Problem 3 - College Physics

Question

\(\bullet\) (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) Angular velocities: second hand = \( \frac{\pi}{30} \) rad/s, minute hand = \( \frac{\pi}{1800} \) rad/s, hour hand = \( \frac{\pi}{21600} \) rad/s. (b) Periods: second hand = 60 s, minute hand = 3600 s, hour hand = 43200 s.

Step 1: Understanding Angular Velocity

Angular velocity is the rate of change of the angular position of a rotating body. It is often measured in radians per second (rad/s). For objects that rotate in a circle, like the hands of a clock, the angular velocity can be calculated using the formula: \( \omega = \frac{2\pi}{T} \), where \( \omega \) is the angular velocity, and \( T \) is the period of rotation in seconds.

Step 2: Calculating Angular Velocity of the Second Hand

The second hand completes one full rotation every 60 seconds. Using the formula \( \omega = \frac{2\pi}{T} \), we substitute \( T = 60 \) seconds: \( \omega = \frac{2\pi}{60} = \frac{\pi}{30} \) rad/s.

Step 3: Calculating Angular Velocity of the Minute Hand

The minute hand completes one full rotation every 3600 seconds (60 minutes * 60 seconds per minute). Using the formula \( \omega = \frac{2\pi}{T} \), we substitute \( T = 3600 \) seconds: \( \omega = \frac{2\pi}{3600} = \frac{\pi}{1800} \) rad/s.

Step 4: Calculating Angular Velocity of the Hour Hand

The hour hand completes one full rotation every 43200 seconds (12 hours * 3600 seconds per hour). Using the formula \( \omega = \frac{2\pi}{T} \), we substitute \( T = 43200 \) seconds: \( \omega = \frac{2\pi}{43200} = \frac{\pi}{21600} \) rad/s.

Step 5: Understanding the Period

The period \( T \) is the time taken to complete one full cycle of rotation. It is the inverse of frequency \( f \), i.e., \( T = \frac{1}{f} \). For the second, minute, and hour hands, these are predetermined by their regular motion in a clock face.

Step 6: Calculating the Period of Each Hand

For the second hand, the period \( T = 60 \) seconds, for the minute hand \( T = 3600 \) seconds, and for the hour hand \( T = 43200 \) seconds. This corresponds to 1 minute, 1 hour, and 12 hours for their respective completion of one full circle.

Key Concepts

Period of Rotation

The period of rotation refers to the time it takes for an object in circular motion to make one complete cycle. For clock hands, the period is how long it takes each hand to return to its starting position. Therefore, understanding the period is crucial in determining the motion characteristics of rotating objects like clock hands.

• The second hand has a period of 60 seconds, implying it completes one full cycle every minute.
• The minute hand's period is 3600 seconds, which equates to one full rotation every hour.
• Finally, the hour hand has a period of 43,200 seconds, meaning it makes one full loop every 12 hours.

Each period reflects the consistent and predictable motion intrinsic to clock hands, embodying the repetitive cycle of time segmentation in everyday life.

Radians per Second

Radians per second is a unit of angular velocity that describes how fast an object is rotating. It's quintessential in circular motion as it quantifies the rotation rate without dealing with linear measurements.
When calculating angular velocity for clock hands, we employ the formula \[ \omega = \frac{2\pi}{T}, \] where \( \omega \) is angular velocity in radians per second and \( T \) is the period of rotation:

• For the second hand, using \( T = 60 \) seconds, we find \( \omega = \frac{\pi}{30} \) rad/s.
• For the minute hand, substituting \( T = 3600 \) seconds results in \( \omega = \frac{\pi}{1800} \) rad/s.
• For the hour hand, \( T = 43200 \) seconds gives \( \omega = \frac{\pi}{21600} \) rad/s.

These values reveal how swiftly each hand moves around the clock face, offering a clear picture of their angular motion.

Clock Hands

Clock hands are a common practical example of circular motion. Each hand moves at different speeds due to varying periods, offering an excellent model for understanding angular velocity and its applications:

• The second hand, relatively quick, finishes a rotation every minute.
• The minute hand, slower, requires a full hour for a complete cycle.
• The hour hand is the slowest, taking half a day to return to its starting position.

Understanding these distinctions helps in grasping the fundamentals of motion where each speed dictates how we measure and perceive time. These consistent motions can provide insight into the dynamics of periodicity and regularity in circular motion theories.

Circular Motion Analysis

Circular motion analysis involves examining objects revolving around a point or axis. Clock hands offer a fantastic illustration of such motion in action. They help demonstrate key principles of circular motion, such as angular velocity and rotational periods.
In a clock face:

• All hands pivot around the same central axis.
• They maintain a consistent orbital path at uniform speeds guided by their respective periods.
• Angular velocity differs, highlighting variations in their rotational motion.

Analyzing clock hands in this way not only elucidates concepts of angular velocity and rotational dynamics but also underscores the importance of periodicity. This analysis aids in understanding more complicated systems by providing a simple, everyday example that connects physics with timekeeping devices.