91Ó°ÊÓ

The period, often represented by the symbol \( T \), is a fundamental concept in the study of rotational motion. It describes the time required for a system to complete one full cycle of motion. In the context of a rotational system, like the fan blades of a jet engine, it indicates the time taken for one complete revolution.
To compute the period, you divide the total time taken by the number of complete cycles or revolutions. In our example, with the jet engine fan blades, the total time is \( 50.0 \) ms, which is equivalent to \( 0.050 \) seconds. With \( 1000 \) revolutions, the period can be calculated using the formula:
\[ T = \frac{t}{n} \]
where:

• \( t = 0.050 \) seconds (total time)
• \( n = 1000 \) revolutions (complete cycles)

Using these values in the formula, you find:
\[ T = \frac{0.050 \, \text{s}}{1000} = 0.00005 \, \text{s} \]
This indicates that each blade takes \( 0.00005 \) seconds to complete one full revolution.

Frequency in the context of rotational motion is defined as the number of cycles or revolutions per unit of time. It is a measure of how often an event occurs within a particular time frame and is expressed in hertz (Hz), where \( 1 \text{ Hz} = 1 \text{ cycle/second} \).
For rotational systems, there's a simple relationship between frequency \( f \) and period \( T \): They are reciprocals of each other. The formula to find frequency is:
\[ f = \frac{1}{T} \]
From the previous calculation, we know that the period \( T \) is \( 0.00005 \text{ s} \). Thus, the frequency is calculated as:
\[ f = \frac{1}{0.00005 \, \text{s}} = 20,000 \, \text{Hz} \]
This means the fan blades rotate \( 20,000 \) times each second, indicating a high-speed rotation typical of jet engines.

Angular frequency, denoted \( \omega \), is a measure of how quickly something rotates or oscillates and is expressed in radians per second. It's directly tied to the frequency that we've already determined.
The relationship is given by the formula:
\[ \omega = 2\pi f \]
where:

• \( \omega \) is the angular frequency in radians per second
• \( f \) is the regular frequency in hertz

The factor of \( 2\pi \) comes from the mathematical constant \( \pi \), since there are \( 2\pi \) radians in one complete cycle of a circle.
By using the calculated frequency, \( f = 20,000 \text{ Hz} \), the angular frequency becomes:
\[ \omega = 2\pi \times 20,000 = 40,000\pi \, \text{rad/s} \]
This high angular frequency reflects the rapid rotation characteristic of machinery such as jet engine fan blades.