Question

The minute hand on a certain tower clock is 2 m long. Find the tangential speed of the tip of this hand.

Answer 1

To find the tangential speed of the tip of the minute hand on the clock, we can use the formula for tangential speed:

\[
v = r \cdot \omega
\]

where:
- \( v \) is the tangential speed,
- \( r \) is the radius (length of the minute hand), and
- \( \omega \) is the angular speed in radians per second.

1. **Finding the radius (r)**:
The length of the minute hand is given as \( r = 2 \) m.

2. **Finding the angular speed (ω)**:
The minute hand completes one full rotation (360 degrees, or \( 2\pi \) radians) in 60 minutes. To find the angular speed in radians per second:

\[
\text{Time for one rotation} = 60 \text{ minutes} = 60 \times 60 = 3600 \text{ seconds}
\]

Thus, the angular speed in radians per second is:

\[
\omega = \frac{2\pi \text{ radians}}{3600 \text{ seconds}} = \frac{\pi}{1800} \text{ radians/second}
\]

3. **Calculating the tangential speed (v)**:

Now we can substitute the values of \( r \) and \( \omega \) into the tangential speed formula:

\[
v = r \cdot \omega = 2 \text{ m} \cdot \frac{\pi}{1800} \text{ radians/second}
\]

Calculating this gives:

\[
v = \frac{2\pi}{1800} \text{ m/s} = \frac{\pi}{900} \text{ m/s}
\]

4. **Evaluating the speed** (optional):
Using \( \pi \approx 3.14159 \):

\[
v \approx \frac{3.14159}{900} \approx 0.00349 \text{ m/s}
\]

So, the tangential speed of the tip of the minute hand is approximately:

\[
\boxed{0.00349} \text{ m/s}
\] (or exactly \(\frac{\pi}{900}\) m/s).