Physics Help >> Index to Physics Homework Help » Waves, Harmonic Motion, Doppler Effect

EXAMPLE PROBLEMS

1. A mass is executing simple harmonic motion with amplitude A. What is the total distance traveled by the mass during one period of the oscillation? Does your answer depend on the time instant from which the period of the oscillation is measured?

2. The following diagram depicts a series of straight waves traveling from shallow water into deeper water. Complete the diagram by drawing in the following:
(a) incident ray; (b) reflected ray; (c) refracted ray; (d) two refracted wave fronts; (e) two reflected wave fronts.

3. The security alarm on a parked car goes off and produces a frequency of 735 Hz. The speed of sound is 343 m/s. As you drive toward this parked car, pass it, and drive away, you observe the frequency to change by 78.4 Hz. At what speed are you driving?

4. Transverse waves traveling across a rope have a frequency of 12.0 Hz and a wavelength of 2.40 m. What is the velocity of the waves?

ANSWERS TO EXAMPLE PROBLEMS

1.

During one complete oscillation, the mass travels a distance of 4A.

Where the cycle is deemed to start does not matter.

2.

3.

The change in frequency can be calculated using the general Doppler equation:

f'=f((V+-Vo) / (V-+Vs))

V is speed of sound (343m/s)

Vo is speed of observer

Vs is speed of source

f' is shifted frequency

f is original frequency

f'=f((V+Vo)/(V-Vs)) used for situations in which the source and the observer are coming closer together

f'=f((V-Vo)/(V+Vs)) used for situations that they are moving away from one another.

So for this question, when you drive toward this parked car

f'1=735*(343+Vo)/343

when you drive away

f'2=735*(343-Vo)/343

The difference in f' will be 735*2*Vo/343=78.4 , solve this one gives the speed of observer

Vo=18.29m/s=65.86km/hr

4.

The universal wave equation applies:

v = λf where v = the speed of propagation, λ = wavelength, and f = frequency.

v = λf = (12)(2.40) = 28.8 m/s

A. PERIODIC (SIMPLE HARMONIC) MOTION PROBLEMS

1. You have a variety of masses available but only one spring. What can be done to double the maximum speed for simple harmonic oscillation at a fixed amplitude A?

2. Both neon (Ne) and helium (He) are monatomic gases and can be assumed to be ideal gases. The fundamental frequency of a tube of neon is 291 Hz. What is the fundamental frequency of the tube if the tube is filled with helium, all other factors remaining the same? The molar masses of neon and helium are 20 grams/mol and 4 grams/mol, respectively.

3. The springs of a 526 kg motorcycle have an effective force constant of 9067 N/m. By what percent does the period of oscillation change when a 110 kg person rides the motorcycle. Enter a positive number for an increase and a negative number for a decrease.

4. A mass attached to a spring oscillates with a period of 3.13 s. If the mass starts from rest at x = 0.0410 m and time t = 0, where is it at time t = 3.21 s?

5. A 0.824 kg mass attached to a vertical spring of force constant 162 N/m oscillates with a maximum speed of 0.372 m/s. Calculate

a. the period related to the motion of the mass.

b. the maximum acceleration of the mass.

6. A huge mass spring system is undergoing simple harmonic motion with angular frequency, ω, and amplitude, A. Find its speed at the point where the kinetic and potential energies are equal.

7. Why is the acceleration of vibrating body is zero at its mean position?

B. WAVES PROBLEMS

1. The following shows a view from above an interference pattern. The two sources, S1 and S2, have the same frequency and are in phase. The point, P, is on the second nodal line. What is the wavelength of the sources?

2. In the next diagrams, label compression, rarefaction, crest, trough, wavelength. Also describe the motion of points A and B.

3. The following diagrams are successive "snapshots" of two pulses travel\ling in opposite directions along a coil. Draw three diagrams to show the shape of the coil at t = 0.03 s, t = 0.06, and 0.09 s.
t = 0 s t = 0.12 s

4. The following 2 pulses on a coil are pictured approaching a second coil. Draw a diagram to demonstrate the behavior of the pulses if the second coil is
a. faster than the first coil
b. slower than the first coil.

5.

When combined with the pulse on the left, which of the pulses below will result in the following situation:

6. A certain vibration has a wavelength of 58.6m and travels at a speed of 1.098⁸ m/s. what is the frequency of the vibration?

7. A siren can be made by blowing a jet of air through 20 equally spaced holes in a rotating disk. The time it takes for successive holes to move past the air jet is the period of the sound. The siren is to produce a 2480-Hz tone. What must be the angular speed in radians per second of the disk?

8. Find the wavelength of a water wave of frequency 40 Hz traveling at 120 cm/s.

9. How long should an antenna be for a receiver with frequency of 30 MHz?

C. DOPPLER EFFECT PROBLEMS

1. A speeder is pulling directly away and increasing his distance from a police car that is moving at 29 m/s

with respect to the ground. The radar gun in the police car emits an electromagnetic wave with a frequency

of 8.0x10⁹ Hz. The wave reflects from the speeder\'s car and returns to the police car where its frequency is

measured to be 318Hz less than the emitted frequency. Find the speeder's speed with respect to the

ground.

2. You are driving 8 m/s on a straight road and sounding a horn which you hear at a frequency of 600Hz. The sound of the horn gets reflected from the high rise building ahead and you hear the echo.(a)What is the frequency of the echo you hear? (b) What beat frequency you hear?

ANSWERS

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A6.

The total mechanical energy of the system is

E = K + U = (1/2)mω²A²

Since K = U

(1/2)mv²+ (1/2)mv² = (1/2)mω²A²

2v² = ω²A²

v = ωA / √2

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B8.

The universal wave equation informs us

speed = frequency * wavelength

120 cm/s = (40 cycles/s)* wavelength

wavelength = 3.0 cm

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