A table clock has its minute hand 4.0 cm long. find the average velocity of the tip of the minute hand (a) between 6am to 6.30am and (b) between 6am to 6.30pm
angular velocity constant = omega = 2 pi radians/3600 seconds
tip speed is also constant = omega * r
To find the average velocity of the tip of the minute hand, we need to calculate the total displacement and divide it by the total time elapsed.
(a) Between 6am to 6.30am:
In 30 minutes, the minute hand completes one-half of a revolution, which is 180 degrees.
The displacement of the minute hand is equal to the arc length it covers.
The circumference of a circle is given by 2πr, where r is the radius of the circle (in this case, the length of the minute hand).
So, the circumference of the clock is 2π(4.0 cm) = 8π cm.
The displacement of the minute hand in radians is therefore (180/360) * 2π = π radians.
To convert this to centimeters, we need to multiply by the radius:
Displacement = π * 4.0 cm = 4π cm.
The time elapsed is 30 minutes.
The average velocity is displacement divided by time: (4π cm) / (30 min) = (4π/30) cm/min. This is the average velocity of the tip of the minute hand between 6am and 6.30am.
(b) Between 6am to 6.30pm:
In 12 hours (or 720 minutes), the minute hand completes 12 revolutions.
The displacement of the minute hand is equal to the arc length it covers.
The circumference of the clock is 8π cm (as calculated earlier).
The displacement of the minute hand in radians is therefore (12/1) * 2π = 24π radians.
To convert this to centimeters, we need to multiply by the radius:
Displacement = 24π * 4.0 cm = 96π cm.
The time elapsed is 720 minutes.
The average velocity is displacement divided by time: (96π cm) / (720 min) = (96π/720) cm/min. This is the average velocity of the tip of the minute hand between 6am and 6.30pm.