Waves Practice Problem Set, Lecture notes of Mechanics

Download Waves Practice Problem Set and more Lecture notes Mechanics in PDF only on Docsity!

ACTIVITY

BASED

PHYSICS

SHM, WAVES,

THERMODYNAMICS AND

E & M PRACTICE

PROBLEM SETS

SOUTHINGTON HIGH SCHOOL

CCP PHYSICS

W

Unit 7 SHM, WAVES AND SOUND 7.1 SIMPLE HARMONIC MOTION 7.1.1 Mass-Spring Systems

1. An ultrasonic transducer used for medical diagnosis oscillates with a frequency of 6. x 10^6 Hz. How much time does each oscillation take and what is the angular frequency? Answer: T=1.5 x 10-^7 s; ω =4.21 x 10 7 rad/s
2. A body of unknown mass is attached to an ideal spring that is mounted horizontally with its left end held stationary. The spring constant of the spring is 120 N/m and it vibrates with a frequency of 6.00 Hz. Assuming that there is no friction, find: a. The Period Answer: 0.17 sec b. The Angular Frequency Answer: 37.7 rad/s c. The Mass of the body Answer: 0.084 kg
3. A spring stretches 0.200 m when a 0.600 kg mass is hung from it. The spring is then stretched 0.150 m from this equilibrium point and released. Find: a. The spring constant (k) Answer: 29.4 N/m b. The amplitude Answer: 0.150 m c. The total energy of the system. Answer: 0.33 J d. Maximum velocity (vo) Answer: 1.05 m/s e. The velocity when the mass is 0.050 m from equilibrium. Answer: .99 m/s f. The equation for the position of the mass x (t) Answer: x (t) = 0.150cos(7t)
4. A proud deep-sea fisherman hangs a 65.0 kg fish from an ideal spring with a negligible mass. The fish stretches the spring 0.120m. What is the period of oscillation of the fish if it is pulled down and released? Answer: 0.70 sec
5. A 0.150 kg toy is undergoing simple harmonic motion on the end of a horizontal spring with a spring constant of 300 N/m. When the object is 0.12m from equilibrium, it has a speed of 0.300 m/s. Find: a. The Total energy of the system Answer: 2.17 J b. The Velocity of the object at equilibrium Answer: 5.4 m/s c. The Amplitude of the motion. Answer: 0.120 m 7.1.2 Simple Pendulum
6. What is the period of a pendulum at sea level with a length of 1.5 m? Answer: 2.45 sec a. What would the period of this pendulum be if the length were shortened to a ¼ of the original length? Answer: 1.23 sec

2. How long would a pendulum have to be if it has a frequency of 2 Hz? Answer: 0.062 m
3. A simple pendulum with a bob mass of 0.5 kg and a string length of 1.0 m and is taken to the moon, which has an acceleration due to gravity that is 1/6 that of the acceleration due to gravity on Earth. If you want to maintain the same period on Moon that you have on Earth, what adjustment will you have to make to the… a. Mass of the pendulum bob? Answer: None b. Length of the string? Answer: Shorten to 0.166 m 7.1.3 SHM Review Problems
4. (G1) When a 65-kg person climbs into a 1000-kg car, the car’s springs compress vertically by 2.8 cm. What will be the frequency of vibration when the car hits a bump? Ignore damping. Answer: 0.74 Hz
5. (G7) A balsa wood block of mass 50 g floats on a lake, bobbing up and down at a frequency of 2.5 Hz. a. What is the value of the effective spring constant of the water? Answer: 12 N/m b. A partially filled water bottle of mass 0.25 kg and almost the same size and shape of the balsa block is tossed into the water. At what frequency would you expect the bottle to bob up and down? Assume SHM. Answer: 1.1 Hz
6. (G9) A 0.50-kg mass at the end of a spring vibrates 3.0 times per second with an amplitude of 0.15 m. Determine: a. The velocity when it passes the equilibrium point. Answer: 2.8 m/s b. The velocity when it is 0.10 m from equilibrium. Answer: 2.1 m/s c. The total energy of the system. Answer: 2.0 J d. The equation describing the motion of the mass. Assume φ = 0. Answer: (0.15 m) cos [2 π (3.0 Hz)t] 4. (G13) It takes a force of 80.0 N to compress the spring of a toy popgun 0.200 m to “load” a 0.150-kg ball. With what speed will the ball leave the gun? Answer: 10.3 m/s
7. (G15) A mass sitting on a horizontal, frictionless surface is attached to one end of a spring; the other end of the spring is fixed to a wall. 3.0 J of work is required to compress the spring by 0.12 m. If the mass is released from rest with the spring compressed, it experiences a maximum acceleration of 15 m/s^2. Find the value of: a. The spring constant. Answer: 4.2 x 10 2 N/m b. The mass. Answer: 3.3 kg
8. (G17) A 0.50-kg mass vibrates according to the equation x=0.45cos(8.40t), where x is in meters and t is in seconds. Determine: a. The amplitude. Answer: 0.45 m

7.2 SOUND

Unless stated otherwise, assume that T = 20°C.

1. Find the frequency of a sound wave moving in air at a temperature of 22°C with a wavelength of 0.667 m. Answer: f = 516 Hz
2. You hear the sound of the firing of a distant cannon 6.0 seconds after seeing the flash. How far are you from the cannon? Answer: d = 2.1 km
3. A sound wave with a frequency of 9800 Hz travels along a copper pipe. If the wavelength is 0.370 m and the density of copper is 8.9 x 10^3 kg/m^3 , what is the elastic modulus of copper? Answer: E= 1.2 x 10^11 N/m^2
4. A certain instant camera determines the distance to the subject by sending out a sound wave and measuring the time needed for the echo to return to the camera. How long will it take the sound wave to return to the camera if the subject were 3.00 m away? Answer: t = 0.017 s
5. If you drop a stone into a mineshaft that is 122.5 m deep, how soon after you drop the stone do you hear it hit the bottom of the shaft? The temperature in the mineshaft is 10°C. Answer: t = 5.36 s
6. With what tension must a rope of length 2.50 m and mass of 0.120 kg be stretched for transverse waves of frequency of 40.0 Hz to have a wavelength of 0.750 m? Answer: T= 43.2 N
7. A ship uses a sonar system to detect underwater objects in the ocean. The system emits underwater sound waves and measures the time interval for the reflected wave to return to the detector. a. Determine the speed of sound in seawater if the bulk modulus is 2.2 x 10^9 N/m^2. Answer: v= 1465 m/s b. The ship is on the continental shelf when it emits a sound wave. It takes the echo 0.203 seconds to be picked up by the detector. What is the depth of the ocean on the continental shelf? Answer: depth = 150 m. 8. A tuning fork of frequency 262 Hz is sounded at the same time as another tuning fork with a frequency of 257 Hz. What is the beat frequency that is heard? Answer: fb = 5 Hz 9. A tuning fork with a frequency of 432 Hz is sounded at the same time as a guitar. If 6 beats are heard in 3 seconds, what are the possible frequencies of the guitar string? Answer: f = 430 Hz or 434 Hz

7.3 DOPPLER EFFECT

1. On a cold wintry day, Bob is late for work. He drives at a speed of 15 m/s toward the factory where he works. The factory whistle is blown with a frequency of 800 Hz to indicate the start of the workday. If it is - 4°C… a. What is the frequency that Bob hears when the whistle is blown? Answer: f’= 837 Hz b. What is the frequency that Bob hears when he passes the building and moves away toward the parking lot behind the factory? Answer: f’= 763 Hz
2. While standing near a railroad crossing, a person hears a distant train horn. According to the train’s engineer, the frequency of the horn is 262 Hz. If the train is traveling at 20.0 m/s toward the crossing and the speed of sound is 346 m/s… a. What would the train horn’s wavelength be at rest? Answer: λ = 1.32 m b. By how much would the horn’s wavelength change as a result of the train’s motion? Answer: Δλ = 0.075 m
3. An ambulance with a siren emitting a whine at 1300 Hz races by a car that was pulled off to the side of the road. After being passed, the driver of the park car hears a frequency of 1220 Hz. How fast was the ambulance moving? Answer: vs = 22.5 m/s 4. In 1845, French Scientist B. Ballot first tested the Doppler shift. He had a trumpet player sound an A, 440 Hz, while riding on a flatcar pulled by a locomotive. At the same time, a stationary trumpeter played the same note. If the locomotive was moving toward Ballot at a speed of 5.0 m/s, what beat frequency would Ballot have heard? Answer: fb = 6.5 Hz

10. (G12.51) The predominant frequency of a certain police car’s siren is 1800 Hz when at rest. What frequency do you detect if you move with a speed of 30.0 m/s… a. Toward the car Answer: 1950 Hz b. Away from the car Answer: 1640 Hz
11. (G12.53) In one of the original Doppler experiments, one tuba was played on a moving platform car at a frequency of 75 Hz, and a second identical one was played on the same tone while at rest in the railway station. What beat frequency was heard if the train approached the station at a speed of 10.0 m/s? Answer: 2 Hz
12. (G12.78) The frequency of a steam train whistle as it approaches you is 522 Hz. After it passes you, its frequency is measured as 486 Hz. How fast was the train moving? Assume constant velocity. Answer: 12.3 m/s
13. (G12.71) A tight guitar string has a frequency of 600 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? Answer: 333 Hz
14. (G12.73) The string of a violin is 32 cm long between fixed points with a fundamental frequency of 440 Hz and a linear density of 5.5 x 10
    • 4 kg/m. a. What are the speed and tension in the string? Answer: 2.8 x 10^2 m/s, 44 N b. What is the frequency of the first overtone? Answer: 880 Hz Material Bulk Modulus Water 2.0 x 10 9 N/m 2 Mercury 2.5 x 10^9 N/m^2 Alcohol (ethyl) 1.0 x 10^9 N/m^2 Air 1.01 x 10 5 N/m 2 Material Elastic Modulus Iron, cast 100 x 10^9 N/m^2 Steel 200 x 10 9 N/m 2 Brass 100 x 10 9 N/m 2 Aluminum 70 x 10^9 N/m^2 Concrete 20 x 10^9 N/m^2 Brick 14 x 10 9 N/m 2 Marble 50 x 10^9 N/m^2 Granite 45 x 10^9 N/m^2 Wood (pine) Parallel to Grain 10 x 10 9 N/m 2 Perpendicular to Grain 1 x 10^9 N/m^2 Nylon 5 x 10^9 N/m^2 Bone (limb) 15 x 10 9 N/m 2 Density of Substances

7.4.2 Mass-Spring System

1. When a family of four with a total mass of 200 kg gets into their 1200 kg car, the car’s springs compress 5 cm. What is the spring constant of the car’s springs assuming they act as one single spring? Answer: 39,200 N/m
2. Imagine that you videotape the motion of a mass attached to a spring and measure the displacement x from the equilibrium position as a function of time t. When you plot position, velocity and acceleration as a function of time you get the following graphs.
3. When a 0.50 kg-object is attached to a vertically supported spring, it stretches 0.10 m. It is then pulled another 0.10 m from its new equilibrium position. Find the period, angular frequency, total energy, and displacement equation for this mass-spring system. Answer: T = 0.63 sec, ω = 9.9 rad/s, ET= .25 J, x(t) = 0.10 cost (9.9t) 4. An object with mass m = 0.60 kg attached to a spring with k = 10 N/m vibrates back and forth along a horizontal frictionless surface. If the amplitude of the motion is 0.050 m, what is the velocity of the object when it is 0.010 m from the equilibrium position? Answer: 0.20 m/s a. Find the amplitude. b. Find the period. c. Find the angular frequency. Answer: ω = 4 rad/s d. Find the magnitude of the maximum velocity. e. Find the magnitude of the maximum acceleration.

3. A mass-less spring with spring constant k = 10.0 N/m is attached to an object of mass m = 0.300 kg. One third of the spring is cut off. What is the frequency of the oscillations when the "new" spring-mass is set into motion? Answer: f = 0.92 Hz
4. Figure 6 below is a plot of the potential energy of a mass-spring system. The total mechanical energy ET of the system = 0.200 J. Find (a) the potential energy PE and (b) the kinetic energy KE at x = 0.025 m. Find (c) the spring constant k and (d) the speed of the particle when x = 0.025 given that the mass of the object m = 0.30 kg. Find (e) the amplitude of the motion and (f) the maximum velocity of the object. Answer: PE = 0.050 J, KE = 0.15 J, k = 160 N/m, v = 1 m/s, xm = 0.0500 m, vmax = 1.15 m/s
5. The motion of a particle is given by x(t) = 4.0 cm cos(πt - π/6). Find the particle’s velocity when x = 2.0 cm. Answer: v = - 10.9 cm/s 6. A wave travels along a stretched rope. The wavelength is 2.0 m. The wave period is 0.1 s. What is the speed of this wave? Answer: 20 m/s 7. A wave has a frequency of 58 Hz and a speed of 31 m/s. What is the wavelength of this wave? Answer: 0.53 m 8. A clothesline with a mass of 0.750 kg is 3.00 meters long. How much tension do you have to apply to produce the observed a wave speed of 12.0 m/s? Answer: 36 N
6. Determine the wavelength of a 6000 - Hz sound wave traveling in Alcohol. Answer: 0. 19 m
7. A sound wave produced by a clock chime is heard 501 meters away, 1.50 sec later. a. What is the temperature of the air through which the sound travels? Answer: T = 5°C b. How long would it take this sound to travel 501 meters away in freshwater? Answer: t = 0.35 s 11. If you clap your hands and hear the echo from a distant wall 0.20 seconds later, how far away is the wall? Answer: d = 34.3 m

12. Billy Bob Joe whistles at 441 Hz and Mary Jane whistles at 462 Hz. What beat frequency does Mary Jane hear? Answer: fb = 21 Hz
13. A railroad train is traveling at 25.0 m/s in still air. The frequency of the note emitted by the locomotive whistle is 400 Hz. a. What is the frequency heard by a stationary observer standing in front of the locomotive? Answer: f’ = 431 Hz b. What is the frequency heard by a stationary observer standing behind the locomotive? Answer: f’ = 373 Hz c. The train comes to a full stop at the nearest station so that new passengers can get onto the train. When it has come to rest, it blows its whistle at another train coming toward the station at 10 m/s. What frequency does a passenger sitting on the moving? Answer: f’ = 412 Hz
14. A train moving at a constant speed is passing a stationary observer on a platform. On one of the train cars, a flute player is continually playing a note known as concert A, which has a frequency of 440 Hz. After the flute has passed, the observer hears the sound as a G, which has a frequency of 392 Hz. What is the speed of the train? Answer: 42 m/s
15. A guitar string has a fundamental frequency of 256 Hz. What is the frequency of the 3 rd harmonic? Answer: 768 Hz a. What would the length of the guitar string have to be to produce a transverse wave in the string with a speed of 405 m/s at the fundamental frequency? Answer: 0.79 m
16. A string that is 2.00 meters long and has a mass of .0025 kg. a. If the fundamental frequency is 120 Hz, what are the frequencies of the first four harmonics? Answer: 120 Hz, 240 Hz, 360 Hz, 480 Hz b. What must the tension be for the string to vibrate at the 4th^ harmonic? Answer: 288 N
17. The speed of sound in a certain metal block is 3.00 x 10 3 m/s. The graph shows the amplitude in meters of a wave traveling through the block versus time in milliseconds. What is the wavelength of this wave? Answer: 6 m

Unit 8 OPTICS 8.1 REFLECTION AND MIRRORS 8.1.1 Plane Mirrors

1. Draw the reflected light ray(s) and position of the observer’s eye where it can see the reflected ray.
2. A bulb is placed in front of a plane mirror. a. Use a ruler and a protractor to construct four rays that travel from the bulb to the mirror and reflect. Include eyes at positions that could see the reflected rays. b. Extend the reflected rays with dotted lines behind the mirror to locate the virtual image. c. Measure and compare the image distance to the object distance.
3. The ray diagram below shows where Observer 1 sees the virtual image of the bulb. Show where, if at all, Observer 2 sees the virtual image.

4. The ray diagram below shows where Observer 1 sees the virtual image of the bulb. Show where, if at all, Observer 2 sees the virtual image.
5. The ray diagram below shows where Observer 1 sees the virtual image of the bulb. Show where, if at all, Observer 2 sees the virtual image.
6. A top view of a mirror and an arrow is shown below.

9. One of Cinderella’s stepsisters, who is 1.60 m tall, is looking at herself in a plane mirror when she spots one of the mice helping Cinderella at the very bottom of the mirror. If the mouse is 5.0 cm away from the wall where the mirror is hung, and the mirror’s edge is 60 cm from the floor, how far away is she from the mouse? Make sure to draw a ray diagram to help solve this problem! Answer: 3.3 cm from the mouse 8.1.2 Spherical Mirrors

1. An action figure that is 8.0 cm tall is placed 23.0 cm in front of a convergent

mirror with a focal length of 10.0 cm. a. Draw a ray diagram for the image of the action figure. b. What are the three characteristics of the image? Answer: Inverted, M<1, Real c. What is the image distance? Answer: 17.7 cm d. What is the image height? Answer: - 6.15 cm

2. If the 8.0 cm tall action figure from Question 1 is now placed 6.0 cm in front of a

divergent mirror with a radius of 24.0 cm. a. Draw a ray diagram for the image of the action figure. b. What are the three characteristics of the image? Answer: Upright, M<1, Virtual c. What is the image distance? Answer: - 4.0 cm d. What is the image height? Answer: 5.3 cm

3. The focal point of a divergent mirror is 20.0 cm behind the mirror. An object is

placed 12 cm from the mirror. a. Draw a ray diagram for the image of the object. b. What is the image distance? Answer: - 7.5 cm c. What is the magnification of the image? Answer: M = 0.625 x

4. A magnified, inverted image is located a distance of 32.0 cm from a spherical

mirror with a focal length of 12.0 cm. a. Is it a convergent or divergent mirror? Explain. b. Is the image real or virtual?

c. How far was the object placed in front of the mirror? Answer: 19.2 cm

5. An inverted image has a magnification of 2 when the object is placed 22 cm from

a convergent mirror. What are the image distance and the focal length of the mirror? Answer: di = 44 cm f = 14.7 cm 8.1.3 Reflection and Mirror Review

1. (G1) Suppose that you want to take a photograph of yourself as you look at your image in a flat mirror 1.5 m away. For what distance should the camera lens be focused? Answer: 3.0 m
2. (G4) A person whose eyes are 1.62 m above the floor stands 2.10 m in front of a vertical plane mirror whose bottom edge is 43 cm above the floor, as shown. What is the horizontal distance x to the base of the wall supporting the mirror of the nearest point on the floor that can be seen reflected in the mirror? Answer: 76 cm
3. (G5) Two mirrors meet at a 135° angle. If light rays strike one mirror at 40° as shown, at what angle do they leave the second mirror? Answer: 5°
4. (G9) A solar cooker, really a convergent mirror pointed at the sun, focuses the Sun’s rays 17.0 cm in front of the mirror. What is the radius of the spherical surface from which the mirror was made? Answer: 34.0 cm
5. (G12) How far from a convergent mirror (radius 27.0 cm) must an object be placed if its image is to be at infinity? Answer: 13.5 cm
6. (G12) If you look at yourself in a shiny Christmas tree ball with a diameter of 9.0 cm when your face is 30.0 cm away from it, where is your image located? Answer: - 2.09 cm a. What is the image’s magnification? Answer: +0. b. Is it real or virtual? Is it upright or inverted?