Problem 3 - College Physics

Question

(a) Calculate the angular velocity (in rad/s) of the second, minute, and hour hands on a wall clock. (b) What is the period of each of these hands?

Step-by-Step Solution

Answer

(a) \( \omega_s = \frac{\pi}{30} \) rad/s, \( \omega_m = \frac{\pi}{1800} \) rad/s, \( \omega_h = \frac{\pi}{21600} \) rad/s. (b) Periods: 60 s, 3600 s, 43200 s.

Step 1: Understanding Angular Velocity

Angular velocity is a measure of how fast an object rotates or revolves relative to another point, typically the center of a circle. It is usually expressed in radians per second (rad/s). For a complete revolution, the angular displacement is \(2\pi\) radians.

Step 2: Calculating Angular Velocity of the Second Hand

The second hand completes one full revolution (\(2\pi\) radians) in 60 seconds. Thus, the angular velocity of the second hand \( \omega_s = \frac{2\pi}{60} \). So, \( \omega_s = \frac{\pi}{30} \) rad/s.

Step 3: Calculating Angular Velocity of the Minute Hand

The minute hand completes one full revolution in 3600 seconds (60 minutes). Thus, the angular velocity of the minute hand \( \omega_m = \frac{2\pi}{3600} \). Therefore, \( \omega_m = \frac{\pi}{1800} \) rad/s.

Step 4: Calculating Angular Velocity of the Hour Hand

The hour hand completes one full revolution in 43200 seconds (12 hours x 3600 seconds/hour). Thus, the angular velocity of the hour hand \( \omega_h = \frac{2\pi}{43200} \). Therefore, \( \omega_h = \frac{\pi}{21600} \) rad/s.

Step 5: Understanding Period

The period is the time it takes for one complete cycle of rotation, usually measured in seconds. It is the reciprocal of the angular speed when the speed is in cycles per second.

Step 6: Calculating the Period of the Second Hand

The period of the second hand is the time for one complete rotation, which is 60 seconds.

Step 7: Calculating the Period of the Minute Hand

The period of the minute hand is the time for one complete rotation, which is 3600 seconds (or 60 minutes).

Step 8: Calculating the Period of the Hour Hand

The period of the hour hand is the time for one complete rotation, which is 43200 seconds (or 12 hours).

Key Concepts

Period of Rotation

The period of rotation is a fundamental concept in rotational motion. It refers to the time taken for an object to complete one full cycle of rotation along its circular path. The period is measured in seconds and is represented by the symbol \( T \). Notably, in circular motion, the period can be determined by finding the inverse of the object's rotational speed, provided that the speed is given in revolutions per second (rps).
In our exercise, the period can vary among different objects based on their speed. For example:

• The second hand of a clock takes 60 seconds for a full revolution, so its period \( T_s = 60 \) seconds.
• The minute hand completes a full cycle in 3600 seconds (or 60 minutes), hence its period \( T_m = 3600 \) seconds.
• The hour hand rotates once every 43200 seconds (equivalent to 12 hours), meaning \( T_h = 43200 \) seconds.

This concept helps understand how rotational speed affects the time needed for one complete cycle, crucial in many applications including engineering and physics.

Radian Measure

Radian measure is an essential unit in describing angles and is widely used in mathematics and physics, particularly in circular motion problems. Unlike degrees, radians directly relate the arc length of a circle to its radius. One full revolution around a circle equates to an angular displacement of \(2\pi\) radians.
The advantage of using radians is clear when dealing with angular velocity calculations, as it provides a natural measure that aligns neatly with the properties of circles. When we describe angular velocities:

• A full circle is \(2\pi\) radians, meaning half a circle is \(\pi\) radians, and a quarter circle is \(\frac{\pi}{2}\) radians.
• Using radians simplifies formulas and conversions, such as going from linear to angular motion where arc length \(s = r \theta\) if \(\theta\) is measured in radians.

Understanding radian measure is vital in analyzing and solving rotational dynamics and motion-related problems.

Circular Motion

Circular motion refers to the movement of an object along a circular path and can be uniform or non-uniform based on the object's speed. In uniform circular motion, the speed of the object remains constant, but its direction changes continuously due to the curvature of the path.
The key aspects of circular motion include:

• Angular Displacement: The amount of rotation an object undergoes, which is measured in radians.
• Angular Velocity: The rate at which angular displacement changes, expressed in radians per second.
• Centripetal Force: The inward force necessary for an object to follow a circular path, dependent on the object's mass, speed, and the circle's radius.

Circular motion concepts are critical in a variety of fields, such as mechanical engineering, astronomy, and amusement park ride design. Understanding these concepts can aid in predicting object behaviors and designing systems involving rotative or cyclic actions.

Revolutions Per Second

Revolutions per second (rps) is a unit of rotational speed used to express the number of complete rotations an object makes around a central axis every second. It's a convenient way to express angular velocity, particularly in engineering and physics applications.
Rotational speed in rps can be easily converted to other units of angular velocity, such as radians per second, by using the understanding that one revolution corresponds to \(2\pi\) radians. Thus, having an object rotating at 1 rps means it has an angular velocity of \(2\pi\) rad/s.
Some key points to remember with rps include:

• To find the angular velocity in rad/s from rps, multiply the rps value by \(2\pi\).
• RPS is often preferred in applications where quick and clear understanding of rotational speed is required, such as in mechanical metering and process control systems.

Understanding revolutions per second and how it correlates with other units of measurement is crucial in the design and analysis of systems involving rotation.