The speed of the tip of a second hand of length 5 cm of a clock is ____ `ms^(-1)`

Options

A. 1
B. 60
C. `5.3 xx 10^(-3)`
D. `3.4 xx 10^(-5)`

Answer

To find the speed of the tip of a second hand of a clock with a length of 5 cm, we can follow these steps:

### Step 1: Understand the motion of the second hand
The second hand of a clock completes one full revolution (360 degrees or \(2\pi\) radians) in 60 seconds.

### Step 2: Calculate the angular speed
The angular speed (\(\omega\)) can be calculated using the formula:
\[ \omega = \frac{\text{Total angle in radians}}{\text{Time in seconds}} = \frac{2\pi \text{ radians}}{60 \text{ seconds}} = \frac{\pi}{30} \text{ radians/second} \]

### Step 3: Use the formula for linear speed
The linear speed (\(v\)) of the tip of the second hand can be calculated using the formula:
\[ v = \omega \times r \]
where \(r\) is the length of the second hand. Here, \(r = 5 \text{ cm} = 0.05 \text{ m}\).

### Step 4: Substitute the values into the formula
Now, substituting the values into the formula for linear speed:
\[ v = \left(\frac{\pi}{30}\right) \times 0.05 \]

### Step 5: Calculate the speed
Calculating the above expression:
\[ v = \frac{\pi \times 0.05}{30} = \frac{0.05\pi}{30} \approx \frac{0.157}{30} \approx 0.00523 \text{ m/s} \]

### Step 6: Final result
Thus, the speed of the tip of the second hand of a clock with a length of 5 cm is approximately:
\[ v \approx 5.23 \times 10^{-3} \text{ m/s} \]

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