A clock has a hour hand of `5 cm` long find the magnitude of avg. velocity of tip of hour hand between `2: 00 a.m.` to `4.00 p.m`

Options

A. `30 cm//hr`
B. `(5pi)/(12) cm//hr`
C. `(5)/(14) cm//hr`
D. `(5pi)/(84). Cm//hr`

Explanation

To find the average velocity of the tip of the hour hand of a clock between 2:00 a.m. and 4:00 p.m., we can follow these steps:

### Step 1: Understand the motion of the hour hand
The hour hand of a clock makes a complete revolution in 12 hours. From 2:00 a.m. to 4:00 p.m., the hour hand moves from the 2 o'clock position to the 4 o'clock position.

### Step 2: Calculate the total time duration
The total time from 2:00 a.m. to 4:00 p.m. is:
- From 2:00 a.m. to 2:00 p.m. = 12 hours
- From 2:00 p.m. to 4:00 p.m. = 2 hours

Total time = 12 hours + 2 hours = 14 hours.

### Step 3: Determine the angular displacement
The hour hand moves from 2 to 4, which is a movement of 2 hours. Each hour corresponds to an angle of 30 degrees (360 degrees/12 hours). Therefore, the angular displacement from 2 to 4 is:
- Angular displacement = 2 hours × 30 degrees/hour = 60 degrees.

### Step 4: Calculate the linear displacement
The length of the hour hand is given as 5 cm. The linear displacement can be calculated using the formula:
\[ \text{Displacement} = \text{Length of hour hand} \times \sin(\theta) \]
Where \( \theta \) is the angle in radians. First, we convert 60 degrees to radians:
\[ 60 \text{ degrees} = \frac{\pi}{3} \text{ radians} \]

Now, calculate the displacement:
\[ \text{Displacement} = 5 \text{ cm} \times \sin\left(\frac{\pi}{3}\right) = 5 \text{ cm} \times \frac{\sqrt{3}}{2} \approx 4.33 \text{ cm} \]

However, since we are looking for the straight-line distance from the starting point (2 o'clock) to the ending point (4 o'clock), the displacement is simply the straight line between these two points, which is the horizontal distance of 5 cm.

### Step 5: Calculate the average velocity
Average velocity is defined as the total displacement divided by the total time taken:
\[ \text{Average Velocity} = \frac{\text{Total Displacement}}{\text{Total Time}} \]
Substituting the values we have:
\[ \text{Average Velocity} = \frac{5 \text{ cm}}{14 \text{ hours}} \approx 0.357 \text{ cm/hour} \]

### Final Answer
The magnitude of the average velocity of the tip of the hour hand between 2:00 a.m. and 4:00 p.m. is approximately **0.357 cm/hour**.

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