What are the angular velocities of the hour-hand and minute-hand of a clock?
Understand the Problem
The question is asking for the rates at which the hour hand and the minute hand of a clock rotate, which are measured in degrees per unit of time. To solve this, we'll calculate the angular velocities for both hands based on the fact that a clock face is a circular motion with specific time intervals for each hand.
Answer
The hour hand rotates at \(30\) degrees per hour, and the minute hand rotates at \(360\) degrees per hour.
Answer for screen readers
The rate of the hour hand is (30) degrees per hour, and the rate of the minute hand is (360) degrees per hour.
Steps to Solve
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Calculate the rotation of the hour hand The hour hand completes one full rotation (360 degrees) in 12 hours. Thus, its rate of rotation is given by: $$ \text{Rate of hour hand} = \frac{360 \text{ degrees}}{12 \text{ hours}} = 30 \text{ degrees per hour} $$
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Calculate the rotation of the minute hand The minute hand completes one full rotation (360 degrees) in 60 minutes (or 1 hour). Therefore, its rate of rotation is: $$ \text{Rate of minute hand} = \frac{360 \text{ degrees}}{60 \text{ minutes}} = 6 \text{ degrees per minute} $$
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Convert the minute hand's rate to degrees per hour Since we often compare both rates in terms of hours, convert the minute hand’s rate: $$ \text{Rate of minute hand in degrees per hour} = 6 \text{ degrees/min} \times 60 \text{ min/hour} = 360 \text{ degrees/hour} $$
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Summarize the rates of rotation Now we have both rates:
- Hour hand: 30 degrees per hour
- Minute hand: 360 degrees per hour
The rate of the hour hand is (30) degrees per hour, and the rate of the minute hand is (360) degrees per hour.
More Information
The hour hand moves much slower than the minute hand, which highlights how quickly the minute hand sweeps around the clock face compared to the hour hand's gradual movement.
Tips
A common mistake is forgetting to convert the units properly, especially when calculating the minute hand’s rate in terms of hours. Ensure that all rates are compared in the same time unit.
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