Clock Angle Problem

Understanding how clock hands move is key to solving problems about the angle between them. Both the hour hand and the minute hand move at constant speeds, but their speeds are different.

Minute Hand Movement

The minute hand completes a full circle (360 degrees) in 60 minutes. Therefore, its speed is:

Speed of minute hand = $\frac{360^\circ}{60 \text{ minutes}} = 6^\circ \text{ per minute}$

At 3:15, the minute hand is exactly at the 15-minute mark, which corresponds to the '3' on the clock face.

Position of minute hand at 15 minutes from 12:00:

Minute hand angle = $15 \text{ minutes} \times 6^\circ/\text{minute} = 90^\circ$ from the 12 o'clock position.

Hour Hand Movement

The hour hand completes a full circle (360 degrees) in 12 hours. Therefore, its speed is:

Speed of hour hand = $\frac{360^\circ}{12 \text{ hours}} = 30^\circ \text{ per hour}$

In minutes, its speed is:

Speed of hour hand = $\frac{30^\circ}{60 \text{ minutes}} = 0.5^\circ \text{ per minute}$

At 3:15, the hour hand has moved past the '3' because of the 15 minutes past the hour. The total time elapsed since 12:00 is 3 hours and 15 minutes.

Angle moved by hour hand from 12:00:

Total minutes past 12:00 = $(3 \text{ hours} \times 60 \text{ minutes/hour}) + 15 \text{ minutes} = 180 + 15 = 195 \text{ minutes}$

Hour hand angle = $195 \text{ minutes} \times 0.5^\circ/\text{minute} = 97.5^\circ$ from the 12 o'clock position.

Calculating the Angle Between Hands

The angle between the hour and minute hand is the absolute difference between their angles measured from the 12 o'clock position.

Angle = $| \text{Hour Hand Angle} - \text{Minute Hand Angle} |$

Angle at 3:15 = $| 97.5^\circ - 90^\circ | = 7.5^\circ$

Thus, the angle between the hour and minute hand of a clock at 3h:15m is 7.5 degrees.

Summary of Angles

The difference is $97.5^\circ - 90^\circ = 7.5^\circ$.