Problem 27 - College Physics

Question

\(\bullet\) The wings of the Blue-throated Hummingbird (Lampornis clemenciae), which inhabits Mexico and the southwestern United States, beat at a rate of up to 900 times per minute. Calculate (a) the period of vibration of the bird's wings, (b) the frequency of the wings' vibration, and (c) the angular frequency of the bird's wingbeats.

Step-by-Step Solution

Answer

(a) 0.067 s, (b) 15 Hz, (c) 94.25 rad/s

Step 1: Calculate the Period of Vibration (a)

The period of vibration (T) is the time it takes for one complete cycle of wingbeats. To find the period, use the formula \( T = \frac{1}{f} \), where \( f \) is the frequency in hertz (cycles per second). Since the wings beat 900 times per minute, the frequency is \( f = \frac{900}{60} = 15 \) Hz. Thus, the period is \( T = \frac{1}{15} \approx 0.067 \) seconds.

Step 2: Confirm Frequency of Vibration (b)

The frequency \( f \) represents the number of wingbeats per second and is already calculated in the previous step as 15 Hz. This confirms that the frequency is \( 15 \) Hz.

Step 3: Calculate the Angular Frequency (c)

The angular frequency (\( \omega \)) is given by the formula \( \omega = 2\pi f \), where \( f \) is the frequency. Substituting \( f = 15 \) Hz, we have \( \omega = 2\pi \times 15 = 30\pi \approx 94.25 \) radians per second.

Key Concepts

Period of Vibration

Understanding the period of vibration is crucial in examining repetitive motions, like the wingbeats of a hummingbird. The period of vibration, denoted as \( T \), tells us how long one complete cycle or wingbeat takes. For example, in the hummingbird with a wingbeat frequency of 900 per minute, we first convert this rate to per second by dividing by 60, which gives us 15 wingbeats per second or 15 Hertz. To find the period \( T \), use the formula:
\[ T = \frac{1}{f} \]
where \( f \) is the frequency in Hertz.For our hummingbird, the calculation becomes:- \( f = 15 \) Hz- \( T = \frac{1}{15} \approx 0.067 \) secondsThis indicates that each wingbeat cycle takes about 0.067 seconds. Knowing the period helps in understanding how quickly a system can return to its starting position, which is vital not just in biology but in physics and engineering as well.

Frequency

Frequency, expressed in Hertz (Hz), measures how many cycles of a motion repeat in one second. It's a straightforward way to quantify how fast a repetitive action occurs, such as the 15 Hz frequency of our hummingbird's wings.
To determine frequency, divide the number of cycles (wingbeat in this case) by the duration in seconds. The steps include:

• Begin with a total rate (900 times per minute for the hummingbird).
• Convert minutes to seconds (\( 900 \text{ cycles/min} = \frac{900}{60} = 15 \text{ Hz} \)).

Frequency not only defines the rate of action but also translates this biological rhythm into a measurable figure, aiding in comparisons across different phenomena. For instance, one can compare the frequency of the hummingbird's wingbeats to a pendulum's swings or a motor's rotations.

Angular Frequency

Angular frequency, represented by \( \omega \), offers a perspective on how quickly an object moves through its angular cycle, useful for circular motions. In oscillatory systems, it's linked to the frequency \( f \) through this relation:
\[ \omega = 2\pi f \]This formula incorporates the mathematical constant \( \pi \) to reflect cycles in terms of radians per second. Radians measure the angle of rotation, making angular frequency indispensable in contexts involving periodic rotations like circular paths or cycles.Using the hummingbird's wing frequency of 15 Hz, calculate \( \omega \):

• \( \omega = 2\pi \times 15 = 30\pi \approx 94.25 \text{ radians/second} \)

This tells us the wings not only vibrate quickly but move through their cyclic path at a rotational speed of about 94.25 radians per second, showcasing the dynamism of their flight pattern. Understanding \( \omega \) is key in areas such as acoustics, electronics, and other harmonic motion applications.