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.
-
Finding the radius (r):
The length of the minute hand is given as \( r = 2 \) m. -
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:
\[ \omega = \frac{2\pi \text{ radians}}{3600 \text{ seconds}} = \frac{\pi}{1800} \text{ radians/second} \]
- 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:
\[ v = \frac{2\pi}{1800} \text{ m/s} = \frac{\pi}{900} \text{ m/s} \]
- 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).