Question

What is the angular speed of \((a)\) the second hand, \((b)\) the minute hand, and \((c)\) the hour hand of a watch?

Short answer

The angular speeds are: (a) second hand: 6 degrees per second, (b) minute hand: 0.1 degrees per second and (c) hour hand: 0.008333 degrees per second.

Step-by-step solution

Understand the Concept

Angular speed (also called angular velocity) is a measure of the speed of a rotation. It expresses the amount of degrees which an object spans in a certain amount of time. It can be calculated using the formula: \[\omega = \frac{Theta}{t}\] where omega (\(\omega\)) is the angular speed, Theta is the total angle through which the hand has moved and t is the time taken for the movement.

Calculate Angular Speed of the Second Hand

The second hand of a watch travels a full circle or 360º in 1 minute i.e. 60 seconds. We substitute these values in the above given formula. Thus, \[\omega = \frac{360}{60} = 6\] degrees per second.

Calculate Angular Speed of the Minute Hand

The minute hand of a watch travels a full circle or 360º in 60 minutes i.e. 3600 seconds. We substitute these values in the above given formula. Thus, \[\omega = \frac{360}{3600} = 0.1\] degrees per second.

Calculate Angular Speed of the Hour Hand

The hour hand of a watch travels a full circle or 360º in 12 hours i.e. 43200 seconds. We substitute this values in the above given formula. Thus, \[\omega = \frac{360}{43200} = 0.008333\] degrees per second.

Key concepts

Angular Velocity

Consider angular velocity as the rotational counterpart to linear velocity; it is the rate at which an object spins around a fixed point. Unlike linear velocity, which measures distance over time, angular velocity measures the angle through which an object rotates over a period of time. It is typically represented by the Greek letter omega \( \omega \).

In physics, angular velocity has critical relevance in systems involving circular or rotational movement, like the gears in a clock or the Earth rotating on its axis. It forms an essential aspect of understanding how quickly objects are rotating in mechanical and astrophysical contexts.

Rotational Motion

Rotational motion is any motion where an object spins around an internal axis or an external point, much like a Ferris wheel turning around its central support. This type of movement is ubiquitous in nature and man-made machines.

Understanding Rotation Periods and Frequencies

In studying rotational motion, two significant aspects are often discussed: the rotation period (the time it takes to make one complete revolution) and the rotational frequency (how many revolutions occur within a unit of time). These concepts help describe the tempo of a rotating system.

Physics of Time Measurement

The physics of time measurement, or horology, is based on periodic events such as the swing of a pendulum or the vibration of a crystal in a quartz watch. For mechanical watches, the rotational movement of the hands around the dial translates to the passage of time. The precise and regulated movement of these hands is essential for the accuracy of timekeeping.

Watches and clocks utilize gears with known rotation rates, governed by the predictable oscillations of the time-measuring element, to provide consistent time measurement. This accuracy is vital for navigation, astronomy, and many other scientific and practical endeavors.

Angular Speed Calculation

The formula for calculating angular speed is \( \omega = \frac{\text{Theta}}{t} \), where \( \omega \) is the angular speed, Theta represents the angular displacement (usually given in degrees or radians), and t denotes the time taken. To calculate the angular speed of a watch's hands, you would look at how far (the angle) the hands move in a certain amount of time.

Real-Life Application

Using the exercise as an example, you determine the angular velocity of a watch's second hand by dividing the full rotation angle (360 degrees) by the period it takes to complete one rotation (60 seconds). This calculation provides practical information for designing and repairing time-measuring devices.