Problem 25 - College Physics

Question

The wings of the blue-throated hummingbird, 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

Period ≈ 0.0667 s; Frequency = 15 Hz; Angular Frequency ≈ 94.2 rad/s.

Step 1: Understand the Problem

To tackle this problem, realize that you need to calculate three things related to the hummingbird wings' beat: the period of vibrations, the frequency, and the angular frequency, given that the wings beat 900 times per minute.

Step 2: Calculate the Period of Vibration

The period of vibration (T) is the reciprocal of the frequency. Since the wings beat 900 times per minute, convert this rate to beats per second to find the frequency.\[ T = \frac{1}{f} \]First, convert beats per minute to beats per second, so:\[ f = \frac{900}{60} = 15 \text{ beats per second} \]Now calculate the period:\[ T = \frac{1}{15} \approx 0.0667 \text{ seconds} \]

Step 3: Calculate the Frequency of Vibration

The frequency (f) of vibration is already calculated during the conversion from beats per minute to beats per second:\[ f = 15 \text{ Hz} \]This means the frequency is 15 hertz, as there are 15 beats of the wings each second.

Step 4: Calculate the Angular Frequency

The angular frequency (ω) is calculated using the formula:\[ \omega = 2\pi f \]Substitute the value of frequency:\[ \omega = 2\pi \times 15 = 30\pi \]So, the angular frequency is approximately 94.2 radians per second (as \( \pi \approx 3.1416 \)).

Key Concepts

Period of Vibration

When we talk about the period of vibration, we are discussing how long it takes to complete one full cycle of motion. For the wings of the blue-throated hummingbird, this means one complete up and down beat of the wings. The period is typically denoted by the symbol \( T \). If you know how many cycles happen in a certain time, you can find the period by taking the reciprocal of the frequency.

Here's a simple way to think about it:

• If something vibrates quickly, it has a short period.
• Conversely, if something vibrates slowly, it has a long period.

In the case of the hummingbird, its wings beat 900 times in one minute, which converts to 15 beats per second when divided by 60 (the number of seconds in a minute). Thus, each beat is roughly every 0.0667 seconds. That's your period! So each wingbeat cycle is fairly quick, indicating the amazing speed of the tiny bird's wings.

Frequency

Frequency is a measure of how often something happens over a set period of time. It's like keeping track of how many ticks a clock makes in a single second. In our context of harmonic motion, frequency tells us how many cycles of motion a wing goes through in one second. The unit for frequency is hertz (Hz), named after Heinrich Hertz, a pioneer in the study of electromagnetism.

For the hummingbird, converting from 900 beats per minute gives us 15 beats every second, hence a frequency of 15 Hz. Here are some useful points to remember:

• High-frequency means many cycles per second.
• Low-frequency means fewer cycles per second.

Understanding frequency helps us appreciate how rapid and energetic the hummingbird's wing movements are. With a frequency of 15 Hz, these birds truly showcase nature's engineering marvel in flight efficiency and speed.

Angular Frequency

Angular frequency, denoted by \( \omega \), adds a twist to our understanding of motion cycles by considering rotation. Instead of thinking linearly, angular frequency helps us to conceptualize rotations in radians per second.

We calculate angular frequency using the formula \( \omega = 2\pi f \), where \( f \) is the frequency.

• \( 2\pi \) comes from the mathematics of circles, as a full circle is \( 2\pi \) radians.
• This means \( \omega \) gives us a circular perspective on the frequency of cycles per second.

For the blue-throated hummingbird with a frequency of 15 Hz, our calculation becomes \( \omega = 2\pi \times 15 \). The resulting angular frequency is approximately 94.2 radians per second. This value not only highlights the pace of the hummingbird's wings but also reflects the elegant connection between linear and rotational motion principles.