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L4 Physics B Vibrations and Waves Practice Guide
4.1 Describe the measurable properties of waves (velocity, frequency, wavelength, amplitude, period) and explain the relationships among them. Recognize examples of simple harmonic motion.
4.2 Distinguish between mechanical and electromagnetic waves.
4.3 Distinguish between the two types of mechanical waves, transverse and longitudinal. 4.4 Describe qualitatively the basic principles of reflection and refraction of waves.
4.5 Recognize that mechanical waves generally move faster through a solid than through a liquid and faster through a liquid than through a gas.
4.6 Describe the apparent change in frequency of waves due to the motion of a source or a receiver (the
Doppler effect).
Essential Questions
1. What are the conditions required to produce SHM?
2. How do you calculate the position, velocity, acceleration, potential, and kinetic energy at any point in the motion of an object undergoing SHM?
3. What conditions are necessary for resonance?
4. What is the difference between a longitudinal wave and a transverse wave?
5. How do longitudinal waves and a transverse waves relate to mechanical and electrical systems? 6. How do the speeds of longitudinal waves and transverse waves depend on the medium it travels
through?
7. What is the energy transmitted by a wave?
8. What are the rules for wave reflection from a barrier? 9. What is refraction and why does it occur?
10. What are constructive and destructive interference? 11. How can standing waves be produced in a string or rope?
12. What are the harmonic frequencies needed to produce standing waves in instruments? 13. What are pitch, frequency, wavelength, sound intensity, and loudness?
14. What is the Doppler effect?
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PHY L4 CHART OF UNDERSTANDING
WAVES AND SOUND UNIT:
(RATE YOUR UNDERSTANDING OF THE OBJECTIVES)
*For signatures, if the student attempts to explain a concept to you, but not thoroughly and not to a point where you feel they understand it, initial NY for not yet. A NY will help a student identify areas of focus. Initial when they describe the concept well enough for you to understand it as well.
1. I h ave n o id ea 2. I k in d o f k n o w what t h is is , b u t co u ld n o t test wel l 3. I h ave a mo d er ate g ras p o f thi s co n ce p t 4. I k n o w what t h is is an d co u ld test w el l 5. I h ave a tho ro u gh u n d er st an d in g a n d co u ld teac h th is to an o th er SIGNATU RE S * in it ia l w h en t h e s tu d en t h as t h o ro u gl y exp la in ed th is c o n cep t t o y o u .
Describe the conditions required to produce simple harmonic motion (SHM). Determine the period of a mass m attached to a spring of force constant k
Determine the period of a simple pendulum of length L
Determine the kinetic and potential energy at any point during SHM Determine a relationship between frequency, wavelength and wave-speed
Define natural frequency, resonance and their relation
Define pitch, frequency, wavelength, sound intensity, and loudness.
describe the relationship between frequency and pitch of sound waves
Describe difference between a longitudinal wave and a transverse wave.
identify/describe reflections and transmissions of waves through mediums
Determine constructive and destructive interference.
Explain how standing waves can be produced in a string or rope.
Describe harmonic frequencies.
Define/describe the Doppler effect.
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Important Terms
amplitude
maximum displacement from equilibrium position; the distance from the midpoint of a wave to its crest or trough.
crestofawave
the highest point on a wave
Doppler effect
apparent change in frequency of a wave due to motion between the source of the wave and the detector of the wave
equilibrium position
the position about which an object in harmonic motion oscillates; the center of vibration
frequency
the number of vibrations per unit of time
Hooke’s law
law that states that the restoring force applied by a spring is proportional to the displacement of the spring and opposite in direction
ideal spring
any spring that obeys Hooke’s law and does not dissipate energy within the spring.
longitudinalwave
wave in which the vibration of the medium is parallel to the direction of motion of
the wave
mechanical resonance
condition in which natural oscillation frequency equals frequency of a driving force
period
the time for one complete cycle of oscillation
periodic motion
motion that repeats itself at regular intervals of time
pitch
the perceived sound characteristic equivalent to frequency
rarefaction
and expansion of the medium in a longitudinal wave
restoring force
the force acting on an oscillating object which is proportional to the displacement and always points toward the equilibrium position.
simple harmonic motion
regular, repeated, friction-free motion in which the restoring force has the form F = - kx.
transversewave
a wave in which the vibration is perpendicular to the velocity of the wave
troughofawave
the low point of wave motion
wavelength
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Properties of Waves
Part One (determining speed)
1. A pulse is a transfer of ____________ through a medium.
2. The greater amount of energy in the pulse, the greater the ____________.
3. Given two mediums: a slinky and a spring coil,
-If each is given a moderate pulse, predict which will carry a pulse the fastest.
4. What was the speed of the pulse in the slinky? (use stop watches)
5. What was the speed of the pulse in the spring coil?
6. Say we give the slinky a greater pulse. Predict: Will it travel faster?
7. Give greater pulses to the slinky and the coil. What was the new speed of the slinky? …the coil?
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Part Two: Wave Properties
1. On a separate sheet do the following:
-Using a pencil at the top of the page, move back and forth in a smooth motion at a rate of .5 cycles/sec. -Simultaneously, your partner must pull the sheet perpendicular to your motion at a speed of 5cm/sec. (Have your instructor check your results before moving on)
2. While making your wave, what was its speed?
3. What was its frequency?
4. What is its wavelength?
5. Repeat these steps to create 2 more waves of different frequencies and/or speeds. Create a data table for frequency, wavelength and speed (record below):
6 6. One of your waves, label the following properties:
-crest -trough -amplitude -wavelength
7. Plot a graph of the function: y = 3 sin(πx) (where unit length of x and y are 1cm)
a. This is a model of a _______________ wave. What are the values for amplitude and wavelength?
b. If this wave were moving at a speed of 8cm/s, what value of frequency would the wave have?
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Wave Motion
Key Concept/Idea(s):
Example 1: A sound wave in air has a frequency of 262 Hz and travels with a speed of 343ms. How far apart are the compressions?
Example 2: A certain ocean wave has a frequency of 0.07 Hertz and a wavelength of 10 meters. What is the wave’s speed?
8 Example 4: A 300-g mass vibrates according to the equation x0.38sin 6.50t, where x is in meters and t
is in seconds. Determine (a) the amplitude, (b) the frequency, (c) the period, (d) the total energy
Example 5: The figure shows two examples of SHM, labeled A and B. For each, what is (a) the amplitude, (b) the frequency, and (c) the period?
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Simple Harmonic Motion with Springs
Key Concept/Idea(s):
Example 1: A block of mass m is supported by two identical parallel vertical springs, each with spring stiffness constant k. What will be the frequency of vibration?
Example 2: At t0, a 755-g mass at rest on the end of a horizontal spring
k124N m
is struck by a hammer, which gives the mass an initial speed of 2.96ms. Determine (a) the period and frequency of the motion, (b) the amplitude, (c) the maximum acceleration.10
Simple Harmonic Motion with Pendulums
Key Concept/Idea(s):
Analyzing the FBD of a pendulum we have Fnet = -mgsinθ…
The x-component of motion is analogous to a SHM spring system… restoring force: F = -kx
So for small angles, where the approximation sinθ ≈ θ can be made…
F ≈ -(mg/L)x
Where x = L θ. Returning to Newton’s 2nd Law and Hooke’s Law we get:
m*a = -(mg/L)x = -kx
Therefore, for small angle pendulum oscillations, the “k” value is effectively mg/L. Thus SHM equations become:
and
Example 1: What is the period of a simple pendulum 80 cm long (a) on the Earth, and (b) when it is in a
freely falling elevator?
Example 2: The length of a simple pendulum is 0.760 m, the pendulum bob has a mass of 365 grams,
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Example 3: A 5kg pendulum has a natural frequency of 2Hz. If it is replaced with another pendulum identical in
every way except the mass is 3kg. What is the period of the new pendulum’s natural cycles?
Example 4: Your grandfather clock’s pendulum has a length of 0.9930 m. If the clock loses half a
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Doppler Effect:
Key Concept/Idea(s):
Example 1:
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Example 2:
A bat at rest sends out ultrasonic sound waves at 50.0 kHz and receives them returned from an object moving directly away from it at 25.0m s. What is the received sound frequency?
Example 3:
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Wave Reflection, Transmission & Superposition
Key Concept/Idea(s):
Examples:
Sketch the resulting waves after the pulse has met the interface.
15 After:
Draw the superposition of the following waves when they fully overlap:
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Standing Waves
Key Concept/Idea(s):
Keywords:
Natural frequency
Resonance
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Example 1:
An open organ pipe has a fundamental frequency of 300 Hz in air. The first overtone produced in a closed organ pipe has the same frequency as the first overtone of the open pipe. How long is each pipe? Use vair = 343 m/s.
Example 2:
A tight guitar string has a frequency of 540 Hz as its third harmonic. What will be its fundamental frequency if it is fingered at a length of only 60% of its original length?
Example 3:
18 SHM Multiple Choice Practice:
1. A block with a mass M is attached to a spring with a spring constant k. The block undergoes SHM. Where is the block located when its velocity is a maximum in magnitude?
A) x = 0 B) x = ±A C) x = +A/2 D) x = -A/2 E) None of the above
2. A block with a mass M is attached to a spring with a spring constant k. The block undergoes SHM. Where is the block located when its potential energy is a maximum? A) x = 0 B) x = ±A C) x = +A/2 D) x = -A/2 E) None of the above
3. A block with a mass M is attached to a spring with a spring constant k. The block undergoes SHM. Where is the block located when its acceleration is a minimum in magnitude?
A) x = 0 B) x = ±A C) x = +A/2 D) x = -A/2 E) None of the above
4. A mass-spring oscillating system undergoes SHM with a period T. What is the period of the system if the amplitude is doubled?
A) 2T B) 4T C) T D) 1/2T E) 1/4T
19 A) 6T B) T/6 C) D) T E) T
6. A block with a mass M is attached to a vertical spring with a spring constant k. When the block is displaced from equilibrium and released its period is T. A second identical spring k is added to the first spring in parallel. What is the period of oscillations when the block is suspended from two springs?
A) 2T B) 4T C) T D) E) T
7. Two oscillating systems: spring-mass and simple pendulum undergo SHM with an identical period T. If the mass in each system is doubled which of the following is true about the new period?
Mass-spring Simple pendulum
A)
B) C) D)
E)
8. An object undergoes SHM and position as a function of time is presented by the following formula: x= (0.1 m) Sin (4πt). What is the period of oscillations?
A) 2 s B) 1 s C) 0.5 s D) 0.1 s E) 4 s
9. An object undergoes SHM and position as a function of time is presented by the following formula: x= (0.5 m) Cos (πt). What is the amplitude of oscillations?
20 10. The position as a function of time of a mass-spring oscillating system is presented by the
graph. Which of the following is true about velocity and acceleration at the time 1.5 s? Velocity Acceleration
21 11. A particle undergoes SHM represented by the graph. Which of the following is true about
the amplitude and period of oscillations? Amplitude Period A) 1 m 0.1 s B) 2 m 0.5 s C) 1 m 0.6 s D) 1 m 0.8 s E) 2 m 0.4 s
12. An object oscillates at the end of a spring. The position as a function of time is presented by the graph. Which of the following formulas represent the position and velocity of the object?
Position Velocity
22 13. A simple pendulum oscillates with a period T. If the mass of the pendulum is doubled
what is the new period of the pendulum?
A) T/2 B) 2T C) T D) T E)
14. A simple pendulum oscillates with a period T. If the length of the pendulum is doubled what is the new period of the pendulum?
A) T/2 B) 2T C) T D) T E)
15. What is the length of a simple pendulum if it oscillates with a period of 2 s? A) 2.0 m B) 1.0 m C) 0.5 m D) 0.4 m E) 0.1 m
16. A simple pendulum consists of a mass M attached to a vertical string L. When the string is displaced to the right the ball moves up by a distance 0.2 m. When the ball is released from rest what is the maximum speed?
A) 1 m/s B) 2 m/s C) 3 m/s D) 4 m/s E) 5 m/s
17. A simple pendulum consists of a mass M attached to a vertical string L. The string is displaced to the right by an angle ϴ. When the pendulum is released from rest what is the speed of the ball at the lowest point?
23 18. A block of mass M is attached to a horizontal spring k. The block undergoes SHM with
amplitude of A. Which of the following graphs represents the elastic potential energy as a function of position x?
A) B)
C) D)
E)
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A) B)
C) D)
E)
20. A 0.9 kg block is attached to an unstretched spring with a spring constant of 10 N/m. The block is released from rest. How long does it take for the block to return to its initial position?
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Sound Multiple Choice Practice:
1. Two sound sources S1 and S2 produce waves with frequencies 500 Hz and 250 Hz. When we compare the speed of wave 1 to the speed of wave 2 the result is:
(A) Twice as greater (B) One-half as greater (C) The same (D) Four times greater (E) One-fourth as greater
2. Which of the following is a true statement about the speed of sound in three different materials: air, water, and steel?
(A) Vair > Vwater > Vsteel (B) Vair > Vwater = Vsteel (C) Vair = Vwater < Vsteel (D) Vair < Vwater > Vsteel (E) Vair < Vwater < Vsteel
3. A sound source S radiates a sound wave in all directions. The relationship between the distances is SA = AB = BC = CD. Which of the following points oscillates at the highest frequency?
26 4. A sound source S radiates a sound wave in all directions. The relationship between the distances
is SA = AB = BC = CD. Which of the following points oscillates with the greatest intensity? (A) Point A (B) Point B (C) Point C (D) Point D
(E) All points have the same intensity
5. The loudness of a sound wave increases with increasing which of the following:
(A) Frequency (B) Amplitude (C) Period (D) Wavelength (E) Speed of sound
6. A sound wave travels from air into water. Which of the following doesn’t change? (A) Frequency (B) Amplitude (C) Speed of Particles (D) Wavelength (E) Speed of sound
7. A sound wave resonates in a tube with two open ends. What are the wavelengths of the three lowest resonating frequencies generated in the tube?
(A) L, 2L, 3L (B) 2L, L, 2L/3 (C) L/2, L/3, L/5 (D) L/3, L/5, L/7 (E) 4L, 4L/3, 4L/5
8. The lowest frequency in an open tube is 300 Hz. What are the three following frequencies will resonate in the tube?
(A) 600Hz, 900Hz, 1200Hz (B) 100Hz, 200Hz, 400Hz (C) 250Hz, 500Hz, 750Hz (D) 150Hz, 450Hz, 850Hz (E) 50Hz, 100Hz, 150Hz
9. The lowest frequency in an open tube is 200 Hz. Which of the following frequencies will resonate in the tube?
(A) 50Hz (B) 100Hz (C) 150Hz (D) 250 Hz (E) 400Hz
10. A sound wave resonates in an open pipe with a length of 2 m. What is the wavelength of the wave?
27 11. A sound wave resonates in an open pipe with a length of 4 m. What is the resonating
frequency? (Vsound = 340 m/s)
(A) 85 Hz (B) 170 Hz (C) 340 Hz (D) 510 Hz (E) 680 Hz
12. A sound wave resonates in an open pipe with a length of 3 m. What is the wavelength of the wave?
(A) 1.5 m (B) 2.0 m (C) 2.5 m (D) 3.0 m (E) 6.0 m
13. A sound wave resonates in an open pipe with a length of 1.5 m. What is the resonating frequency? (Vsound = 340 m/s)
(A) 85 Hz (B) 170 Hz (C) 340 Hz (D) 510 Hz (E) 680 Hz
14. A sound wave resonates in a tube with one open end. What are the wavelengths of the three lowest resonating frequencies generated in the tube?
28 15. The lowest frequency in a closed tube is 300 Hz. What are the three following frequencies will
resonate in the tube? (A) 600Hz, 900Hz, 1200Hz (B) 100Hz, 200Hz, 400Hz (C) 250Hz, 500Hz, 750Hz (D) 900Hz, 1500Hz, 2100Hz (E) 50Hz, 100Hz, 150Hz
16. The lowest frequency in a closed tube is 400 Hz. Which of the following frequencies will resonate in the tube?
(A) 500Hz (B) 1000Hz (C) 1200Hz (D) 2500 Hz (E) 3000Hz
17. Two sound sources generate pure tones of 70 Hz and 80 Hz. What is the beat frequency? (A) 5Hz (B) 10Hz (C) 15Hz (D) 20Hz (E) 25Hz
18. A sound wave resonates in a closed pipe with a length of 1.5 m. What is the wavelength of the wave?
(A) 1.5 m (B) 2.0 m (C) 2.5 m (D) 3.0 m (E) 6.0 m
19. A sound wave resonates in a closed pipe with a length of 3.5 m. What is the wavelength of the wave?
29 20. A sound wave resonates in a closed pipe with a length of 2.5 m. What is the resonating
frequency? (Vsound = 340 m/s)
(A) 85 Hz (B) 170 Hz (C) 340 Hz (D) 510 Hz (E) 680 Hz
21. Two sound sources generate pure tones of 115 Hz and 130 Hz. What is the beat frequency? (A) 5Hz (B) 10Hz (C) 15Hz (D) 20Hz (E) 25Hz
22. Two sound sources produce waves with slightly different frequencies. What happens with the beat frequency if the frequency of the lowest tone increases?
(A) Increases (B) Decreases (C) Stays the same (D) Increases and then decreases (E) Decreases and then increases
23. A sound source approaches a stationary observer at a constant speed of 34 m/s. If the frequency of the stationary source is 90 Hz, what is the frequency heard by the observer?
(A) 90 Hz (B) 100 Hz (C) 180 Hz (D) 270 Hz (E) 360 Hz
24. An airplane moves away from a stationary observer at a constant speed of 340 m/s. The frequency of the sound wave of the stationary airplane is 780 Hz. What is the frequency heard by the observer? (Vsound = 340 m/s)
(A) 1560 Hz (B) 780 Hz (C) 390 Hz (D) 195 Hz (E) 0 Hz
25. **Two loudspeakers generate sound waves with frequencies of 680 Hz. What is the extra distance traveled by the second wave if a stationary observer detects maximum intensity of sound at point P?
30 26. Two loudspeakers generate sound waves with frequencies of 680 Hz. What is the extra distance
traveled by the second wave if a stationary observer detects maximum intensity of sound at point P?
(A) 0.75 m (B) 1.20 m (C) 1.50 m (D) 1.60 m (E) 2.00 m
26. A sound source moves at a constant velocity Vobj and generates a sound wave. The speed of sound is Vsound. Which of the following is true about the direction and magnitude of the source velocity?
