PHYS 240 Waves, Thermo, & Modern, instructed by William Caudy

Homework ( jump to set 1 2 3 4 5 6 7 8 9 10 11 )

There are ten Homework sets with three to four questions each. Your homework solutions must be hand written and drawn on paper with no torn edges. Legibility is your responsibility if the grader can't read it, it's zero credit. Questions may ask for graphs, written explanations, diagrams, or calculations.

Your solution to any homework calculation must be in the form of an equation, written in terms of the variables given in the question and any fundamental constants. Your job is to show how the equation is derived algebraically from the basic physical principles and equations covered in class notes.

Work with your classmates to solve homework questions! Attempt homework questions on your own referring to your book and notes, then work with a group and come to office hours. Also you are free to attend the Help Sessions of all lab instructors, not just your own.

Homework set 1 due 9/6   Solutions
Question 1:

Express the image distance si of a convex mirror in terms of the object distance so and the radius of curvature of the mirror R. Keep in mind that the focal length is considered negative for a convex mirror.

Hint: si = -R/2 - so is incorrect!

Question 2:

Express the image distance si of a concave mirror in terms of the object distance so and the radius of curvature of the mirror R. The focal length is positive for a concave mirror. For what range of si is the image virtual? For what range is the image real?

Hint: The image is virtual for 0 < so < f and real for so > f.
What are the corresponding ranges for si?

Question 3: Lea & Burke 18.38

A photographer takes a self-portrait of his own reflections in a bubble. The front surface of the bubble (closest to the photographer) acts as a convex mirror and the back surface (farther from the photographer) acts as a concave mirror. If the photographer's face is a distance d from the front surface of the bubble, and the radius of the bubble is r << d, then what are the locations of the images created by these two mirrors? Draw a ray diagram for each image.

Sample Answer: For d = 28 cm, r = 2 cm,
the virtual image formed by the front surface is (-)28/29 cm behind (negative) the front surface.
the real image formed by back surface is 32/31 cm in front of the back surface.

Question 4:

A reflecting telescope makes a nebula 900 times brighter than it would appear to the naked eye. If the maximum radius of your pupil is r, express the radius of the telescope's mirror Rmirror as a multiple of r. (If you thought the mirror radius was 900 times greater than the pupil radius, you would write Rmirror = 900r. This is not the correct answer!)

Answer: Rmirror = 30r

Homework set 2 due 9/13   Solutions

Question 1:

An equilateral glass prism is to be designed so that a light beam entering the prism perpendicular to the first face will be totally internally reflected once off the second face and then emerge perpendicular to the third face. Find the minimum necessary index of refraction.

Question 2:

Use the thin lens equation to find a general expression for image distance in terms of object distance and focal length. Using the convention that focal length is positive for a converging (convex) lens and negative for diverging (concave) lens, under what conditions can a lens produce a real image?

Question 3:

An object is so = 2 m from the front side of a converging lens with focal length f1 = 1 m. A second converging lens with focal length f2 = 2 m is placed a distance d = 1 m from the first on the opposite side as the object. Draw a ray diagram of the situation. What is the distance of the final image from the second lens? What is the magnification of the lens system? The object is now moved inward until it is at so = 1 m from the front of the first lens. Where is the final image now? What is the new magnification? You do not need to draw a ray diagram of the second situation.

Hint: For an object on the transmission side of the lens 1/so < 0. (This only happens with multiple lens systems. It is a virtual object, because the second lens bends the rays before this object every forms.)

Answers: For so1 = 2 m, si2 = 2/3 m and m1&2 = -1/3. For so1 = 1 m, si2 = 2 m and m1&2 = -2.

Question 4 updated:

A telescope consists of two converging lenses of focal lengths f1 and f2 separated by a distance d. An object is a distnace so1 from lens 1. The image createed by lens 1, at a distance si1 from lens 1, becomes the object for lens 2, at a distance so2 = d - si1 from lens 2. The image created by lens 2 (the final image) is a distance si2 from lens 2.

Show that, for an object at infinity (such as the Moon) with so1 → +∞, the final image location becomes si2 → (d - f1)f2/(d - f1 - f2), and this image is virtual when the lens separation d is in the range f1 < df1 + f2.

Still assuming an object at infinity, so1 → +∞, show that for df1 + f2 (limit from the left), si2 → -∞, and the magnification of the telescope becomes m → -f1/f2 (so for example, in the case f1 > f2, the image of the Moon would be inverted, magnified, and still appear to be very far away "at infinty"). So a virtual image appears at -∞. I leave it to you to decide whether a real image is formed at +∞ in actuality (limit from the right).

 Homework set 3 due 9/27   Solutions
Question 1:

Mork (observer 1) is on a train car moving with speed v = 2c/3 relative to Mindy (observer 2). Mork measures the positions of the front and rear of the car simultaneously (relative to him) to determine the car's length. He measures the rear at position xrear1 at time trear1, and the front at
xfront1 = xrear1 + L1 at time tfront1 = trear1 (the car's length is L1 and Mork's measurements are simultaneous in reference frame 1).
(a) What is the car's length L2 according to Mindy in reference frame 2?
(b) Relative to Mindy, how much time passed between Mork's measurements of the car's rear and front positions? Use the Lorentz coordinate transformation rule to find tfront2 - trear2, where trear2 is the time Mindy sees Mork measure the car's rear position and tfront2 the time she sees Mork measure the car's front position.

The Lorentz coordinate transformation rule:
t2 = γ ( t1 - x1v21/c2 )
x2 = γ ( x1 - v21t1 )
for observer 2 moving with velocity v21 relative to observer 1, or
t1 = γ ( t2 - x2v12/c2 )
x1 = γ ( x2 - v12t2 )
for 1 moving in with velocity v12 = -v21 relative to 2.

Hint: You are given Mork and Mindy's relative speed. Because xfront1 = xrear1 + L1 > xrear1, we know Mork (observer 1) is moving in the positive x direction relative to Mindy (observer 2), and Mindy is moving in the...

Question 2:

Observer 1 sees observer 2 moving with speed v = 2c/3 to the right and observer 3 moving with speed v = 2c/3 to the left.
(a) By how much is observer 2's clock dialated relative to observer 1?
(b) What is observer 2's speed relative to observer 3?
(c) By how much is observer 2's clock dialated relative to observer 3?

Question 3:

An astronaut with an identical twin leaves Earth on her 20th birthday travelling away with speed v = 2c/3 (reference frame 1). The twin on Earth (reference frame 2) sends out a signal to her astronaut sibling at 12AM everyday (the time between signals is one day relative to reference frame 2); the signal is a simple light pulse. The astronaut travels in a straight line until her 21st birthday (1 year after leaving Earth relative to reference frame 1), and then turns around and returns home at speed v = 2c/3 (reference frame 3) along the same straight line.
(a) How frequently does the astronaut receive the light pulse signal on the way out? Find the frequency relative to reference frame 1.
(b) What is the Earth twin's age according to the astronaut just before turning around?
(c) What is the Earth twin's age according to the astronaut just after turning around?
(d) How frequently does the astronaut receive the light pulse signal on the way back? Find the frequency relative to reference frame 3.
(e) What are the ages of the astronaut and Earth twin, according to both twins, on the day the astronaut returns to Earth?

 Homework set 4 due 10/4   Solutions
Question 1

A wave of amplitude Ai and wave number ki is incident on the junction of two strings. If the wave number in the second string is kt = 2ki , find the amplitudes of the transmitted and reflected waves. By what fraction is the speed of the wave reduced as it crosses the junction? What fraction of the incident power is transmitted? (Power is proportional to amplitude square but also depends on wavespeed, so it is easiest to compare the power reflected to that of the incident wave, and then find the transmitted power by requiring energy be conserved across the boundary.)

Answers: At = (2/3)Ai , Ar = (-1/3)Ai , vt/vi = 1/2, Pt/Pi = 8/9

Question 2

A resonance tube for a pipe-organ is closed at the bottom by a speaker (pressure antinode) and open to the atmosphere at the top (pressure node). How long should the tube be built so that its fundamental resonant frequency is 64 Hz (lowest C)? Assume the speed of sound is 340 m/s. What are the next two harmonic frequencies above the fundamental frequency that would create resonance in the same tube? What length tubes (also closed at one end) would have as their fundamental frequencies 128 Hz, 192 Hz, and 256 Hz?

Answers: 1.33m, 192Hz, 320Hz, 66.4cm, 44.3cm, 33.2cm

Question 3

An amplifier supplies 5.0 Watts to a perfectly efficient loudspeaker at one end of a 10.0 cm diamter tube filled with air. What is the intensity of the sound at the other end of the tube? What is the intensity 10 m in front of the tube (approximate the wavefront to be a hemisphere moving outward in the direction that the loudspeaker is pointing). If the frequency of the sound is 250 Hz, how far apart are the wavefronts? The speed of sound is 340 m/s.

Answers: 637 W/m2, 0.00796 W/m2, 1.36 m

Question 4:

A siren creates a 800 Hz tone while at rest. It is mounted on an ambulance and turned on while traveling at 20.0 m/s. What frequency is heard by an observer at rest in front of the ambulance? By an observer at rest behind the ambulance? By an observer in front of the ambulance moving towards it at 20.0 m/s? By an observer behind the ambulance moving towards it at 20.0 m/s? The speed of sound is 340 m/s.

Answers: 850 Hz, 756 Hz, 900 Hz, 800 Hz

 Homework set 5 due 10/18   Solutions
Question 1

A laser with light of wavelength 500 nm shines through two parallel thin slits separated by 0.5 mm onto a screen, and an interference pattern is seen. At what distance should the screen be placed so that nearest maxima of the interference pattern are 5 cm apart?

Answer: 50 m

Question 2:

A diffraction grating with 5.00 102 lines/mm is used to observe a spectral line at 447.6 nm. How far from the central maximum is the next primary maximum?

Answer: angle from center to first primary maximum is 12.9

Question 3:

If the 447.6 nm line from Question 3 is a blend of He (447.1 nm) and MgII (448.1 nm), then what N is required to seprate the two wavelengths? (This is when the first off-center maximum of one wavelength, with phase difference 2pi between nearest slits, overlaps a minimum of the other wavelength, with phase difference 2pi(1 +or- 1/N) between nearest slits.)

Answer: N = 448 is needed to resolve the two wavelengths

Question 4:

One glass plate rests on top of another. The plates touch on one side and are separated on the other side by a fine wire of diameter d. The plates are illuminated by light of wavelength 540 nm and six bright fringes are observed. This corresponds to six intensity maxima between by seven intensity minima. The minimum along the side where the plates touch is due to the 180 phase difference between rays reflected of the bottom of the top plate (free end) and the top of the lower plate (fixed end). What is the diameter of the wire?

Answer: d = 1.6210-6 m

 Homework set 6 due 10/25   Solutions
Question 1

Copper has a work function of 4.7eV, while gold has a work function of 5.1eV. What are the minimum frequencies of light needed to emit electrons off each metal? Suppose a lamp is made which shines light of the exact minimum frequency needed to emit electrons off of gold. If this lamp shines onto copper, with what speed are electrons emitted from the copper?

Hint: Use Planck's constant in electron Volts, h = 4.135710-15 (eV)s. The mass of an electron is always the same (look it up).

Question 2

What is the minimum frequency of light needed to free an electron from a neon nucleus when the electron is in its lowest energy level n=1? What frequency is emitted by an electron that falls from the n=7 to the n=2 energy level of a neon nucleus?

Question 3

Consider a single photon of orange light with wavelength 600nm. What are the momentum, energy, and frequency of this photon?

Apply the Heisenberg uncertainly relation for position and momentum to this orange photon, ΔxΔph/4π where h is Planck's constant. Assume the wavelength is a good approximation of the uncertainty of the photon's position. What are the minimum uncertainties of the photon's momentum, energy, and frequency? (Consider what momentum uncertainty means for a photon, a particle which necessarily has rest mass exactly equal to zero and speed exactly equal to c = 3×108m/s.) Rewrite the Heisenberg position/momentum uncertainty relation for photons as an uncertainty relation for position and frequency with the appropriate constants.

Question 4

Consider the Heisenberg uncertainty relation for energy and time ΔEΔth/4π where h is Planck's constant.

Apply this uncertainty relation to electron/positron pair production, a process by which an electron and positron can appear out of vacuum so long as the two particles exist for a time shorter than Δt. (A positron is the electron's antiparticle, having equal mass but opposite charge.) The two particle's are created moving apart with approximately equal and opposite velocities, but in a brief time (less than or equal to Δt) they pull each other back together and annhilate upon colliding. For electron/positron pair production, ΔE is the uncertainty in the energy of the vacuum, which may vary by as much as the total energy of an electron/positron pair.

An electron and positron are created with speeds much less than c, so that the pair's total energy is approximately equal to their combined rest energy. What is the maximum time that the pair can exist, from when they are created out of vacuum to when they collide and annhilate?

 Homework set 7 due 11/03   Solutions
Question 1 Lea & Burke 13.17

A U-shaped pipe contains water and oil - water filling one arm of the U and oil filling the other. The two fluids meet at the bottom of the U. The water level is 40 cm high and the oil level is 50 cm high. Find the density of the oil.

800 kg/m3

Question 2 Lea & Burke 13.65

The Goodyear blimp is filled with 5740 m3 of helium with density pHe = 0.18 kg/m3. The blimp has a mass of 5430 kg when empty. How much weight can the blimp carry? Take pair = 1.3 kg/m3.

9788 N

Question 3

A 100 m tall water tower holds a huge amount of water, so that the water level in the tower remains approximately constant even when some water flows out through a faucet at the bottom of the tower. With what speed does water come out the faucet when opened? What is the flow rate of water out of the faucet, volume of water per time, if the faucet has radius 1 cm?

Question 4

A pump is designed to pump water at a flow rate of 3 gallons per second through a 1 inch diameter hose at zero elevation. The water flows into a very long hose (of constant cross-section) that can withstand large pressures. You walk the end of the hose uphill and notice that the flow rate decreases. You continue walking uphill until the flow rate drops to zero. To what elevation have you walked, and what is the pressure at the pump? (Hint: the pump is supplying the same energy per volume to the water, but now that energy is going into maintaining a pressure difference at the pump, rather than to accelerating the water.) You use this pump to pump water up 100 ft. What is the flow rate through the end of the hose at this elevation?

Practice Question 5 (optional) Lea & Burke 13.62

A piece of rock weighs 301 N in air and has an apparent weight of 202 N when suspended under water. What are the volume and density of the rock?

0.0101 m3, 3040 kg/m3

Practice Question 6 (optional)

Blood flows through the aorta at speed 30 cm/s. The cross-sectional area of the aorta is initially 4 cm2. Though a single capillary is very small, the total cross-sectional area of all the capillaries fed by the aorta is 6000 cm2, over half a square meter! What is the flow rate of blood through the capillaries?

0.02 cm/s

 Homework set 8 due 11/29   Solutions to 0, 1, & 2 and Solutions to 3
Practice Question 0 (optional) Lea & Burke 19.23
Mercury freezes at -38.87C. What is this temperature on the Kelvin and Fahrenheit scales?

234.28 K and -37.97F
Question 1 Lea & Burke 20.31

A household thermometer contains 2.5 cm3 of alcohol. The diameter of the stem is 1.5 mm. How far apart are the marks for (a) C?, (b) F, and (c) K?

(a) 1.5610-3 m,
(b) 8.6710-4 m, and
(c) 1.5610-3 m
Question 2 Lea & Burke 21.19

By how much does a 10-cm-thick layer of glass wool insulation decrease the heat loss through the roof of a house (area 100 m2, Rf = 0.15 m2K/W for the roof alone, Rf = 2.5 m2K/W for 10-cm-thick glass wool) if the temperature difference between the interior and the exterior is 25 K?

1.57104 W
Question 3 Lea & Burke 20.38

An insulated metal vessel of mass 5.0 kg contains 20.1 kg of water. The vessel and its contents are in thermal contact, but are thermally insulated from their environment. A 2.5 kg piece of the same metal, intitally at a temperature of 190C is dropped into the water. If the water temperature rises from 16.0C to 17.8C, calculate the specific heat of the metal.

359 J/kgK

 Homework set 9 due 12/6   Solutions to 0 & 1 and Solutions to 2 & 3
Question 0 (optional) Lea & Burke 19.37

A souffle is made from ingredients at 50F and contains trapped air. When put into a 375F oven, the souffle rises by a factor of 1.4 (Vf = 1.4V0). What fraction of the souffle's original volume is air? What fraction of its final volume is air? (Hint: Vtotal = Vair + Vother , where air expands as an ideal gas, and the other stuff in the souffle does not expand, Vother0 = Votherf .)

0.626, 0.733
Question 1 Lea & Burke 19.40

As morning sunlight falls on a tethered helium balloon, it expands from a volume of 1.05 103 m3 to 1.15 103 m3. How much work does the helium do on its environment (assuming the balloon's elastic force can be ignored)? If the initial temperature of the helium is 270K, how many moles of gas are in the balloon, how much does the helium's internal energy increase, and how many joules of solar energy are absorbed?

1.01 107 J, 4.73 104 mol, 1.52 107 J, 2.53 107 J    
Question 2 Lea & Burke 19.47

A tank of argon gas has a volume of 0.0160 m3. The gas has a pressure of 5.00 atm and a temperature of 293 K. How much argon is in the tank? The tank is fitted with a piston and the gas is allowed to expand isothermally until its pressure is 3.00 atm. Subsequently, the pressure drops to 1.00 atm at constant volume. Find the final temperature and volume of the gas. How much work was done by the gas? Graph the pressure versus volume of the gas for this two-step process. The graph may be drawn, but use a ruler and correctly measure the distance between points along both the P and V axes.

N = 1.998 1024 atoms, Wby = 4127 J
Question 3 Lea & Burke Figure 19.17

Calculate the work done by 10 moles of an ideal gas in each step of the following cycle: The gas is initially in state A with pressure PA = 6 104 Pa and volume VA = 0.15 m3. It expands isobarically to state B with volume VB = 0.30 m3. Then the gas' pressure is reduced isochorically to PC = 3 104 Pa. Finally the gas returns to state A isothermally. How much heat is added to or lost by the gas in each step?

WA = 9000 J, QA = 22500 J,
WB = 0 J, QB = -13500 J,
WC = -6238 J, QC = -6238 J

 Homework set 10 due 12/13   Solutions to 0 , Solutions to 1 & 4 , and Solutions to 2 & 3
Question 0 (optional) Lea & Burke Figure 22.3

N molecules of ideal gas initially at volume VA and temperature TA expands isothermally to volume VB and then expands adiabatically to volume VC . Find VB in terms of N, VA , TA , VC , TC .
The gas then is compressed isothermally to state D volume VD and then compressed adiabatically to state A. Find VD in terms of the same variables.

VB = (TC/TA)3/2VC , VD = (TA/TC)3/2VA
Question 1 Lea & Burke 22.26
A Carnot cycle uses a cold reservoir at 280 K. In each cycle, 2500 J of heat is supplied and 1200 J of work is done. What is the temperature of the hot reservoir? If the cold reservoir could be lowered to 210 K (while the hot reservoir remains at the same temperature found above), how much work could be done for 2500 J of heat supplied?

538.5 K, 1525 J
Question 2 Lea & Burke 22.20

An engine uses a fixed amount of ideal monatomic gas in the cycle shown above. Find the efficiency of the engine.

e = 13.38%
Question 3 Lea & Burke 22.38
You put a kettle on to boil and forget about it so that all the water boils away. If initially there was 4.0 10-3 m3 of water at 20C in the kettle, what is the change in entropy of the water once it has all turned to steam?

Sf - Si = 2.841 104 J/K
Question 4 (optional) Lea & Burke 22.41

Show that for N molecules of an ideal gas S2 - S1 = Mcvln(T2/T1) + Nkln(V2/V1) where M is the total mass of the system. Does change in entropy equal zero for adiabatic expansion of an ideal gas according to this equation?

 Homework set 11 (optional) due 12/13
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