Waves and Vibration Problems

ACTIVITY 1: Cut out the boxes below and re-arrange them so that the solution to the problem makes sense. Tape or glue or staple the re-arranged boxes onto another piece of paper. Use the front and back of the extra paper, if possible, to conserve paper.

Example Problem 1: A hummingbird beats its wings up and down with a frequency of 80.0Hz. What is the period of the hummingbird’s flaps?

To solve this problem, we use the period/frequency formula. Since we want to find period, we use the version that is solved for T: . Thus, we find that it takes a hummingbird 0.0125s to complete one flap of its wings. We want to determine the period of a hummingbird’s wing flap. Plugging into the formula we get: . We know that the frequency of the hummingbird’s wing flap is 80.0Hz.

Example Problem 2: A hypnotist swings her watch from 20.0cm chain in front of a subject’s eyes. What is the period of the swing of the watch.

Thus, we see that the pendulum used by the hypnotist has a period of 0.898s. Before we can use this formula, however, we must ensure all our variables are in the correct units. The length variable in this formula should be in meters, however our length is in centimeters. Converting centimeters to meters we get: . We know that a watch swinging back and forth on a chain is a form of a pendulum and that the length of that pendulum is 20.0cm. Also, since we are assuming that the hypnotist is on Earth, we know that the acceleration due to gravity is 9.8m/s2. Now we are ready to plug into the formula and simplify to get our answer: .` Fortunately, there is a formula specifically for finding the period of a pendulum. We will use the original version of this formula, since it is already solved for T: . We want to find the period at which a watch swings back and forth on the end of a chain.

Example Problem 3: A spider swings in the breeze from a silk thread with a period of 0.6s. How long is the spider’s strand of silk?

To solve this we use the formula for the period of a pendulum. Since we are trying to find the length, we use the version of this formula that is solved for L: . We can see that the spider at the end of the strand makes one complete swing, back and forth, every 0.6s. Plugging in and simplifying, we get:

Thus, we now know, that the length of the spider’s silk is 0.089m. We want to know how long a strand of spider’s silk is.

Example Problem 4: Radio Station WKLB in Boston broadcasts at a frequency of 99.5 MHz. If there are 106 Hertz in every MHz, and the speed of radio waves is 3.00x108m/s, then what is the wavelength of these waves emitted by WKLB? In order to determine the wavelength of a wave, we use the wavespeed formula. Since we are seeking wavelength, we use the version solved for : .

Therefore, we find that the radio waves emitted by WKLB have a wavelength of 3.02m.

However, before we use this formula, we need to convert the frequency from MHz to Hz. To do so, we use the unit conversion 1MHz=106Hz: .

We know that the speed of these waves is 3.00x108m/s and that the frequency of the waves is 99.5 MHz.

Now that the units are all correct, we plug in and simplify to find the answer: .

We want to figure out the wavelength of radio waves emitted by station WKLB.

ACTIVITY 2: For each of the example problems below, the mathematical steps to find the answer are shown. On your own paper, write up the formal solution to each problem, complete with written explanations like the ones you see in Activity 1.

Example Problem 5: Terry jumps up and down on a trampoline with a frequency of 1.5Hz. What is the period of Terry’s jumping?

Example Problem 6: A pendulum swings with a frequency of 0.5Hz. How long is the pendulum? Example Problem 7: The note middle C, as a sound wave in air, has a wavelength of approximately 1.3m. If sound waves in air have a wavespeed of 340m/s, what is the period of this wave?

ACTIVITY 3: Write a formal solution to each problem below. Make sure your formal solutions have a clear introduction, body and conclusion.

1. A piano tuner taps his 440Hx tuning fork with a mallet. What is the period of the vibrating tuning fork? 2. Denny jumps up and down on his bed, taking 0.75s for each jump. What is the frequency of Denny’s jumping? 3. A monkey swings from a jungle vine by his 0.30m-long tail. A) What is the period of the monkey’s swing? B) With what frequency does the monkey swing? (Write parts A and B together in one solution, stating both goals in the introduction and stating both answers in the conclusion.) 4. A wrecking ball used to demolish buildings swings from a 10.0m-long cable. What is the period of the wrecking ball as it swings? 5. A crow attempts to land on a small bird feeder, causing it to swing back and forth with a frequency of 0.350Hz. How long is the wire from which the feeder hangs? 6. The acceleration due to gravity on the moon is 1/6 that on Earth.

(gmoon=gearth/6) If a pendulum was originally 24.0cm long on Earth, how long should it be on the moon so that it has the same period as it does on Earth? 7. Dogs are able to hear much higher frequencies than humans are capable of detecting. For this reason, dog whistles that are inaudible to the human ear can be heard easily by a dog. If a dog whistle has a frequency of 3.0x104Hz, and the speed of a sound wave in air is 340m/s, what is the wavelength of the sound emitted by a dog whistle? 8. A popular pastime at sporting events is “the wave,” a phenomenon where individuals in the crowd stand up and sit down in sequence, causing a giant ripple of people. If a continuous “wave” passes through a stadium of people with a speed of 20m/s and has a frequency of 0.5Hz, what is the distance from “crest” to “crest” in the wave? (In other words, what is the wavelength of this wave?) 9. A radio station KSON in San Diego broadcasts at both 1240kHz (AM) and 97.3MHz (FM). A) which of these signals, AM or FM, has a longer wavelength? B) What is the wavelength of each, if the speed of these waves is 3.00x108m/s? (1kHz=1000Hz and 1MHz = 1000000Hz) 10. Find the wavelength of the ultrasonic wave emitted by a bat if it has a frequency of 4.0x104Hz and a wavespeed of 340m/s.