Question

The length of the second's hand in a watch is 1 cm. The magnitude of the change in the velocity of its tip in 30 seconds (in cm/s) is:

• A. \( \frac{\pi}{30} \)
• B. \( \frac{\sqrt{2\pi}}{30} \)
• C. \( \frac{\sqrt{2\pi}}{15} \)
• D. \( \frac{\pi}{15} \)
• E. \( \frac{\pi}{30\sqrt{2}} \)

Hint:

For circular motion, the tangential velocity can be found by multiplying the radius with angular velocity.

Answer 1

The Correct Option is D

We are given that the length of the second's hand of the watch is \( r = 1 \) cm. The second's hand describes a circular motion, so the velocity of its tip can be calculated using the formula for the magnitude of the linear velocity in circular motion: \[ v = r \omega \] where \( r \) is the radius (length of the second's hand), and \( \omega \) is the angular velocity. 
Step 1: Calculate the angular velocity \( \omega \). Since the second's hand completes one full revolution (i.e., \( 2\pi \) radians) in 60 seconds, the angular velocity is: \[ \omega = \frac{2\pi}{T} \] where \( T = 60 \) seconds is the time period for one revolution. Therefore, \[ \omega = \frac{2\pi}{60} = \frac{\pi}{30} { radians per second}. \] 
Step 2: Find the velocity at \( t = 0 \) and \( t = 30 \) seconds. The velocity of the tip of the second's hand at any time \( t \) is given by: \[ v(t) = r \cdot \omega \] At \( t = 0 \), the velocity is: \[ v(0) = 1 \times \frac{\pi}{30} = \frac{\pi}{30} { cm/s}. \] At \( t = 30 \), the second's hand has moved halfway around the circle, and the direction of the velocity has changed. Since the hand has rotated by \( \pi \) radians, the magnitude of the velocity is still \( \frac{\pi}{30} { cm/s} \), but the direction has changed. 
Step 3: Calculate the change in velocity. The change in velocity, \( \Delta v \), is the difference in the vectors of velocity at \( t = 0 \) and \( t = 30 \). Since the direction of the velocity vector has changed by \( 180^\circ \) (half a circle), the change in velocity is given by: \[ \Delta v = 2v = 2 \times \frac{\pi}{30} = \frac{\pi}{15}. \] 
Thus, the magnitude of the change in the velocity of the tip of the second's hand in 30 seconds is \( \frac{\pi}{15} { cm/s} \).