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

1 answer

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:

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

  1. 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:

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

  1. 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).