Problem 24 - College Physics

Question

\(\bullet\) Find the period, frequency, and angular frequency of (a) the second hand and (b) the minute hand of a wall clock.

Step-by-Step Solution

Answer

The period of the second hand is 60 seconds, the frequency is \( \frac{1}{60} \) Hz, and the angular frequency is \( \frac{\pi}{30} \) rad/s. The period of the minute hand is 3600 seconds, the frequency is \( \frac{1}{3600} \) Hz, and the angular frequency is \( \frac{\pi}{1800} \) rad/s.

Step 1: Understanding the Concept of Period

The period of a repeating event is the duration of one complete cycle. In the case of a clock, the second hand completes one full cycle in 60 seconds, and the minute hand completes one full cycle in 60 minutes, or 3600 seconds.

Step 2: Calculating the Period of the Second Hand

The second hand of a clock completes a full cycle every 60 seconds. Therefore, its period is 60 seconds.

Step 3: Calculating the Period of the Minute Hand

The minute hand of a clock completes a full cycle every 3600 seconds (60 minutes multiplied by 60 seconds per minute). Therefore, its period is 3600 seconds.

Step 4: Determining Frequency of the Second Hand

The frequency is the reciprocal of the period. For the second hand, frequency \( f \) can be calculated as \( f = \frac{1}{T} = \frac{1}{60} \) Hz.

Step 5: Determining Frequency of the Minute Hand

For the minute hand, the frequency \( f \) is \( f = \frac{1}{T} = \frac{1}{3600} \) Hz.

Step 6: Calculating Angular Frequency of the Second Hand

Angular frequency \( \omega \) is given by the formula \( \omega = 2\pi f \). For the second hand, \( \omega = 2\pi \times \frac{1}{60} \), which simplifies to \( \frac{\pi}{30} \) radians per second.

Step 7: Calculating Angular Frequency of the Minute Hand

For the minute hand, \( \omega = 2\pi \times \frac{1}{3600} \), which simplifies to \( \frac{\pi}{1800} \) radians per second.

Key Concepts

Frequency Calculation

Calculating frequency involves understanding how often a repeating event occurs within a specific timeframe. Frequency is denoted by the symbol \( f \) and is measured in hertz (Hz), where one hertz equates to one cycle per second. In the context of a clock:

• The second hand completes a full rotation every 60 seconds. Therefore, its frequency is determined using the formula \( f = \frac{1}{T} \), where \( T \) is the period. So, for the second hand, \( f = \frac{1}{60} \) Hz.
• Similarly, the minute hand takes 3600 seconds to complete a cycle, making its frequency \( f = \frac{1}{3600} \) Hz.

And here's a tip: the larger the period, the smaller the frequency, as they are inversely proportional. Using these principles, you can calculate the frequency of any oscillating system by first determining its period and then applying the simple reciprocal formula.

Angular Frequency

Angular frequency is a concept that describes how fast something rotates or oscillates in relation to a full circle, measured in radians per second. It is given by the formula \( \omega = 2\pi f \), where \( \omega \) is angular frequency and \( f \) is the frequency.

• For the second hand of a clock, with a frequency of \( \frac{1}{60} \) Hz, the angular frequency becomes \( \omega = 2\pi \times \frac{1}{60} = \frac{\pi}{30} \) radians per second.
• For the minute hand, \( \omega = 2\pi \times \frac{1}{3600} = \frac{\pi}{1800} \) radians per second.

Angular frequency provides insight into how quickly an object is oscillating in terms of angles, which is particularly useful in systems involving circular motion or harmonic oscillators. It offers a different perspective from linear frequency, emphasizing the rotational aspect of motion.

Clocks and Timekeeping

Clocks are excellent examples of oscillatory systems in everyday life, utilizing consistent oscillations to keep track of time. The second and minute hands exhibit periodic motion, making it easier to calculate not only their period but also their frequency and angular frequency.

• The second hand's 60-second period indicates a slow but steady pace, allowing us to measure seconds accurately using its known frequency and angular frequency.
• The minute hand's 3600-second period aids in tracking longer intervals, such as minutes.

Understanding the mechanism of clocks helps in appreciating the accuracy and precision provided by these devices. The regulated motion of the hands, governed by precise periods and frequencies, ensures that time is measured uniformly across various timepieces, aiding everything from global communication to daily scheduling tasks.