Question

Is the angular speed of the hour hand of a clock greater than, less than, or equal to the angular speed of the minute hand?

Short answer

The angular speed of the minute hand is greater than that of the hour hand.

Step-by-step solution

Understand Angular Speed

Angular speed refers to the rate at which an object rotates or revolves relative to another point, in this case, the center of the clock. Angular speed is generally measured in radians per unit of time.

Calculate the Angular Speed of the Hour Hand

The hour hand completes a full rotation (360 degrees or \(2\pi\) radians) in 12 hours. Therefore, the angular speed of the hour hand can be calculated using the formula: \( \text{angular speed} = \frac{\text{total radians}}{\text{time}} \). Substituting the values, we get \( \frac{2\pi}{12} \) radians per hour.

Calculate the Angular Speed of the Minute Hand

The minute hand completes a full rotation (360 degrees or \(2\pi\) radians) in 1 hour. Using the formula \( \text{angular speed} = \frac{\text{total radians}}{\text{time}} \), we get \( \frac{2\pi}{1} \) radians per hour for the minute hand.

Compare the Angular Speeds

By comparing the two angular speeds, the hour hand's speed is \( \frac{2\pi}{12} = \frac{\pi}{6} \) radians per hour, while the minute hand's speed is \(2\pi\) radians per hour. Clearly, \(2\pi\) is greater than \(\frac{\pi}{6}\), indicating that the angular speed of the minute hand is greater.

Key concepts

Radians

Radians are a unit of angular measure used in many areas of mathematics, particularly in trigonometry and calculus. One radian is the angle created when the radius of a circle is wrapped along its circumference. This relationship provides a direct link between the linear and rotational motion of objects.
In mathematical terms, a full circle has an angle of 360 degrees, which is equivalent to \(2\pi\) radians. This equivalence is crucial when calculating angular speed, as radians provide a simpler and more natural way to handle these measurements than degrees.
When calculating angular speed, particularly in exercises related to circular motion, using radians simplifies the calculations and provides more accurate results. For instance, when determining the angular speed of clock hands, converting the rotations into radians allows us to easily compare speeds using fundamental formulas like \( \text{angular speed} = \frac{\text{total radians}}{\text{time}} \).

Clock Mechanics

Clocks are a great example of understanding angular speed and motion in a practical context. A typical analog clock includes three hands: hour, minute, and sometimes second. Each hand travels in a circular path around the clock face.
The mechanics of a clock dictate that the hands complete their respective cycles at different rates. The hour hand takes 12 hours to complete one full revolution (360 degrees or \(2\pi\) radians), while the minute hand completes this rotation every hour. This difference is due to the job each hand is meant to perform—indicating hours, minutes, and sometimes seconds.
Through understanding these mechanics, we can calculate and compare the angular speeds of the different hands. Since the minute hand completes its cycle much faster than the hour hand, its angular speed is higher, demonstrating the application of angular motion principles in everyday devices.

Angular Motion

Angular motion involves the rotation of an object around a central point or axis. It is a fundamental concept in physics and engineering that helps explain how objects behave when they rotate or revolve.
There are several key terms related to angular motion:

• Angular Position: The angle at which an object is oriented relative to a reference direction at a given point in time.
• Angular Speed: The rate of change of the angle with respect to time, usually measured in radians per unit time.
• Angular Acceleration: The rate of change of angular speed over time.

Understanding these concepts is essential in calculating angular speed, like when determining which hand of a clock moves faster based on its angular velocity. When comparing angular speeds, as in the clock hand example, observe that the minute hand, with its quick completion of a full rotation in one hour, has a significantly higher angular speed than the slow-moving hour hand.