\omega\))**: \(\frac{\pi}{30} \text{ radians/second}\)
- **Linear Velocity (\(v\))**: \(\frac{\pi}{750} \text{ m/s}\)