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:

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