Physics 1B Additional Problems Instructor: Eyal Goldmann 1.1 Fluid Statics (1.1.05) According to the 1976 U.S. standard atmosphere (you may look up the Wikipedia article on this) typical atmospheric pressure at 11 km altitude is 22.6 kPa, and typical atmospheric pressure at 20 km altitude is 5.5 kPa. What is the mass of air in this layer above an area on the surface of the earth 2.0 miles long and 1.0 mile wide? (1.1.10) The diagram below shows a U-shaped tube. The fluids in the tube are glycerine (blue shading) and mercury (gray shading). The height of the glycerine column on the left side of the tube is h1 = 5.00 cm. The density of the mercury is 13.53 g/cm3. (a) What is h2? (b) Oil with a density 0.800 g/cm3 is added to the right side of the tube, so that the right side of the tube has an oil column sitting on top of the mercury. What must be the depth of this oil column so that the top of the glycerine column on the left and the top of the oil column on the right are at the same height? Keep four significant digits in any intermediate steps.

h1

h2

(1.1.20) The diagram below shows a balloon which is connected to a massless string, which then passes through a pulley and connects to a spring which has one end attached to a fixed wall. The balloon has radius R = 40.0 cm, and is filled with helium. The balloon skin has a mass of 30.0 g. The spring is stretched 20.0 cm. Take the density of air to be 1.29 kg/m3 and the density of helium to be 0.178 kg/m3. (a) Assuming that the string is massless, what is the spring constant? Include a free-body diagram for the balloon in your analysis of this problem. (b) The massless string is replaced with a steel cable. The linear mass density of the steel cable is 2.0 g/cm. It is noticed that when the spring is held at its “resting” length, the vertical portion of the cable has length h = 10.0 cm. When the spring is released, and allowed to stretch in response to the tension in the cable, how far does it extend? (If you haven’t encountered linear mass density before, check the index of your textbook.) (1.1.30) In the picture below, a 7.00 kg piece of aluminum hanging from a spring scale is immersed in water. The total mass of the water is 3.00 kg, and it is contained in a beaker with a mass of 2.00 kg. The beaker sits on top of another scale. Find the readings on both scales. Take the density of aluminum to be 2700 kg/m3 and the density of water to be 1000 kg/m3.

R

h

(1.1.40) An iron cylinder and a silver cylinder have the same length, L = 10cm, and different cross-sectional areas, AFe = 4cm2, and AAg = 5.0cm2. The cylinders are connected to each other by a string which passes over a pulley. Initially they are held in place so that each is half-submerged in mercury. When the cylinders are released, the iron cylinder moves upward through a distance x and the silver cylinder moves downward through a distance x, allowing the system to come to equilibrium. Find x. You will need the densities of iron, silver, and mercury, which can be found in your textbook. (1.1.50) The diagram below depicts a dam after an oil spill has taken place. The dam extends 80 m into the page. The oil layer is 10 m deep and the water layer is 30 m deep. Assume that the density of the oil is 850 kg/m3. (a) Set up a single y-axis which has its origin at the top of the oil level and points downward. Write down two functions: one which gives pressure in the oil as a function of y, and another which gives pressure in the water as a function of y. (b) Find the net force on the dam due to fluid pressure. (c) Find the net torque on the dam with respect to an axis passing through the bottom of the dam. (Hint: Each part of the problem requires two integrals: one for the oil layer and one for the water layer. The torque exerted by the oil layer is 1.11 × 109 N∙m and the torque exerted by the water layer is 6.53 × 109N∙m. This is probably the most computationally intense problem of the semester.)

Mercury

Fe Ag

Mercury

Fe

Ag

cylinders are held in place system is in equilibrium; no

external forces on cylinders

oil

water

(1.2) Surface Tension (1.2.10) A child blows a spherical soap bubble of radius R. The soap bubble is made of a film of soap water with surface tension 𝛾𝛾. There is air both outside and inside the soap film. Starting from first principles, show that the pressure difference between the air inside the film and the air outside the film is

∆𝑃𝑃 = 𝑃𝑃𝑖𝑖𝑖𝑖 − 𝑃𝑃𝑜𝑜𝑜𝑜𝑜𝑜 =4𝛾𝛾𝑅𝑅

. (1.2.20) A plastic cylinder with diameter 5.0mm and length 4.0cm is floating in water (density 1.0g/cm3) so that 2.5cm of the cylinder’s length is beneath the water surface. The wetting angle between this particular type of plastic and water is 90°. (a) What is the density of the cylinder? (b) A thin, massless coating is applied to the cylinder. This changes the wetting angle between the cylinder and the water to 20°. When the cylinder is placed back in the water, how deep is the bottom of the cylinder relative to the surface level? The surface tension of water is 72 dynes/cm. (Hint: It’s probably better in this problem to compute the force the water exerts on the cylinder directly from the formula for pressure vs. depth.)

1.5cm

2.5cm ? 20°

(1.2.30) A small amount of liquid is in contact with a horizontal solid surface. If gravity is neglected, the liquid will be in a shape we can call a “spherical cap.” That is, the surface of the liquid has a spherical shape, but the sphere is incomplete. Assume that the contact angle between the droplet and the surface is θ, where θ < 90°. Then the center of curvature for the surface of the droplet will be beneath the solid surface, as shown. Supposing that the surface tension is γ, and the radius of curvature is R, determine the pressure difference between the inside of the liquid and the air outside of the liquid. Ignore the effects of gravity. (Hint: A good first step would be to determine the contact area between the liquid and the solid.) (1.2.40) A straw with a 1.20-mm diameter is inserted into a tub of water, as shown. The wetting angle between water and the surface of the straw is 130°. Consequently, the liquid surface inside the straw is lower than the liquid surface outside the straw. Starting with a free-body diagram, determine the vertical distance h between the meniscus in the straw and the bulk surface level of the water. (Hint: Draw the free-body diagram for the column of water in the straw.) The surface tension of water is 72 dyne/cm.

θ θ

R

center of curvature

air

liquid

solid

h

Accelerating Fluids (1.3.10) (a) A fish tank filled with water and open on the top is accelerating upward at 3.00 m/s2. Starting from first principles, derive an equation which gives the pressure in the tank as a function of the depth h below the liquid surface. Assume that the pressure at the liquid surface is P0. (b) A cylinder of uniform cross section is floating in the fish tank, partially submerged as shown in the picture. The cylinder has the same upward acceleration as the water. Starting from a free-body diagram drawn for the cylinder, derive the density of the cylinder.

(1.3.20) In the diagram below, a tank of water open at the top surface is accelerating to the right at 2.50 m/s2, resulting in a sloped upper surface, as we saw in class. At the bottom of the tank is a steel sphere of radius R = 0.80 cm, which is connected to the right side of the tank by a light cord. The contact between the steel sphere and the bottom of the tank is frictionless. (a) What is the net horizontal force exerted on the steel sphere by the pressure of the surrounding water? (Hint: What would be your answer if the steel sphere was replaced by a bag of water with the same radius?) (b) Starting from a free-body diagram analysis, determine the tension in the cord. Only horizontal forces need to be included in the free-body diagram. The density of steel is 7.85 g/cm3.

Water surface

3.00 cm

5.00 cm

�⃗�𝑎

air

water

2.1 Fluid Dynamics (no viscosity) (2.1.05) A nozzle with a 3.00-mm exit diameter is placed on a garden hose. With the nozzle in place, a 1.50 liter bucket is filled in 80.0 seconds. What is the speed at which water exits the nozzle? (2.1.10) The following text is taken from howstuffworks.com: (see link http://science.howstuffworks.com/fire-engine2.htm)

The [fire] truck also has at least three lines called preconnects. These lines are preconnected to the truck in order to save time at the fire scene. There's one preconnect on the driver's side, one on the back and one on the captain's side of the truck. These lines are between 1.5 and 2.5 inches in diameter, and can put out 250 gallons (946 liters) per minute.

Suppose that the fire hose in the picture below has a 2.5-in diameter and is producing 250 gallons of water per minute. Suppose also that the fire fighter in the picture is bending the hose from the horizontal to a direction 30° above the horizontal. (A simplified sketch of the hose is given to the side.) Find the magnitude of the force the fire fighter is exerting on the hose.

(2.1.15) A liquid jet is made of liquid of density ρ traveling at speed v. The cross-sectional area of the jet is A. The jet is intercepted by a solid board. Upon hitting the board, the liquid is diverted equally in all directions parallel to the board. The board is held in place by a hand. Starting from first principles, derive the force that the hand must exert on the board.

30°

v

(2.1.20) Air is blowing through a horizontal tube which is connected to a U-shaped tube, as shown below. The left side of the U-tube is connected to a part of the horizontal tube with a diameter d1 = 3.00 cm. The right side of the U-tube is connected to a part of the horizontal tube with a diameter d2 = 2.00 cm. The liquid in the U-tube is water, except for a column of oil of height h = 5.00 cm on the left side of the U-tube. The liquid levels on the right and left sides of the U-tube are equal. What is the speed of the air in the narrow part of the horizontal tube? Use the following densities: 𝜌𝜌𝑎𝑎𝑖𝑖𝑎𝑎 = 1.29 kg/m3, 𝜌𝜌𝑤𝑤𝑎𝑎𝑜𝑜𝑤𝑤𝑎𝑎 = 1000 kg/m3, and 𝜌𝜌𝑜𝑜𝑖𝑖𝑜𝑜 = 800 kg/m3. (2.1.30) A vertical cylinder filled with water has a 5.0-cm inner diameter. There is a 1.0 cm hole near the bottom of the cylinder. The hole is a distance h beneath the water surface. (a) As a function of h, what is the speed at which water exits the hole? (b) As a function of h, what is the rate at which the water level in the cylinder drops? (c) If the water level in the cylinder is initially 40.0 cm above the hole, how much time elapses before the water level is 30.0 cm above the hole?

v1 v2

h

x

(2.1.40) Water flows out of a garden hose which has an opening 2.20 cm in diameter. The flow is such that the water fills a 2.50 gallon bucket in 30.0 seconds. The hose is supplied by an underground pipe 1.10 meter below the exit of the hose. The diameter of the underground pipe is 2.0 inches. (a) What is the flow speed of the water as it exits the hose? (b) What is the flow speed in the underground pipe? (c) What is the gauge pressure in the underground pipe? (2.1.50) Air is blowing through a horizontal tube which is connected to the right arm of a U-shaped tube, as shown below. The liquid in the U-tube is water, except for a column of oil of height h = 4.00 cm on the left side of the U-tube. The liquid levels on the right and left sides of the U-tube are equal. What is the speed of the air in the horizontal tube? Use the following densities: 𝜌𝜌𝑎𝑎𝑖𝑖𝑎𝑎 = 1.29 kg/m3, 𝜌𝜌𝑤𝑤𝑎𝑎𝑜𝑜𝑤𝑤𝑎𝑎 = 1000 kg/m3, and 𝜌𝜌𝑜𝑜𝑖𝑖𝑜𝑜 = 800 kg/m3.

h

v

(2.1.60) At Whale World Amusement Park, guests are given the opportunity to be lifted by a stream of water shot from the blowhole of a concrete whale. A horizontal 2.0-inch diameter pipe runs into mouth of the whale, and then narrows to a 1.0-inch diameter as it bends upward, shooting a stream of water into the air. Park visitors may stand on a board that is held in the air by the impact of the stream. (a) In the diagram below, a 30-kg child is on a 3-kg board held aloft by the stream. What is the speed of the water as it strikes the board? Assume that the speed of the water immediately after striking the board is zero, and that the distance between the end of the pipe and the board is negligible. Begin your analysis with a free-body diagram for the board. (b) What is the flow speed in the horizontal part of the pipe? (c) The horizontal part of the pipe is 5.0m below the board. What is the gauge pressure in the horizontal part of the pipe?

2.2 Fluid Dynamics (with viscosity) (2.2.20) The signature dish of the world-famous French Laundromat restaurant is the bologna-honey-molasses sandwich. The sandwich, shown in cross-section below, includes a square piece of bologna which is 12 cm on each side and 3.0 mm thick. The bologna is between two pieces of bread with the same dimensions as the bologna. Between the bologna and the top piece of bread is a layer of molasses with viscosity 5000 cP. Between the bologna and the bottom piece of bread is a layer of honey with viscosity 10000 cP. The distance between the top and bottom layers of bread is 6.00 mm. A diner attempts to remove the bologna from the bread by holding the two slices of bread in place and applying a 13N force to the bologna. The result is that the bologna begins to slide out from between the slices of bread at 5 cm/s. Given that the honey layer must be thicker than the molasses layer in order for the flavors to balance, what is the thickness of the molasses layer? The sandwich is shown in cross section below. (2.2.25) In the sandwich below, the mustard has a viscosity of 50,000 cP, and the tomato paste has a 150,000 cP. The mustard layer has a thickness of 3.00 mm and the tomato paste layer has a thickness of 2.00 mm. The entire sandwich is 10 cm long and extends 10 cm in the direction perpendicular to the page. With the bottom cracker held in place, a 5.00 N force is applied to the upper cracker. Find the speed of the upper cracker and the speed of the cheese. Any free body diagrams in your solution should follow the steps demonstratred in class.

bread

bread

bologna

molasses

honey

6.00 mm 3.00 mm

tomato paste

cracker

cheese

cracker

mustard

(2.2.30) Glycerin, with density 1.26 g/cm3 and viscosity 950 cP, is flowing through a horizontal tube with inner diameter 0.80 cm. The flow speed at the center of the tube is 1.00 cm/s. As shown below, the horizontal tube is connected to two vertical tubes, which are separated by L = 20 cm. (a) Use the Reynolds number to show that the flow in the horizontal tube is laminar. (This is necessary to apply the result derived in class.) (b) What is Δh, the difference between the surface levels in the two vertical tubes? Start your calculation from a result obtained in class. (c) What is the rate at which fluid mass exits the right of the apparatus. Again, you may start your calculation with a result obtained in class.

v

L

Δh

(2.2.40) Fluid with viscosity η is driven between two horizontal plates by a pressure difference2 1P P− . The distance between the two plates is h, and the dimensions of the plates are shown in

the diagram. The two plates are held in place by equal external forces, as shown. (a) Draw a free-body diagram for the region shown in the lower diagram. (This region includes both the lower plate and all of the fluid between y = 0 and a generic coordinate y.) (b) Apply 0xF =∑ to the results of your free-body diagram and integrate to get an expression for ( )xv y . (c) Plug y h= into the result of part (b), and solve for Fext in terms of P2, P1, h, and D. (d) Find the flow velocity as a function of the y-coordinate. Other parameters which appear in your result may include P2, P1, η, h, and D.

h

D

L

flow direction

P2 P1

extF

y

x

draw free body diagram for this region

extF

(2.2.50) In class, we studied the flow of a viscous fluid moving downward between two vertical plates. Suppose again that we have a fluid of viscosity η between two parallel vertical plates separated by distance d, each with length L, except that this time, the plate on the left is stationary, and the plate on the right moves straight down with speed vd. Assume also that

,L d so that we can assume uniform flow over the length of the plate. (a) Find vx(y), the fluid speed as a function of the y-coordinate, as indicated below. (b) Find the rate Q at which fluid flows through an imaginary horizontal surface passing through the system.

(2.2.60) Two stationary, square horizontal plates of length and width L are separated by a distance D (D<<L). The pressure at the left end of the plates is P2, and the pressure at the right end of the plates is P1, where P2 > P1. The pressure difference drives a fluid of viscosity η and density ρ to flow through the plates from left to right. The fluid flow is steady. (a) Find the fluid speed as a function of the distance from the bottom plate. Begin your solution by marking off a region within the fluid and applying Newton’s second law to this region. (b) Find the rate at which fluid mass exits the region between the two plates.

y = 0 y = d

v = 0

y

vd

L

x

L

D L

Fluid flow P2 P1

(2.2.70) A ping pong ball is moving through air at 1.00 atm pressure and temperature 288 K.The viscosity of air under these conditions is η = 18.3 µPa·s. The diameter of the ping pong ball is 4.00 cm and its mass is 2.70 g. (a) What is the density of the air? Take the molar mass of air to be 29 g/mol. (b) Using the plot provided at https://www.grc.nasa.gov/www/k-12/airplane/dragsphere.html, approximately what is the range of velocities for which the drag coefficient associated with this motion is about 0.5 ± 0.1? Neglect the region of the plot where Re > 106. (c) A ping pong ball is dropped off of a very tall building on a day with little wind. What will be the speed of the ping pong ball just before it hits the ground? (BIG Hint: Assume it reaches terminal velocity.) 3.1 Kinetic Theory – Temperature, Degrees of Freedom (3.1.05) A single molecule with initial velocity 𝑣𝑣 = 𝑣𝑣𝑥𝑥�̂�𝚤+ 𝑣𝑣𝑦𝑦𝚥𝚥̂ + 𝑣𝑣𝑧𝑧𝑘𝑘� is moving in a three-dimensional rectangular box whose sides have lengths Lx, Ly, and Lz. (a) Assuming that the molecule has only elastic collisions with the sides of the box, what is the average force that the molecule exerts on one side of the box whose area is Ly Lz? (b) Suppose instead that the box is populated by N molecules whose mean-square x-velocity is ⟨𝑣𝑣𝑥𝑥2⟩. What then is the average force exerted on one side of the box whose area is Ly Lz? (c) Starting from your result for (b), show that the pressure on this side of the box is related to the temperature by PV = NkBT. (3.1.10) A two-dimensional gas of xenon atoms moves on the surface of a platinum solid at a temperature of 105K. Find vrms of the xenon atoms. (3.1.15) Following the example in class, estimate the temperature at which the rotational degrees of freedom “turn on” in a nitrogen gas. Take the distance between the nitrogen nuclei to be 0.10975 nm. (Source: http://www.wiredchemist.com/chemistry/data/nitrogen-compounds). (3.1.20) It was mentioned in class that the rotational motion of air molecules is governed by quantum mechanics. But ignore quantum mechanics for this problem. (a) Compute the moment of inertia of a nitrogen molecule for rotation about an axis perpendicular to the bond joining the two nitrogen atoms. Assume that the distance between the atoms is 110 × 10-12 m.

Lx

Ly

Lz

(b) For a gas of nitrogen molecules at 300K, what is the average rotational kinetic energy per molecule? (c) For a nitrogen molecule with this much rotational kinetic energy, what is the magnitude of the angular velocity vector? (3.1.30) A spherical balloon with a 35.0 cm diameter is filled with helium gas. The average kinetic energy of the helium atoms is 8.00 × 10-21 J. If the pressure inside the balloon is 1.10 atm, what is the total mass of the helium? (3.1.40) (a) How many air molecules are in a room 14.0 feet long, 12.0 feet wide, and 10.0 feet high, if the temperature in the room is 20.0 °C and the pressure is 1.00 atm? (b) On average, which are moving faster: The nitrogen molecules in this room or the oxygen molecules? Answer in one or two grammatically correct sentences without doing any computations. (c) Compute the root-mean-square speed of the nitrogen molecules and of the oxygen molecules. (d) What is the total kinetic energy of all the air molecules in the room? 4. Ideal Gas Law (4.10) A certain room is 20.0 ft long, 15.0 ft wide, and 10.0 ft high. The temperature in the room is 20 °C, the pressure is 1.0 atm, and a door to the room is cracked open. (a) As a heater in the room is turned on and the temperature of the room begins to increase, , how do each of the following quantities change (or not change): pressure in the room, number of air molecules in the room, mass of air in the room. Explain briefly. (b) To what temperature must the room be raised in order to expel 50.0 kg of air? You may need to look up the molar mass of air to answer this.

(4.20) A methane bubble 1.00 cm in diameter is released from the bottom of a lake 5.00 m deep. The temperature at the bottom of the lake is 4 °C. (a) How many methane molecules are in the bubble? (b) What is the mass of the methane in the bubble? (If you don’t know the chemical formula for methane, look it up.) (c) The bubble rises to the surface of the lake, where the temperature is 15 °C. (Assume that no gas diffuses in or out of the bubble as it rises.) What is the diameter of the bubble just before it breaks the surface? (4.30) Quote from hotairballoon.org: “ A typical balloon system - envelope, gondola, fuel tanks, and 40 gallons of fuel - will weigh about 600 pounds, deflated on the ground. In the air, the complete system, including the weight of the air inside the envelope, will weigh about 2 ½ tons.” Assume the information stated in the quote, and that additionally, there are three passengers in the balloon with a total mass of 200 kg. Also assume that the air surrounding the balloon is at temperature 10 °C and pressure 0.90 atm. If the balloon is in static equilibrium, what is the temperature of the air inside the balloon? Neglect the volume of all parts of the balloon system other than the hot air inside the balloon envelope. (4.40) The diagram below show a container of neon gas. Air outside the container is at 1.00 atm. The top lid of the container is a massless piston connected to a spring. When the spring is in its “resting” position, the volume of the container is 3.00 L. The piston is circular with a diameter of 5.00 cm. When the neon in the container is at 30 °C, the spring is at its resting position, and when the temperature is increased to 150 °C, the spring compresses 1.30 cm. (a) What is the spring constant? (b) The temperature of the neon is raised to 250° C. What is the new compression of the spring?

gas

piston

5.1 Thermal Expansion (5.1.10) A circus performer’s act involves placing a horizontal pole above an open-top circular aluminum cage filled with hungry lions. When the cage is at 0°C it has an inner diameter of 10.000m. The circus performer’s pole, which is made of length LSt of steel, and length LCu of copper, has length 10.003m at 0°C. The diagram below illustrates the situation at 0°C. One hot day, the temperature inside the circus tent reaches 35°C. At this temperature, the length of the pole matches the inner diameter of the cage, and the pole falls into the cage along with the performer. Find LSt and LCu. The coefficients of thermal expansion for these materials are

6 -123.1 10 KAlα −= × , 6 -111.7 10 KSteelα −= × , 6 -116.5 10 KCuα −= × .

(5.1.20) (a) If β is the volumetric expansion coefficient of a liquid, and the density of the liquid at temperature T0 is ρ0, show that at temperature T, the density of the liquid is given by

𝜌𝜌 = 𝜌𝜌0 (1 − 𝛽𝛽(𝑇𝑇 − 𝑇𝑇0) ).

(b) A solid cylinder of length L and density ρs is floating in a liquid with density ρl and thermal expansion coefficient β. The cylinder is floating in such a way that its symmetry axis is perpendicular to the surface of the liquid. The length of the submerged portion of the cylinder is z. Using a free-body diagram and Newton’s second law, solve for z in terms of L, ρs, and ρl. (Neglect the effects of surface tension.) (c) The linear expansion coefficient of the cylinder is 𝛽𝛽

12. If the temperature of the system

increases by ΔT, what is the fractional change in z? In your computation, you may drop any term containing powers of βΔT of quadratic or higher order, provided that you explain why you are allowed to do this. (Note: In this problem, the approximation

11 − 𝑥𝑥

≈ 1 + 𝑥𝑥, 𝑥𝑥 ≪ 1

may be useful.)

LSt LCu

6.1 Specific Heat – Solids and Liquids (6.1.10) According to quantum mechanics, atoms in crystalline solids vibrate collectively in patterns known as “phonons.” (You can also think of a phonon as a quantum-mechanical packet of sound energy, analogous to a photon, which is a quantum-mechanical packet of light energy.) At temperatures much less than what is known as the Debye temperature, TD, the heat capacity associated with the phonons in a solid is given by

3

234ph BD

TC NkT

=

where N is the number of atoms in the solid. For carbon, TD = 2230K. (a) How much heat is needed to increase the temperature of 10g of carbon from T = 10K to T = 80K? (b) Assuming that there are no phonons present at T = 0K, what is the total energy associated with atomic vibration when the temperature of the 10g sample of carbon is 50K? (6.1.20) In a metal, by definition, there is a high concentration of “conduction” electrons which can freely move through the metal, and are not bound to any atom in particular. The conduction electrons are said to collectively comprise what is called a Fermi liquid. For temperatures much less than what is called Fermi temperature, TF, the specific heat capacity of the Fermi liquid is given by

2

2B

elF

Nk TCT

π=

where N is the number of electrons in the Fermi liquid. In copper, for example, each copper atom contributes one electron to the Fermi liquid, and 48.16 10 K.FT = × Additionally, the Debye temperature of copper is 343K. If a 5g piece of copper is heated from 20K to 100K how much energy goes into creating phonons (i.e. quantum-mechanical atomic vibrations – see previous question) and how much energy goes into the motion of the electrons? (6.1.30) 2.50 kg of beef rump at 20 °C and 2.00 kg of chopped carrots at 5 °C are added to 0.80 gallons of water at 100°C. What is the equilibrium temperature of the combination? The specific heat of carrots is 3.81 kJ/(kg °C) and the specific heat of the beef rump is 2.6 kJ/(kg °C) (From engineeringtoolbox.com.) (6.1.40) A solid aluminum disk with a 10.0 cm diameter and 3.00 cm thickness is spinning around its symmetry axis at 500 rev/s. The disk is dropped into 1.00 L of water. If both the aluminum and the water start at 20°C, what is the temperature after viscous forces have brought the aluminum disk to rest?

(6.1.50) Somewhere in outer space, a 20.0 kg piece of gold moving to the left at 50.0 m/s collides with a 10.0 kg piece of lead moving to the right at 80.0 m/s. The objects stick together. (a) Using momentum conservation, determine the final velocity of the combined object, and how much mechanical energy was lost in the collision. (b) If both are at a temperature of 300 K before the collision, what is the temperature of the combined object after the collision? Ignore radiative heat transfer between the objects and their surroundings. (6.1.60) A gold cylinder with thickness 2.00 cm and radius 10.0 cm (the lower mass in the pictures below) is rotating counterclockwise at 700 rev/s. A silver cylinder with 4.00 cm thickness, radius 5.00 cm (the upper cylinder in the pictures below) , and rotating clockwise at 500 rev/s is dropped onto the gold cylinder and sticks to it in such a way that the two cylinders are coaxial after the collision. (a) Friction between the gold and silver cylinders causes them to come to a common final angular velocity. What is this final angular velocity? (b) Both cylinders begin the process with a temperature of 30°C. What is the final temperature? 6.2 Specific Heat - Gases (6.2.10) 11.0g of helium gas at 300K is combined with 2.0 mol of N2 at 280K. Assuming that the gas mixture does not exchange heat with anything else, find the final temperature of the combination. (6.2.30) 0.0500 mol of gas occupies a cylinder which is sealed on top by a moveable piston. The piston is circular, with a mass of 30.0 kg and diameter of 5.00 cm. It is supported only by the pressure of the gas in the cylinder. Outside of the cylinder, there is air at 1.00 atm. Initially, the piston is 30.0 cm above the bottom of the cylinder. Heat is then added, causing the gas to expand until the piston is 40.0 cm above the bottom of the cylinder. For this process, find the change in internal energy, the work done by the gas, and the heat which flows into the gas (a) assuming that the gas is N2. (b) assuming that the gas is neon (6.2.40) In the scenario shown below, 3.0 mol of (diatomic) N2 gas initially at 400K is in enclosed in a cylindrical container with a frictionless piston of fixed weight. It is brought into contact with a container of fixed volume containing 2.0 mol of (monatomic) helium gas at 300K.

M1 M1

initial final

The contact allows heat to flow very gradually between the two gases, but the volume of the helium gas remains constant. (a) Find the final temperature of the combination. (b) How much heat flows from the N2 to the helium during the equilibration? (c) What is the change in the internal energy of the N2 during this process? (d) How much work does the piston do on the N2 during this process? 7. Phase Changes (7.1.10) In the diagram below, the lower half of a container is occupied by water at 25º C, and the upper half is initially vacuum. The volume of the evacuated region is 3.00 L. The two halves are separated by a partition. The entire containter is immersed in a thermal reservoir, also at 25º C. The equilibrium vapor pressure at this temperature is 3.169 kPa (wikipedia). When the partition is removed, (a) What mass of liquid water evaporates? (Neglect the change in the volume occupied by the liquid phase.) (b) How much heat flows from the thermal reservoir into the container (in order to maintain constant temperature?)

non-moving partition

N2

He

vacuum V = 3.00 L

(7.1.20) In class we estimated the mass of liquid water which would need to evaporate to cool a cup of tea to room temperature. Our computation used the unlikely assumption that the tea would cool only through evaporation. We also neglected the decrease in the liquid mass due to the ongoing evaporation process. Here we will repeat the problem, but also account for the change in the liquid mass of the tea due to the evaporation. (a) Suppose that a mass m of tea (assumed to have the thermodynamic properties of water) is at temperature T. A mass dm of the liquid evaporates, changing the temperature by an amount dT. Find a relation between dm and dT. (b) 355 g of tea is at initial temperature 90 ºC. What mass of the tea evaporates as the tea cools to 20 ºC? Assume that the tea cools only by evaporation. Rather than using the crude calculation from class, obtain your result by integrating your answer from part (a). (7.1.30) A parcel of air near the ground is at a temperature of 30 ºC and relative humidity 70 %. Suppose that the parcel becomes slightly warmer than the nearby air, and begins to rise. As the parcel rises, it cools at the adiabatic lapse rate. To what height must the parcel rise for the water vapor within to condense into liquid droplets? (This height is called the lifting condensation level, i.e. the lowest level at which a cloud can form.) (Hint: Integrate the Clausius-Clapeyron equation, as demonstrated in class.) (7.1.40) A mass of steam ms at 100°C is combined with 50.0 g of ice at 0°C. The system is allowed to come to thermal equilibrium. (a) If ms = 10.0 g, what is the final temperature? (b) Repeat part (a) with ms = 20.0 g. When equilibrium is reached in this case, what masses of steam, liquid water and ice are present? 8.1 Thermal Transport – Conduction (8.1.10) In the picture below, a tank of ice water, which is maintained at 0°C, and a tank of boiling water, which is maintained at 100°C, are thermally connected by a copper bar. The copper bar does not have a uniform cross-section. Rather, it is tapered, with the narrow end in the cold tank and the wide end in the hot tank. The radius of the bar is given, as a function of x (the x-axis is shown below) by

0r r bx= + ,

where r0 = 1.0 cm and b = 0.05. Find the rate at which heat flows from the hot tank to the cold tank. Assume that the temperature of the bar changes only in the x-direction.

0°C 100°C

x 0 25cm

8.2 Thermal Transport – Radiation (8.2.10) A concrete spherical shell has outer radius 20 cm, and an inner cavity with radius 15 cm. Inside the inner cavity, there is a liquid. A constant temperature is maintained throughout the liquid by an electric heater connected to a power source outside the sphere. The outer surface of the concrete sphere has been painted with a perfectly absorbent coating (making the sphere a blackbody). The sphere is in an evacuated chamber whose walls are at 300K, and can cool only through radiation. Once steady state has been achieved, the temperature of the surface of the sphere is 320K. What is the temperature of the liquid in the inner cavity? Take the thermal conductivity of concrete to be 0.8 W/(m·K). (Hint 1: You may use the result from Problem 2096 in the Lab Manual.)

(8.2.20) (a) Examine the mathematical form of the blackbody spectral emittance that we studied in class. Obtain a simplification of this expression which is valid when λT is sufficiently small that 𝑒𝑒ℎ𝑐𝑐/(𝜆𝜆𝜆𝜆𝜆𝜆) ≫ 1. (b) Show that the approximation 𝑒𝑒ℎ𝑐𝑐/(𝜆𝜆𝜆𝜆𝜆𝜆) ≫ 1 applies when T = 1000K and λ is in the visible range. (c) Suppose a heating element on an electric stove has a surface area of 20 in2, and is at a temperature of 1200 K. Estimate the power radiated into wavelengths between 690 nm and 700 nm. Also, estimate the power radiated into wavelengths between 400 nm and 410 nm. What is the apparent color of the heating element, and why? (8.2.30) Suppose that a blackbody at temperature Te and surface area A is enclosed in a large sphere whose inner walls are at temperature Te. The inner walls of the large sphere are covered with perfectly absorbing paint. The space between the blackbody and the large sphere is evacuated, so that the blackbody and the large sphere may only exchange energy through radiation. (a) What is the rate at which the blackbody absorbs radiation from its environment? (Hint: The blackbody is in thermal equilibrium with its environment.) (b) The temperature of the blackbody is raised to T > Te, while the temperature of the surrounding sphere remains at Te. What is the rate at which the blackbody absorbs radiation from the environemnt?

(c) What is the net rate of heat exchange between the blackbody and its environment? (That is, what is �𝑑𝑑𝑑𝑑𝑑𝑑𝑜𝑜�𝑤𝑤𝑒𝑒𝑖𝑖𝑜𝑜

− �𝑑𝑑𝑑𝑑𝑑𝑑𝑜𝑜�𝑎𝑎𝑎𝑎𝑎𝑎𝑜𝑜𝑎𝑎𝑎𝑎

?)

(8.2.40) (Much of this problem was already done in class. I would still like you to go through this on your own. For any astronomical information not provided, you may use a reliable online source, such as nasa.gov.) (a) Taking the surface temperature of the sun to be 5770 K, what is the radiative energy per time per area passing through a sphere centered on the sun, and the surface of which passes through Earth? (b) Assuming that the earth absorbs 70% of the radiation incident upon it, what is the rate at which the earth absorbs solar radiation? (c) Assuming that the earth radiaties as a black body, calculate the surface temperature of the earth assuming that absorbed radiation = emitted radiation. (Otherwise, the earth might experience wild changes in temperature. We don’t want that.) (d) Is the surface temperature you found in (c) cooler or warmer than the actual average surface temperature of the earth? Offer a few sentences to explain the discrepancy. This may require a bit of web-searching. (Hint: Although the earth’s surface doesn’t strictly radiate as a blackbody, the emissivity is quite close to 1. That isn’t the issue.) (8.2.50) A planet with radius 8.00 × 106 m orbits a star with surface temperature 7000 K and radius 4.60 × 108 m. The orbit is circular, with orbital radius 2.50 × 1011 m. The planet rotates, so that both sides of the planet are exposed to radiation from the star, and the surface temperature of the planet may be assumed uniform for the purposes of this problem. The emissivity of the planetary surface is 0.80. (a) Assuming that the star emits as a blackbody, what is the rate at which radiative energy is produced by the star? (b) What is the rate at which the planet absorbs radiative energy from the star? (Assume that all stellar radiation incident on the planet is absorbed.) (c) Suppose that the planet has no atmosphere. Calculate the surface temperature of the planet. (Remember that the emissivity of the planetary surface is 0.80.) (d) Suppose instead that the planet has an atmosphere which absorbs all radiation emitted by the surface of the planet (but radiation from the star passes through the atmosphere without absorption). The emissivity of the atmosphere is 1.00. Under these assumptions, calculate the temperature of the atmosphere and the temperature of the surface of the planet. (Hint: Treat the atmosphere as a thin shell whose surface area is approximately that of the planet. The inner surface of the atmosphere radiates back towards the surface of the planet, and the outer surface of the atmosphere radiates into deep space.)

9.1 Work (9.1.10) 0.90 mol of argon is expanded from a volume of 20.0 L to a volume of 40.0 L in such a fashion that the pressure and volume are related by 𝑃𝑃 = 𝛽𝛽𝑉𝑉3, where 𝛽𝛽 is a constant. The initial temperature of the argon is 300 K. (a) Sketch this process on a P-V diagram. On the axes, show the values of P and V for the initial and final states. (b) For this process, find the work done by the gas on its surroundings, and the work done by the surroundings on the gas. On your P-V diagram, shade in the area which represents the work done by the gas. (9.1.20) 5.00 mol of N2 is taken through the cycle shown below. Find the work done by the gas during a single cycle A B C. The temperature at A is 600 K and the temperature at B is 300 K.

P

V

A

B C 1.0 atm

3.0 atm

(9.1.30) 3.00 moles of helium is taken through the cycle below. Steps AB and CD are isobaric, and steps BC and DA are isothermal. The temperature at A is 300 K, the temperature at B is 500 K, the volume at A is 40. 0 L, and the volume at D is 50.0 L. (a) As shown in class, set up a chart with space for the pressure, volume, and temperature at each point in the cycle. Fill in the chart (b) Find the work done by the gas during a single cycle. 10.1 First Law of Thermodynamics 10.2 Cycles in P-V Plane w/o Adiabatic Steps (10.2.10) 0.07 mol of air is taken through the cycle shown below. A B is an isothermal expansion. (a) As shown in class, set up a grid showing P, V, and T at points A, B, and C, and fill in the values. (b) For step A B, find the work done by the gas, the change in internal energy, and heat flow into the gas. (c) For step B C, find the work done by the gas, the change in internal energy, and heat flow into the gas. (d) For step C A, find the work done by the gas, the change in internal energy, and heat flow into the gas.

P

V

A B

C D

V

P A

B C

2 atm

1 atm

2000 cm3

(10.2.20) 2.00 mol of Helium gas is taken through the cycle A B C, as shown below. AB is an isotherm at TAB = 350 K. (a) As shown in class, set up a grid showing P, V, and T at points A, B, and C, and fill in the values. (b) For step AB, find the work done by the gas, the heat entering the gas, and the change in the internal energy of the gas. (c) For step BC, find the work done by the gas, the heat entering the gas, and the change in the internal energy of the gas. (d) For step CA, find the work done by the gas, the heat entering the gas, and the change in the internal energy of the gas.

V

A

B

C

P

V1

P1 P1 = 2.50 atm V1 = 0.0330 m3

11.1 Adiabatic Processes (11.1.10) A 200 g piston can move frictionlessly in a horizontal cylinder with cross-sectional area 4.0 cm2 and length 20 cm. One end of the cylinder is open, and the other is closed. n moles of air are enclosed between the piston and the closed end of the cylinder. (Assume that there is no way for the air to leak around the piston.) Initially, the piston is 10 cm from the closed end of the cylinder, and all air (both inside and outside of the enclosure) is at atmospheric pressure and T = 300 K. (a) How many moles of air are enclosed by the piston? (b) The piston is held in place, and heat Q is added to the air enclosed air, raising its temperature to 1000 K. Find Q. (The temperature and pressure of the air to the right of the piston do not change.) (c) The piston is released, allowing the enclosed air to expand adiabatically. How much work does the enclosed air do on the piston as it pushes the piston to the open end of the cylinder? (d) How much work does the air to the right of the piston do to the piston as the piston is pushed to the open end of the cylinder? (e) What is the speed of the piston as it is pushed outside the cylinder? (11.1.25) In class we derived the adiabatic lapse rate, which (approximately) is the rate at which a parcel cools as it rises through the atmosphere adiabatically, that is, without exchanging heat with the surrounding air. (11.1.30) In class we derived the adiabatic lapse rate, which is the rate at which temperature decreases with height in a well-mixed atmospheric layer. To do so, we assumed that the acceleration of gravity is constant near the earth’s surface. This is actually quite a good approximation, but for variety, let’s take into account the inverse-square dependence for the

acceleration of gravity, i.e. 2

2( ) Es

Rg r g

r

=

, where gs is the acceleration of gravity near the

earth’s surface, r is the distance from the center of the earth, and RE is the radius of the earth. Using this expression for the acceleration of gravity, find an equation giving the temperature as a function of the height above the earth’s surface, assuming that the temperature at the earth’s surface is T0.

piston

10 cm 20 cm

(11.1.40) In Strombolian volcanic eruptions, giant gas bubbles rise to the top of volcanic conduits which have been plugged by cooled lava. When the pressure of a gas bubble becomes large enough, it expands adiabatically, breaking the lava into fragments and propelling the fragments into the air. (a) Suppose that a cylindrical bubble with 10.0m diameter, 10.0 m height, and pressure of 3.3 atm bursts, and expands until it has reached atmospheric pressure. What are the initial and final volumes of the bubble? The gas inside the bubble is steam. (NEED TO LOOK UP GAMMA.) (b) How much work does the gas do as it expands? (c) Suppose that all of this work is done on a cylindrical film of lava, 0.50 m thick, 10.0m in diameter, with a density of 2000 kg/m3. If all of this lava ends up with the same final velocity, what is that velocity? (Note: This problem was inspired by a paper I wrote when I took Geology 1 here a couple of years ago. The paper is posted on the course web site.)

10 m

10 m

rising magma below bubble

gas bubble

cooled lava

(11.1.50) In the diagram below, a disc of mass m has a diameter very slightly smaller than the inner diameter of a cylinder of diameter d. The inside of the cylinder is occupied by an ideal gas. The mass can move frictionlessly up and down in the cylinder while maintaining its orientation. Initially, the mass is in equilibrium at a height h above the bottom of the cylinder. The pressure above the disc is Patm and the pressure inside the cylinder is P0. The mass is displaced very slightly downward from its equilibrium position and released. The cylinder is insulated so that there is no transport of heat between the gas and its environment. (a) What must be the relation between P0, Patm, and m for the disc to be in mechanical equilibrium? A brief free-body diagram analysis will suffice. (b) Suppose that the the piston is depressed by a tiny amount δh. In terms of δh and other quantities mentioned above, what is the increment δP in the pressure of the gas in the cylinder? (Hint: Start with 𝑃𝑃𝑉𝑉𝛾𝛾 = const, and take a logarithmic derivate with respect to either P or V.) (c) Show that when the disc is displaced a small distance δh from its equilibrium position and released, it executes simple harmonic motion about its equilibrium position. (To do so, begin by applying Newton’s second law to the disc when it has been displaced by δh. You may need to review your Physics 1A notes on SHM.) (d) What is the oscillation period if d = 3.0 cm, h = 10 cm, m = 50 g, and the gas inside the cylinder is helium?

Patm

P0

m

h

11.2 Cycles in P-V Plane w/adiabatic steps (11.2.10) 3.00 mol of a diatomic gas is taken through the cycle shown below. B C is adiabatic and C A is isothermal. (a) As shown in class, set up a grid showing P, V, and T at points A, B, and C. Fill in the grid. (b) For step A B, find the work done on the gas, the change in internal energy, and heat flow into the gas. (c) For step B C, find the work done on the gas, the change in internal energy, and heat flow into the gas. (d) For step C A, find the work done on the gas, the change in internal energy, and heat flow into the gas. 12.1 Engines (12.1.10) A certain engine uses two identically constructed objects as its thermal reservoirs. Both objects have mass m and specific heat c. In this problem, as the engine extracts heat from the thermal reservoirs, their temperatures will change. Assume that the objects start at temperatures Thi and Tci, and that as the process evoles, the temperatures change to Th and Tc. Assume that the engine always operates at the maximum possible efficiency. Show that (a) as the process proceeds, Th Tc = Thi Tci, (a) the engine will operate until both objects reach a final temperature of (ThiTci)1/2. (b) The work done by the engine as it comes to the final temperature is

𝑚𝑚𝑚𝑚��𝑇𝑇ℎ𝑖𝑖 − �𝑇𝑇𝑐𝑐𝑖𝑖�2

.

(Hint for part (a): If heat dQ is extracted from the high-temperature reservoir, the temperatures of the reservoirs change by amounts dTh and dTc. Find equations relating dTh and dTc to dQ and integrate to the final temperature.) (12.1.20) A certain power station produces an electrical output of 700 MW, operating at 42% efficiency. This plant uses a closed-circuit “dry” cooling system in which heat is expelled by cycling the cooling water through pipes which are in direct contact with air taken in from outside the plant. The fuel for the plant is lignite, which has a typical energy content of 15 MJ/kg. (a) What is the mass of lignite consumed in 1.00 hour of operation of the plant?

P

V

PA

VA VB

PA = 3.00 atm VA = 0.033 m3

VB = 0.038 m3

VC

A B

C

(b) If the cooling water operates between a temperature of 90 °C and a temperature of 30 °C, what mass of water must be cycled through the cooling system in 1.00 hour?

12.2 Carnot Processes (12.2.10) 3.00 mol of air are taken through a Carnot cycle, as illustrated below. AB and CD are isothermal processes, and BC and DA are adiabatic. At point A, the pressure is 3.00 atm. At point B, the pressure is 2.30 atm and the temperature is 500 K. At point C, the temperature is 350 K. (a) As shown in class, set up a grid showing P, V, and T at points A, B, C, and D. (b) For step A B, find the work done by the gas, the change in internal energy, and heat flow into the gas. (c) For step B C, find the work done by the gas, the change in internal energy, and heat flow into the gas. (d) For step C D, find the work done by the gas, the change in internal energy, and heat flow into the gas. (e) For step D A, find the work done by the gas, the change in internal energy, and heat flow into the gas. (12.2.20) (a) Calculuate the efficiency of a Carnot engine acting between a high-temperature reservoir at 400 K and a low-temperature reservoir of 300 K. (b) Consider an engine with an efficiency one-fifth of your result from part (a). Show that such an engine, used in combination with a perfect refrigerator, could be made to function as a perfect engine. (c) What conclusions can you draw from your result in part (b)? 12.3 Fridges (12.3.10) Show that a Carnot engine run in reverse acts as a refrigerator with coefficient of performance

COP= 𝑇𝑇𝑚𝑚𝑇𝑇ℎ−𝑇𝑇𝑚𝑚

.

P

V

A

B

C D

(12.3.20) A certain studio apartment is cooled by an air conditioner. The temperature outside the apartment is 35°C and the temperature inside the apartment is maintained at 25°C. The air conditioner extracts heat from the inside of the apartment at a rate of 8.0 × 103J/s. Assuming that the COP of the air conditioner is 30% of the ideal value, find (a) the rate at which electrical energy must be consumed to run the air conditioner (b) the rate at which heat is expelled to the outside of the apartment (c) the amount of entropy created in 2.00 hours. (Exclude whatever entropy had to be created at the power plant to generate the electrical energy input.) (12.3.30) An ice tray is shaped to produce 12 ice cubes, each of which has dimensions 4.0 cm × 2.5 cm × 2.5 cm. Five such ice trays are filled with water at 22°C and placed in a freezer at −5 °C. (a) How much heat must be extracted from the freezer so that all of the water turns to ice at −5 °C? (b) Suppose that the COP of the freezer is 3.00 and the ice cubes must be produced in 1.00 hour. What average electrical power must be supplied to the freezer? (c) As the cubes are forming, what is the average rate at which heat is expelled to the outside of the freezer? 13.1 Distribution Functions (13.1.10) Consider a gas made up of molecules which are restricted to moving in a single plane. Molecules may move freely in the two directions of the plane, but may not move at all in the direction perpendicular to the plane. In this situation, assuming that the molecules obey Maxwell-Boltmann statistics, the density of states is proportional to the speed v (as opposed to v2, which is the case for molecules moving in three dimensions). (a) Explain, using a diagram or words, why this should be the case. (b) Suppose that a two-dimensional gas of N molecules of mass m is at temperature T. Find the Maxwell-Boltzmann speed distribution for the molecular speeds. (This will be analogous to the three-dimensional Maxwell-Boltzmann speed distribution given in your text, Eqn. 21.26.) Make sure to find the correct constant prefactor. (c) What is the most likely speed? (d) What is the average speed? (You will need the integrals given in problem 2578 of the lab manual.) (13.1.20) Compute the average speed of the atoms in a 1D helium gas at T = 50K. (13.1.30) What fraction of nitrogen molecules in a 2D nitrogen gas at T = 100K have speeds greater than 200 m/s?

14.1 Second Law of Thermodynamics (14.1.10) (I) The Kelvin statement of the second law of thermodynamics: It is impossible to create a cyclic engine whose only effect is to extract energy from a single thermal reservoir and convert that energy completely into work. (II) The Clausius statement of the second law of thermodynamics: It is impossible to create a cyclic refrigerator which extracts a quantity of heat from a low-temperature reservoir and deposits that heat into a high temperature reservoir without an input of work. (III) It is impossible to create an engine operating between a given high-temperature reservoir and low-temperature reservoir whose efficiency is greater than that of a Carnot engine operating between the same two temperatures. There are very clever and elegant arguments deriving these various statements from one another that I enjoy going through in class. For this problem, you will generate simpler proofs based on the second law of thermodynamics as we studied it this semester. Each of the following proofs may be accomplished by drawing a “heat engine” cartoon, as from class, considering the inputs/outputs of heat and work, and computing the total entropy change associated with these. (i) Prove statement (I) above. (Hint: Assume that such an engine exists. Show that the existence of such an engine would violate the second law of thermodynamics as stated in class.) (ii) Prove statement (II) above. (Hint: Assume that such a refrigerator exists. Show that the existence of such a refrigerator would violate the second law of thermodynamics as stated in class.) (iii) Prove statement (III) above. (Hint: Assume that such an engine exists. Show that the existence of such an engine would violate the second law of thermodynamics as stated in class.) (14.1.20) Show that the COP of a Carnot refrigerator (i.e. a Carnot engine run backwards) is

CarnotCOP c

h c

TT T

=−

(14.1.30) It is impossible for a refrigerator operating between Tc and Th to have a COP greater than COPCarnot. Illustrate this as follows: (i) Compute for COPCarnot for Tc = 300K and Th = 400K. (ii) Arbitrarily pick a value COPBogus such that COPBogus > COPCarnot. (iii) Show that the existence of such a refrigerator with this coefficient of performance would violate the second law of thermodynamics.

(14.1.40) Prove that any two reversible engines using the same high- and low-temperature reservoirs must have the same efficiency. (This efficiency, of course, is the Carnot efficiency,

1 cC

h

Te T = −

, but that formula is not involved in the proof.) Hint: Do a proof by

contradiction, starting with the (false) assumption that two reversible engines operating between the same two temperatures have different efficiencies, 1 2e e≠ , and reach a contradiction with something else we’ve learned in class. The proof is similar to the one that shows that no engine can have an efficiency exceeding that of a Carnot engine. (14.1.50) 0.03 mol of helium are enclosed in a cylinder with a piston which maintains a constant pressure in the helium. Initially the helium is immersed in thermal bath with at 250K. Suddenly, the helium is removed from this thermal bath and plunged into a thermal bath at temperature 350K. Determine (a) the increase in the entropy of the helium. (Hint the entropy change for the helium is the same as if it had been taken through this temperature change by a reversible process.) (b) the increase in entropy of the universe. (Answer: 0.032 J/K) (14.1.60) A family recipe handed down from my great-great-grandmother involves heating 1.5 kg of coconut milk to 90°C, adding 0.6 kg of goose meat at 5°C, 1.0 kg of candied figs at 30°C, turning off the heat, insulating the pot, and allowing the mixture to reach an equilibrium temperature. The specific heats (taken from engineeringtoolbox.com) are coconut milk: 3.98 J/g°C goose meat: 2.55 J/g°C candied figs: 1.55 J/g°C (a) What is the equilibrium temperature? (b) How much does the entropy of the universe increase during equilibration?

15.1 Waves on Strings (15.1.10) A 750-g mass is attached to a string 65.0 cm long. The string is held at the top and, and swung so that the mass traces out one horizontal circle every 0.70 seconds (i.e. a conical pendulum. See the diagram.) (a) Using the free-body diagram analysis you learned in Physics 1A, determine the angle the string makes with the vertical and the tension in the string. (b) If the mass of the string is 0.150 g, how much time does it take for a pulse to get from the top of the string to the bottom? (15.1.20) A 4.00 kg rock hangs from a 2.00 kg chain which is 1.50 m long. (a) Find the tension in the chain as a function of the distance from the bottom. (b) How much time does it take a pulse to travel from the bottom of the chain to the top of the chain? (Hint: The tension in the chain is NOT constant.) (c) As the pulse travels up the chain, does it become wider or narrower? Explain using grammatically correct sentences and a diagram or two. 15.2 Wave Functions (15.2.05) A certain string, when at rest, lies along the x-axis. A wave is travelling leftward through the string so that, at time t = 0.80s, the shape of the string is given by

2( ,0.8 ) Axy x sBx D

=+

.

The tension in this string is 50N, and its linear mass density is 0.004 g/cm. The parameters appearing in the equation above are A = 0.05, B = 4.0 m-2, D = 12.0. (a) Using your knowledge of differential calculus, find the values of x where this function takes on its maximum and minumum values. What are the values of y for these values of x? (b) Sketch y(x,0.8s).

(c) Determine a functional form for y(x,t). In whatever expression you come up with, numerical values should be plugged in for all parameters except for x and t. You may include whatever symbolic parameters you like, but numerical values should be given for any symbols included in your final expression. (d) Sketch y(x, 2.3s). Your sketch should clearly show where y(x, 2.3s) takes on its maximum and minimum values, and what these maximum and minimum values are. (15.2.10) A wave on a string is described by the wavefunction

𝑦𝑦(𝑥𝑥, 𝑡𝑡) = 𝐴𝐴 sin(𝑘𝑘𝑥𝑥 − 𝜔𝜔𝑡𝑡 + 𝜑𝜑) where k = 20.0 m-1, φ = 0.70, and the wave speed is 40.0 m/s. The linear mass density of the string is 0.0085 g/cm. The amplitude of the wave is 3.00 cm. (a) As a function of time, what is the rate at which energy is transported through the point x = 0.065 m? (b) What is the average rate at which energy passes through this point? (c) How much energy passes through this point between t = 0.001 s and t = 0.004 s? (15.2.20) A transverse sinusoidal wave is traveling rightward along a string stretched along the x-direction. The wavelength is 70.0 cm and 8.00 wave peaks per second pass any fixed point on the string. The maximum speed of the string particles in the direction perpendicular to the string is 3.20 m/s. At time t = 0, the element of string at x = 20.0 cm is 4.00 cm above its resting position. (a) Construct two distinct wave functions y(x,t) consistent with the given information. (If the only difference between two wavefunctions is that their phase constants differ by an integer multiple of 2π, then they are not distinct. ) (b) Sketch both of these wavefunctions for time t = 0 between x = 0 and x = λ. (15.2.30) A transverse sinusoidal wave is traveling leftward along a string. The period of the wave is 0.030 s, the wave speed is 40.0 m/s. At time t = 0.021 s, the string element at x = 0 is at its resting position, but is moving downward at 4.50 m/s. Find the wave function y(x,t). (15.2.40) Show that any function of the form ( , ) ( )y x t y x ct= + is a solution to the general wave equation

2 22

2 2

y yct x

∂ ∂=

∂ ∂.

Do not use the “similarly” shortcut that I used in class. (15.2.50) (a)Show by direct substitution that 2 2 2( , ) ( )g x t A x c t= + is a solution to the wave equation. (b) Does g(x,t) represent a left-traveling wave, a right-traveling wave, neither, or both? You will need to do some algebra to support your assertion.

15.3 Wave Energy (15.3.10) A string stretched along the x direction has tension T and linear mass density μ. A pulse described by

2( )( , ) b x cty x t Ae− −= travels down the string. (a) Find the total energy carried by this pulse. (We discussed how to do this in class.) (b) How much of this energy is kinetic energy? (Hint: At a given moment in time, say t = 0, find the speed at every point on the string by taking 0( / )y tv y t == ∂ ∂ . The mass of a piece of string of length dx is then μdx. Find the kinetic energy of the string by integrating over the kinetic energies of all the mass elements at t = 0.) (15.3.20) A wave traveling through a string is described by

𝑦𝑦(𝑥𝑥, 𝑡𝑡) = 𝐴𝐴 cos(𝑘𝑘𝑥𝑥 + 𝜔𝜔𝑡𝑡 + 𝜑𝜑)

where A = 3.0 cm, λ = 40 cm, c = 10.0 m/s, and φ = 0.30. The linear mass density of the string is 𝜇𝜇 = 0.05 g/cm. (a) What are k and ω? (b) Sketch y(x,t) at t = 0, showing two complete wavelengths of the wave. The amplitude must be shown accurately, and all peaks and troughs must be correctly located along the x-axis. (c) Derive a formula which gives the rate at which energy passes through x = 1.00 m. (d) How much energy passes through x = 1.00 m between t = 0 s and t = 0.15 s? 15.4 Standing Waves in Strings (15.4.10) The diagram below shows a segment of string of fixed linear mass density µ. At the left end, the segment is connected to a vibrator which vibrates at frequency 200 s-1. At the right end, the string passes over a pulley to a hanging mass M. The length of the horizontal part of the string is 80 cm. When M = 612 g or M = 833 g, a standing wave pattern appears on the string. No standing wave patterns appear when M is between these values. What is µ?

M

80 cm

(15.4.20) (a) Suppose that the note “A4” on a piano (also known as the above middle C) is to be tuned so that its fundamental (i.e. lowest) frequency is 440 s-1. The length of the string is 25.0 inches and is described by the manufacturer as “234 feet per pound.” What must be the tension in the string? (b) The note “E5” is to be tuned so that its 2nd harmonic (i.e. 2nd lowest frequency) matches the third harmonic of “A4.” If the string used for E5 is listed as “260 feet per pound,” and the length of the string is 20.5 inches, what tension should be applied? (15.4.30) (Revised 11/30/2017) The following functions represent waves traveling on the same string in opposite directions:

𝑦𝑦1(𝑥𝑥, 𝑡𝑡) = 𝐴𝐴 sin �𝑘𝑘𝑥𝑥 − 𝜔𝜔𝑡𝑡 +𝜋𝜋4�

𝑦𝑦2(𝑥𝑥, 𝑡𝑡) = 𝐴𝐴 sin �𝑘𝑘𝑥𝑥+𝜔𝜔𝑡𝑡+𝜋𝜋6�

Show that these functions add to a single standing wave pattern. If k = 24.0 m-1, find three values of x for which nodes in the standing wave pattern appear. (Hint: There are various ways to do this, but one way would be to find two expressions α and β such that one of these can be written as 𝐴𝐴 sin(𝛼𝛼 + 𝛽𝛽) and the other can be written as 𝐴𝐴 sin(𝛼𝛼 − 𝛽𝛽).) 16.1 Sound Waves (16.1.10) In class we learned how to relate the δP(x,t), the deviation from ambient pressure caused by a sound wave, to s(x,t), the displacement of air molecules caused by the sound wave. We would like to learn how to relate δρ(x,t) and δT(x,t), the deviations from ambient density and temperature caused by a sound wave, to δP(x,t). (a) We know, for example, that PVγ = const for an adiabatic process. Using the ideal gas law, and anything else you know about adiabatic processes, find a combination of pressure and density which is constant for adiabatic processes. Use this result to show that 2P cδ δρ= for a sound wave.

(b) Show that, for a sound wave, the relation between δT and δP is ( )2 1TT Pc

δ γ δρ

= − .

(16.1.20) A tube of air of cross section A lies along the +x direction, with one end at x = 0. A speaker located at x = 0 vibrates, its displacement relative to its resting position given as a function of time by

max( ) cos( )u t s tω=

where smax = 2.0 × 10-7m and ω = 1200s-1. The temperature is 300K and the density of air is 1.16 kg/m3. (a) What is the frequency of the sound generated by the speaker? (b) Find the wavefunction s(x,t) which describes the displacement of air in the tube as a function of both position and time. (c) Find the wavefunctions for pressure and density, δP(x,t), and δρ(x,t). (d) What is the sound level of this wave? Give an example of a real-life sound about as loud as this. (16.1.30) Mr. Snidely, a pretentious lover of avant-garde music, attends a concert where a single note with a frequency of 300 Hz is played for a whole hour at a sound level of 120 dB (comparable to a loud rock concert). The pressure in the concert hall is 1.10 atm (there’s a lot of atmosphere in avant-garde performance venues) and the temperature is 0°C. (Actually, the concert hall could have been heated to a comfortable temperature, but that would be pandering to the masses.) (a) Assuming that his eardrum has an area of 5 25.00 10 m−× , how much acoustic energy does Mr. Snidely’s eardrum receive in that hour? (b) Find the maximum pressure and density deviations caused by the sound wave. (c) Find the maximum increase in temperature due to the sound wave. (16.1.40) A tube of air of cross section A lies along the +x direction, with one end at x = 0. A speaker located at x = 0 makes a single motion into and away from the tube. The speaker’s position as a function of time is given by 2 2( ) exp( / )u t t τ= Γ − . (a) Find the displacement of air in the tube as a function of position and time, i.e. s(x,t). (b) Find the pressure deviation of the air in the tube as a function of position and time, i.e. δP(x,t). (c) Find the velocity of the air in the tube as a function of position and time. (d) Set up an integral that you could use to determine the total energy given to the air by the speaker. (Hint: Power is energy/time. What quantity must be integrated over time to get energy?) (e) Evaluate the integral you found in part (d). Refer to the integrals given after Problem 2578 in the Lab Manual. 16.2 Doppler Effect (16.2.10) As you take your morning jog, running eastward on Manhattan Beach Blvd at 6.0 mi/hr, you see an ambulance approaching you, moving westward. As the ambulance approaches, 440 compressions per second reach your ear, and as the ambulance moves away from you, 400 compressions per second reach your ear. What is the speed of the ambulance, and how many compressions per second does it produce? The temperature is 20 °C.

(16.2.20) One fine summer day, when the temperature is a balmy 30 °C, an ice cream truck parks in front of my house, its speaker system blasting “Turkey in the Straw” over and over. A 10 mi/hr wind is blowing towards the east. Melvin drives westward towards the ice cream truck at 20 mi/hr to buy a Fudgsicle, and Horace drives eastward towards the ice cream truck at 15 mi/hr to buy a box of Bon Bons. When the speaker on the truck plays middle C, with a frequency of 256 s-1, (a) how many compressions per second reach Melvin’s ear? (b) and how many compressions per second reach Horace’s ear? 16.3 Standing Sound Waves (16.3.10) The resonating air column in the open-closed pipe shown below is 0.635 m long. (a) If the lowest frequency that can be produced by the pipe is 129 s-1, what is the temperature? (b) The pipe is cut a distance d from the closed end (shown below by the dotted line) so that we now have two shorter pieces: an open-open piece and an open-closed piece. If the second-lowest frequency that can be produced by the open-open piece equals the third-lowest frequency that can be produced by the open-closed piece, what is d? (c) Sketch the displacement amplitude variation for the second-lowest mode in the open-open piece and the pressure amplitude variation for the third-lowest mode in the open-closed piece. (16.3.20) A segment of piano wire with length 1.10 m is held above an open-open pipe of length 0.635 m. The manufacturer’s specification for the piano wire says “248 feet per pound.” (This is not how a physicist would describe it, but with a bit of thought you can translate this into the language of Physics 1B.) (a) What tension in the wire is necessary so that the second harmonic of the wire has the same frequency as the third harmonic of the open-open pipe? Sketch the displacement amplitude patterns for both the wire and the pipe for this frequency. Take the speed of sound to be 345 m/s. (b) Same as part (a), except that one end of the pipe has been closed, so that the pipe is now open-closed.

Melvin Horace ICE CREAM

WIND

open closed

d

(16.3.30) The length of the open-closed pipe shown below can be adjusted by changing the position of the movable piston at the bottom. A tuning fork vibrating at 440 s-1 is held over the top of the tube. When the piston starts at the top of the tube and begins to move down, the first resonance is produced when the piston is distance L from the top of the tube, and the second resonance is produced when the piston is 54.9 cm from the top. (a) What is the temperature? (b) What is L? (c) At what other piston positions will resonances occur? (16.3.40) If you are handed an open-open pipe, you can turn it into an open-closed pipe by covering one of the open ends with the palm of your hand. (a) If a certain pipe of length 0.35 m is open at both ends, what is the second lowest frequency that can be produced both when the pipe is open at both ends, and also when one of the ends is covered? Take the sound speed to be 345 m/s. (b) For the frequency you found in part (a), sketch the pressure amplitude variation in the pipe for the open-open case. (c) For the frequency you found in part (a), sketch the pressure amplitude variation in the pipe for the open-closed case.

2.00 m

(16.3.45) (Review of fluid mechanics) A vertical tube of with a 10.0 cm diameter has a hole at the bottom 1.00 cm in diameter. The tube is filled to the top with water, and water begins to flow out through the hole. Above the tube is a tuning fork vibrating at 256 s-1. We will denote the distance between the top of the tube and the water level by x. As the water level in the tube lowers, the first resonance is heard when x = 35.0 cm. (a) Determine, as a function of x, the speed of the water exiting the bottom of the tube. (b) Determine, as a function of x, the rate at which the top surface of the water lowers. (c) How much time elapses before another resonance is heard?

x

1.50 m

16.4 Sound Interference (16.4.50). Two speakers separated by 4.0m, as shown below, vibrate in phase, producing sound at a frequency of 250s-1. (a) At what distances from the speaker on the left would a cockroach crawling between the two speakers experience a maximum in the sound intensity? (b) At what distance from the speaker on the left would a cockroach crawling between the two speakers experience a minimum in the sound intensity? (16.4.60) Two speakers are separated by 5.0m. A cockroach crawling between the speakers notices that there is a maximum in the intensity at point A. The cockroach then begins to crawl in the direction perpendicular to the speakers, and notices a minimum at point B. He does not notice any other minima between A and B. (a) What is the frequency of the sound? (b) If the cockroach continues crawling in the same direction, how much further does he need to crawl to find another maximum?

4.0m

1.6m A

B

(16.4.70) Two speakers vibrate in phase, producing sound waves with a 20-cm wavelength. One speaker is placed at the center of a circle of radius 4.0m, and the other is placed on the perimeter, as shown below. Find the values of θ for which a cockroach crawling around the perimeter of the circle will experience (a) a minimum in sound intensity. (b) a maximum in the sound intensity.

17. Statistical Mechanics (17.1) In a certain two-level quantum well, it is found that 61.00 10× electrons reside in the higher-energy well when the temperature is 100K, and 61.30 10× electrons reside in the higher-energy well when the temperature is 200K. Taking the energy of the lower well to be E1=0, and the energy of the upper well to be E2, find the total number of electrons shared between the two wells and the energy difference between the two levels. (Note for future reference: A standard problem encountered in introductory quantum mechanics courses – you might see this in 1D – is the truly one-dimensional quantum well, in which the electrons move only in the direction which goes from one wall of the well to the other, shown above as the x-direction. In such a well, no more than two electrons may have the same energy. Although I didn’t think to say this explicitly, the wells we’ve been discussing in class can accommodate multiple electrons at the same energy because the electrons are allowed to move freely in the two directions perpendicular to the x-axis.)

θ

speakers

E1=0

E

E2

x

(17.2) Consider a distribution function given by 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

= 𝑘𝑘𝑣𝑣 for 0 ≤ 𝑣𝑣 ≤ 𝑣𝑣𝑒𝑒 and zero otherwise. (i.e. the probability density increases linearly with v up to a maximum velocity 𝑣𝑣𝑒𝑒. No particles have velocities above 𝑣𝑣𝑒𝑒. (a) Find an expression for k in terms of 𝑣𝑣𝑒𝑒. (b) If there are 50,000 particles, and 𝑣𝑣𝑒𝑒 = 500 m/s, how many particles have velocities between 300 m/s and 350 m/s? (c) Using the assumptions of part (b), find the average particle velocity. (17.3) Compute the average speed of the atoms in a 1D helium gas at T = 50K. (17.4) What fraction of nitrogen molecules in a 2D nitrogen gas at T = 100K have speeds greater than 200 m/s? (17.5) Following the methods used in class, show that the distribution function for molecular velocities in a 3D gas is

23/ 2222

Bmv

k T

B

dP m v edv k Tπ

− =

.