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Physics 202 Examples 1Apr 1, 2013

1. When responding to sound, the human eardrum vibrates about its equilibrium (a) 730 hertz(b) 13 m/s2position. Suppose an eardrum is vibrating with an amplitude of 6.3×10−7 meters

and a maximum speed of 2.9× 10−3 m/s. (a) What is the frequency (in hertz) ofthe eardrum’s vibration? (b) What is the maximum acceleration of the eardrum?

2. Astronauts on a distant planet set up a simple pendulum of length 1.2 6.0 m/s2

meters. The pendulum executes simple harmonic motion and makes 100 completevibrations in 280 seconds. What is the magnitude of the acceleration due togravity?

3. A loudspeaker diaphragm is producing a sound for 2.5 seconds by moving 30,000 times

back and forth in simple harmonic motion. The angular frequency of the motionis 75,400 rad/s. How many times does the diaphragm move back and forth?

4. For an oscillator with a mass of 0.010 kilograms and a spring constant of 36 (a) 0.02 kg/s(b) 1.2 kg/sN/m, (a) what value of b would make the amplitude decrease to 37% of its original

value in 1.0 second? (b) What value of b would produce critical damping?

5. The damping factor of a spring suspension system is one-tenth the criticalvalue. If the undamped frequency is ω0, find the resonant frequency, and thequality factor of the resonant peak.

6. Calculate (a) the complex amplitude, (b) the amplitude, and (c) the phase (a) 10− 17i(b) 20(c) -60◦

constant, for the vibration given by

x = 10 cosωt+ 17 sinωt

7. The graph in Figure 1 shows the mean power absorption Pavg in watts, as a (a) 600 rad/s(b) 19 rad/s(c) 0.079 s

function of driving frequency in hertz. Find the numerical values of (a) ω0, and(b) γ. (c) If the driving force is removed, how many seconds will it take for theenergy of the system to reach 5% of its initial value?

Figure 1: Power absorption

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1. A car is hauling a 92-kilogram trailer, to which it is connected by a spring. 0.012 meters

The spring constant is 2300 N/m. The car accelerates with an acceleration of0.30 m/s2. By how much does the spring stretch?

2. A vertical spring with a spring constant of 450 N/m is mounted on the floor. 4.8 cm

From directly above the spring, which is unstrained, a 0.30-kilogram block isdropped from rest. It collides with and sticks to the spring, which is compressedby 2.5 cm in bringing the block to a momentary halt. Assuming air resistance isnegligible, from what height (in cm) above the compressed spring was the blockdropped?

3. A vertical ideal spring is mounted on the floor and has a spring constant (a) 0.037 meters(b) 0.074 metersof 170 N/m. A 0.64-kilogram block is placed on the unstrained spring in two

different ways. (a) The block is placed on the spring and not released until itrests stationary on the spring in its equilibrium position. Determine the amount(magnitude only) by which the spring is compressed (b) In a second situation, theblock is released from rest and falls downward until it comes to a momentary halt.Determine the amount (magnitude only) by which the spring is now compressed.

4. Figure 2 shows a 160-kilogram crate hanging from the end of a steel bar. The (a) 4.9× 106 N/m(b) 6.0× 10−6 meterslength of the bar is 0.10 meters, and its cross-sectional area is 3.2 × 10−4 square

meters. Neglect the weight of the bar itself and determine (a) the shear stress onthe bar and (b) the vertical deflection of the right end of the bar.

Figure 2: Problem 10.56

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1. When an object moves through a fluid, the fluid exerts a viscous force on (a) 2.8× 10−5 newtons(b) 10 m/sthe object that tends to slow it down. For a small sphere of radius r, moving

slowly with a speed v, the magnitude of the viscous force is given by Stokes’ law,F = 6πηrv, where η is the viscosity of the fluid. (a) What is the viscous force ona sphere of radius 0.50 mm falling through water (η = 1.00 × 10−3 Pa-s) whenthe sphere has a speed of 3.0 m/s? (b) The speed of the falling sphere increasesuntil the viscous force balances the weight of the sphere. Thereafter, no net forceacts on the sphere, and it falls with a constant speed called the “terminal speed.”If the sphere has a mass of 1.0 × 10−5 kilograms, what is its terminal speed?

2. A gold prospector finds a solid rock that is composed solely of quartz and gold. 1.6 kilograms

The mass and volume of the rock are, respectively, 12.0 kilograms and 0.0040 m3.Find the mass of the gold in the rock.

3. A hot-air balloon is accelerating upward under the influence of two forces, its 3.8 m/s2

weight and the buoyant force. For simplicity, consider the weight to be only thatof the hot air within the balloon, thus ignoring the balloon fabric and the basket.The hot air inside the balloon has a density of 0.93 kg/m3, and that of the coolair outside is 1.29 kg/m3. What is the acceleration of the rising balloon?

4. Suppose that a 15-m/s wind is blowing across the roof of your house. The (a) 0.150 kPa(b) Pressure differentialsdensity of air is 1.29 kg/m3. (a) Determine the reduction in pressure (below

atmospheric pressure of stationary air) that accompanies this wind (b) Explainwhy some roofs are “blown outward” during high winds.

5. An airplane wing has an effective wing surface of 16 m2 that is generating the 9600 newtons

lift force. In level flight the air speed over the top of the wings is 62.0 m/s, whilethe air speed beneath the wings is 54.0 m/s. What is the weight of the plane?

6. The logistic map, See solution

xn+1 = kxn(1 − xn)

is a nonlinear system sometimes used to model population growth where k isrelated to birth and death rates. When k ¡ 1, the system is “attracted” to zeroregardless of the initial state, x1. Between k = 1 and 3, zero no longer attracts.Between k = 3.0 and 3.5 the attractor “bifurcates”— it oscillates between twostates. The attractor continues to bifurcate until a value of approximately k= 3.57. Beyond this point, the iteration does not settle down. In this chaoticregion the solution is extremely sensitive to initial conditions (as opposed to the“attractive” characteristic when k ¡ 3.0).

(a) Let k = 2. Start with x1 = 0.25 and iterate until n = 10 to show that thesystem is attracted to 0.5.

(b) Let k = 3.2. Start with x1 = 0.25 and iterate until n = 15 to show that thesteady state solution oscillates between 0.51 and 0.80.

(c) Start with x1 = 0.2500 and let k = 3.8. Iterate until n = 20.

(d) Start with x1 = 0.2501 and let k = 3.8. Iterate until n = 20. Notice how thepatterns diverges significantly from that in part (c) after n = 10.

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1. A car parked in the sun absorbs energy at a rate of 560 watts per square meter 320 K

of surface area. The car reaches a temperature at which it radiates energy at thissame rate. Treating the car as a perfect radiator (e = 1), find the temperature.

2. An ice chest at a beach party contains 12 cans of soda at 5.0 ◦C. Each can of 19 ◦C

soda has a mass of 0.35 kilograms and a specific heat capacity of 3800 J/kg-◦C.Someone adds a 6.5-kilogram watermelon at 27 ◦C to the chest. The specific heatcapacity of watermelon is nearly the same as that of water. Ignore the specificheat capacity of the chest and determine the final temperature of the soda andwatermelon.

3. Find the mass of water that vaporizes when 2.10 kilograms of mercury at 205 9.49 grams◦C is added to 0.110 kilograms of water at 80.0 ◦C.

4. A steel rod (ρ = 7860 kg/m3) has a length of 2.0 meters. It is bolted at both 1100 newtons

ends between immobile supports. Initially there is no tension in the rod, becausethe rod just fits between the supports. Find the tension that develops when therod loses 3300 joules of heat.

5. Sirius B is a white star that has a surface temperature (in kelvin) that is four 8.7× 106 meters

times that of our sun. Sirius B radiates only 0.040 times the power radiated byour sun. Our sun has a radius of 6.96 × 108 meters. Assuming that Sirius B hasthe same emissivity as the sun, find the radius of Sirius B.

6. One end of a 0.25-meter copper rod with a cross-sectional area of 1.2 × 10−4 35%

m2 is driven into the center of a sphere of ice at 0 ◦C (radius = 0.15 meters). Theportion of the rod that is embedded in the ice is also at 0 ◦C. The rod is horizontaland its other end is fastened to a wall in a room. The rod and the room are keptat a constant temperature of 24 ◦C. The emissivity of the ice is 0.90. What isthe ratio of the heat per second gained by the sphere through conduction to thenet heat per second gained by the ice due to radiation? Neglect any heat gainedthorugh the sides of the rod.

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1. Figure 3 shows two thermally insulated tanks. They are connected by a valve (a) 330 K(b) 280 kPathat is initially closed. Each tank contains neon gas at the pressure, temperature,

and volume indicated in the drawing. When the valve is opened, the contents ofthe two tanks mix, and the pressure becomes constant throughout. (a) What isthe final temperature? Ignore any change in temperature of the tanks themselves.(Hint: The heat gained by the gas in one tank is equal to that lost by the other.)(b) What is the final pressure?


2. The artificial sweetener NutraSweet is a chemical called aspartame (C14H18N2O5). (a) 294.307 u.(b) 4.887× 10−25 kilogramsWhat is (a) its molecular mass (in atomic mass units) and (b) the mass (in

kilograms) of an aspartame molecule?

3. Oxygen for hospital patients is kept in special tanks, where the oxygen has a 67.0 m3

pressure of 65.0 atmospheres and a temperature of 288 K. The tanks are storedin a separate room, and the oxygen is pumped to the patient’s room, where itis administered at a pressure of 1.00 atmosphere and a temperature of 297 K.What volume does 1.00 m3 of oxygen in the tanks occupy at the conditions inthe patient’s room?

4. A clown at a birthday party has brought along a helium cylinder, with which 12 balloons

he intends to fill balloons. When full, each balloon contains 0.034 m3 of heliumat an absolute pressure of 120 kPa. The cylinder contains helium at an absolutepressure of 16,000 kPa and has a volume of 0.0031 m3. The temperature of thehelium in the tank and in the balloons is the same and remains constant. Whatis the maximum number of balloons that can be filled?

5. The diffusion constant for ethanol in water is 1.24×10−9 m2/s. A cylinder has 1.34× 10−7 kilograms

a cross-sectional area of 4.00 cm2 and a length of 2.00 cm. A difference in ethanolconcentration of 1.50 kg/m3 is maintained between the ends of the cylinder. Inone hour, what mass of ethanol diffuses through the cylinder?

6. What is the density (in kg/m3) of nitrogen gas (molecular mass = 28 u) at a 2.2 kg/m3

pressure of 2.0 atmospheres and a temperature of 310 K?

7. Estimate the height at which atmospheric pressure is half the value at sea 5.8 meters

level, making clear your assumptions.

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1. A power plant taps steam superheated by geothermal energy to 505 K (the (a) 36%(b) 1.3× 1013 joulestemperature of the hot reservoir) and uses the steam to do work in turning the

turbine of an electric generator. The steam is then converted back into water in acondenser at 323 K (the temperature of the cold reservoir), after which the wateris pumped back down into the earth where it is heated again. The output power(work per unit time) of the plant is 84000 kilowatts. Determine (a) the maximumefficiency at which this plant can operate and (b) the minimum amount of rejectedheat that must be removed from the condenser every twenty-four hours.

2. Three moles of an ideal monatomic gas are at a temperature of 345 kelvin. 436 kelvin

Then, 2438 joules of heat is added to the gas, and 962 joules of work is done onit. What is the final temperature of the gas?

3. A Carnot engine operates with an efficiency of 27.0% when the temperature 256 kelvin

of its cold reservoir is 275 kelvin. Assuming that the temperature of the hotreservoir remains the same, what must be the temperature of the cold reservoirin order to increase the efficiency to 32.0%?

4. Heat Q flows spontaneously from a reservoir at 394 K into a reservoir that 267 K

has a lower temperature T . Because of the spontaneous flow, thirty percent ofQ is rendered unavailable for work when a Carnot engine operates between thereservoir at temperature T and a reservoir at 248 K. Find the temperature T .

5. The water in a deep underground well is used as the cold reservoir of a Carnot 285 K

heat pump that maintains the temperature of a house at 301 kelvin. To deposit14200 joules of heat in the house, the heat pump requires 800 joules of work.Determine the temperature of the well water.

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Physics 202 Examples 7May 8, 2013

1. A person lying on an air mattress in the ocean rises and falls through one (a) 0.200 hertz(b) 4.00 m/scomplete cycle every five seconds. The crests of the wave causing the motion are

20.0 meters apart. Determine (a) the frequency and (b) the speed of the wave.

2. A spider hangs from a strand of silk whose radius is 4.0 × 10−6 meters. The 0.52 grams

density of the silk is 1300 kg/m3. When the spider moves, waves travel along thestrand of silk at a speed of 280 m/s. Determine the mass of the spider.

3. A wave traveling along the x axis is described mathematically by the equation 15 m/s

y = 0.17 sin(8.2πt+ 0.54πx)

where y is the displacement (in meters), t is in seconds, and x is in meters. Whatis the speed of the wave?

4. A team of geophysicists is standing on the ground. Beneath their feet, at 3.23 meters

an unknown distance, is the ceiling of a cavern. The floor of the cavern is adistance h below this ceiling. To measure h, the team places microphones on theground. At t = 0 seconds, a sound pulse is sent straight downward through theground and into the cavern. When this pulse reaches the ceiling of the cavern, onepart of it is reflected back toward the microphones, and a second part continuesdownward, eventually to be reflected from the cavern floor. The sound reflectedfrom the cavern ceiling reaches the microphones at t = 0.0245 seconds, and thesound reflected from the cavern floor arrives at t = 0.0437 seconds. The cavernis presumed to be filled with air at a temperature of 9 ◦C. Assuming that airbehaves like an ideal gas, what is the height h of the cavern?

5. A middle-aged man typically has poorer hearing than a middle-aged woman. 17%

In one case a woman can just begin to hear a musical tone, while a man can justbegin to hear the tone only when its intensity level is increased by 7.8 dB relativeto that for the woman. What is the ratio of the sound intensity just detected bythe man to that just detected by the woman?

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1. Figure 4 shows a loudspeaker A and point C, where a listener is positioned. 3.89 meters

A second loud-speaker B is located somewhere to the right of A. Both speakersvibrate in phase and are playing a 68.6-Hz tone. The speed of sound is 343 m/s.What is the closest to speaker A that speaker B can be located, so that the listenerhears no sound?


2. Two waves are traveling in opposite directions on the same string. The (a) +13.3 units(b) +48.8 unitsdisplacements caused by the individual waves are given by

y1 = (24.0) sin(9.00πt− 1.25πx)

andy2 = (35.0) sin(2.88πt+ 0.400πx)

Note that the phase angles are in radians, t is in seconds, and x is in meters.At t = 4.00 seconds, what is the net displacement of the string at (a) x = 2.16meters and (b) x = 2.56 meters? Be sure to include the algebraic sign with youranswers.

3. The two speakers in Figure 5 are vibrating in phase, and a listener is standing (a) Destructive(b) Constructiveat point P. Does constructive or destructive interference occur at P when the

speakers produce sound waves whose frequency is (a) 1466 Hz and (b) 977 Hz?Justify your answers with appropriate calculations. Take the speed of sound tobe 343 m/s.


4. Sound exits a diffraction horn loudspeaker through a rectangular opening like 16.0◦

a small doorway. Such a loudspeaker is mounted outside on a pole. In winter,

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when the temperature is 273 K, the diffraction angle has a value of 15.0◦. What isthe diffraction angle for the same sound on a summer day when the temperatureis 311 K?

5. A 41-cm length of wire has a mass of 6.0 grams. It is stretched between two 130 hertz

fixed supports and is under a tension of 160 newtons. What is the fundamentalfrequency of this wire?

6. A thin 1.2-meter aluminum rod sustains a longitudinal standing wave with 2100 hertz

vibration antinodes at each end of the rod. There are no other antinodes. Thedensity and Young’s modulus of aluminum are, respectively, 2700 kg/m3 and6.9 × 1010 N/m2. What is the frequency of the rod’s vibration?

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1. The image behind a convex mirror (radius of curvature = 68 cm) is located (a) 62 cm(b) 0.35(c) Upright(d) Smaller

22 cm from the mirror. (a) Where is the object located and (b) what is themagnification of the mirror? Determine whether the image is (c) upright orinverted and (d) larger or smaller than the object.

2. An object is placed 20.0 cm to the left of a diverging lens (f = -8.00 cm). A (a) 18.1 cm(b) Real(c) Inverted

concave mirror (f = 12.0 cm) is placed 30.0 cm to the right of the lens. (a) Findthe final image distance, measured relative to the mirror (b) Is the final image realor virtual? (c) Is the final image upright or inverted with respect to the originalobject?

3. A ray of light strikes a plane mirror at a 45◦ angle of incidence. The mirror is (a) 30◦

(b) 30◦then rotated by 15◦ into the position shown in red in Figure 6, while the incidentray is kept fixed. (a) Through what angle φ does the reflected ray rotate? (b)What is the answer to part (a) if the angle of incidence is 60◦ instead of 45◦?


4. Figure 7 shows a laser beam shining on a plane mirror that is perpendicular 1.67 meters

to the floor. The beam’s angle of incidence is 33.0◦. The beam emerges from thelaser at a point that is 1.10 meters from the mirror and 1.80 meters above thefloor. After reflection, how far from the base of the mirror does the beam strikethe floor?


5. This problem relates to Figure 8, which illustrates the dispersion of light by 52.7◦ and 56.3◦, respectively

a prism. The prism is made from glass, and its cross section is an equilateraltriangle. The indices of refraction for the red and violet light are 1.662 and 1.698,respectively. The angle of incidence for both the red and violet light is 60.0◦.Find the angles of refraction at which the red and violet rays emerge into the airfrom the prism.

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6. A converging lens (f = 12.0 cm) is located 30.0 cm to the left of a diverging lens (a) 4.00 cm to the left(b) -0.167 (c) Virtual (d) Inverted (e)Smaller

(f = -6.00 cm). A postage stamp is placed 36.0 cm to the left of the converginglens. (a) Locate the final image of the stamp relative to the diverging lens (b)Find the overall magnification, (c) Is the final image real or virtual? With respectto the original object, is the final image (d) upright or inverted, and is it (e) largeror smaller?

7. An example of a gravitational lens is the “Einstein Cross” depicted in Figure 91 7.8× 1041 kilograms

located in the Pegasus constellation. The four outer points of light are images ofa quazar 8.0 billion light years away that is directly behind some galaxy. Thelight from the quazar bends around the galaxy and reaches earth from fourdifferent directions. The angular deflection of the images is about 7.8 × 10−6

radians. Assume the deflecting galaxy is in the middle and estimate the mass ofthe deflecting galaxy.

Figure 9: Gravitational Lens G2237 + 0305, Einstein Cross

1Source: http://hubblesite.org/newscenter/archive/releases/1990/20/

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http://hubblesite.org/newscenter/archive/releases/1990/20/


1. In a Young’s double-slit experiment, the seventh dark fringe is located 0.025 470 nm

meters to the side of the central bright fringe on a flat screen, which is 1.1 metersaway from the slits. The separation between the slits is 1.4× 10−4 meters. Whatis the wavelength of the light being used?

2. The diffraction gratings discussed in the text are transmission gratings because (a) 37◦

(b) 22◦light passes through them. There are also gratings in which the light reflects fromthe grating to form a pattern of fringes. Equation 27.7 also applies to a reflectiongrating with straight parallel lines when the incident light shines perpendicularlyon the grating. The surface of a compact disc (CD) has a multicolored appearancebecause it acts like a reflection grating and spreads sunlight into its colors. Thearms of the spiral track on the CD are separated by 1.1 × 10−6 meters. UsingEquation 27.7, estimate the angle that corresponds to the first-order maximumfor a wavelength of (a) 660 nm (red) and (b) 410 nm (violet).

3. A nonreflective coating of magnesium fluoride (n = 1.38) covers the glass (n = 102 nm

1.52) of a camera lens. Assuming that the coating prevents reflection of yellow-green light (wavelength in vacuum = 565 nm), determine the minimum nonzerothickness that the coating can have.

4. A soap film (n = 1.33) is 465 nm thick and lies on a glass plate (n = 1.52). Only at 495 nm

Sunlight, whose wavelengths (in vacuum) extend from 380 to 750 nm, travelsthrough the air and strikes the film perpendicularly. For which wavelength(s) inthis range does destructive interference cause the film to look dark in reflectedlight?

5. Astronomers have discovered a planetary system orbiting the star Upsilon 2.3 meters

Andromedae, which is at a distance of 4.2 × 1017 meters from the earth. Oneplanet is believed to be located at a distance of 1.2 × 1011 meters from the star.Using visible light with a vacuum wavelength of 550 nm, what is the minimumnecessary aperture diameter that a telescope must have so that it can resolve theplanet and the star?

6. The pupil of an eagle’s eye has a diameter of 6.0 mm. Two field mice are 5.1 seconds

separated by 0.010 meters. From a distance of 176 meters, the eagle sees them asone unresolved object and dives toward them at a speed of 17 m/s. Assume thatthe eagle’s eye detects light that has a wavelength of 550 nm in vacuum. Howmuch time passes until the eagle sees the mice as separate objects?

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Physics 202 Examples 11Jun 5, 2013

1. When an electron and a positron (its antimatter twin) collide they typically 2.43× 10−12 meters

annihillate in a burst of electromagnetic energy. Assume all the energy from themass of the pair (each equal to 9.11 × 10−31 kilograms) is converted into twophotons of energy. What is the wavelength of these photons?

2. Use Wien’s law to estimate the temperature at the surface of the sun given 5800 kelvin

that its radiation spectrum peaks at 500 nm.

3. The diameter of the pupil of the eye is about 5 mm. Find the intensity of light 1.7× 10−15 W/m2

of wavelength 600 nm such that one photon per second passes through the pupil.

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