Physics 1B

Additional Problems

Instructor: Eyal Goldmann

1.1 Fluid Statics

(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.

(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.)

h1

h2

R

h

(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/m

3.

(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 = 5.0cm

2. 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.

Mercury

Fe Ag

Mercury

Fe

Ag

cylinders are

held in place system is in equilibrium; no

external forces on cylinders

(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) Find the pressure behind the dam as a function of the depth below the top surface of the oil.

(You will need two functions, really.)

(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.

(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 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

(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.)

oil

water

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.)

θ θ

R

center of

curvature

air

liquid

solid

(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.

2.1 Fluid Dynamics (no viscosity)

(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.

Water

surface

ar

3.00 cm

5.00 cm

30°

(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.

(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: , , and

v1 v2

h

v

(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?

(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?

x

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.

bread

bread

bologna

molasses

honey

6.00 mm 3.00 mm

(2.2.30) Air is flowing to the right through a horizontal tube with an inner diameter of 0.5 cm.

The tube is connected to the ends of a U-tube which is partially filled with water. The two points

of connection with the U-tube are separated by 25.0 cm. The vertical column of water on the

right is 0.8 cm higher than the vertical column of water on the left. Find

(a) the speed of the air at the center of the horizontal tube

(b) the mass flow rate

Take the viscosity of air to be 181 µP. (This is the value at 20° C.)

25.0 cm flow direction 0.5 cm

0.8 cm

(2.2.40) Fluid with viscosity η is driven between two horizontal plates by a pressure

difference 2 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.

(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, η, h, and L.

(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 L.

h

D

L

flow

direction

P2 P1

extFr

y

x

draw free body

diagram for

this region

extFr

(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

3.1 Kinetic Theory – Temperature, Degrees of Freedom

(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.

(3.1.20) 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.

(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.

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 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 −= × .

(Partial answer: Original length of copper is 5.886 m.)

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 B

D

TC Nk

T

=

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?

LSt LCu

(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

2

Bel

F

Nk TC

T

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.

6.2 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.

(6.2.20) A pressure cooker with a capacity of 6.0 quarts is filled with steam at 100 °C and

atmospheric pressure. The standard operating gauge pressure for a pressure cooker is 15 psi.

How much heat must be added to the steam to reach this pressure? Assume that a water

molecule has six degrees of freedom.

(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.

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

(7.10) 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?

non-moving

partition

N2

He

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. (Answer:

112 J/s.Q =& This uses the textbook’s value for the thermal conductivity of copper.)

0°C 100°C

x 0 25cm

8.2 Thermal Transport – Radiation

(8.1.10) 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?

(Hint 1: You may use the result from Problem 2096 in the Lab Manual.)

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 where 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.

(9.1.30) 3.00 moles of helium is taken through the cycle below. Steps A�B and C�D are

isobaric, and steps B�C and D�A 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

P

V

A

B C 1.0 atm

3.0 atm

P

V

A B

C D

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.

(10.2.20) 2.00 mol of Helium gas is taken through the cycle A� B � C, as shown below.

A�B 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 A�B, 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 B�C, 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 C�A, find the work done by the gas, the heat entering the gas, and the change in the

internal energy of the gas.

V

P A

B C

2 atm

1 atm

2000 cm3

11.1 Adiabatic Processes

(11.1.10) In class we derived the rate at which temperature decreases with altitude in the

troposphere (the layer of the atmosphere nearest the Earth’s surface).

(a) Starting with the result from class, find an equation which expresses temperature as a

function of height within the troposphere, assuming that T0 is the temperature at the earth’s

surface.

(b) Starting with your result from (a), find an expression for pressure as a function of height in

the troposphere, assuming that the pressure at the earth’s surface is Patm.

(c) What is the pressure difference between the surface and a point 5 km above the surface?

Assume that the surface temperature is 300K.

(11.1.20) In class we derived the rate at which temperature decreases with altitude near the

earth’s surface. To do so, we assumed that the acceleration of gravity is constant. Now, suppose

that we dig a very deep, narrow hole. We would like to know the temperature in the hole as a

function of depth beneath the earth’s surface. This is complicated by the fact that the

acceleration of gravity decreases linearly as you go to the center of the earth. (You may have

learned this in Physics 1A.) That is, the acceleration of gravity at a distance r from the center of

the earth is ( ) s

E

rg r g

R= where gs is the acceleration of gravity at the surface and RE is the radius

of the earth.

(a) Following the derivation shown in class (but using the different expression for g) find an

equation expressing the temperature of air in the hole as a function of distance from the earth’s

center. Take the temperature at the surface of the earth to be T0 = 300K. Ignore the possibility

that the increasing temperature will activate extra degrees of freedom in the air molecules.

(b) According to the formula you found in (a), what would be the temperature in the hole 51.00 10 m× below the surface of the earth?

(11.1.30) In class we derived the rate at which temperature decreases with altitude in the

troposphere (the layer of the atmosphere closest to the earth’s surface). To do so, we assumed

V

A

B

C

P

V1

P1 P1 = 2.50 atm

V1 = 0.0330 m3

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( ) E

s

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.

11.2 Cycles in P-V Place 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.

P

V

PA

VA VB

PA = 3.00 atm

VA = 0.033 m3

VB = 0.038 m3

VC

A B

C

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. Assuming that the objects start at

temperatures Thi and Tci, where Thi > Tci , and that the engine always operates at the Carnot

efficiency, show that

(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

(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 expressing 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?

(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. A�B and C�D

are isothermal processes, and B�C and D�A 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.

12.3 Fridges

(12.3.10) Show that a Carnot engine run in reverse acts as a refrigerator with coefficient of

performance

.

(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 . 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.)

P

V

A

B

C

D

(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

C.

(a) How much heat must be extracted from the freezer so that all of the water turns to ice at

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) In class, we proved that if the Clausius statement of the second law of

thermodynamics is true, then the Kelvin statement of the second law of thermodynamics must be

true as well. Prove the converse of this, i.e., that if the Kelvin statement is true, then the

Claudius statement is true. Your response should include a picture and well-written English

explaining your reasoning.

(14.1.20) Show that the COP of a Carnot refrigerator (i.e. a Carnot engine run backwards) is

CarnotCOP c

h c

T

T 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

(15.2.10) A wave on a string is described by the wavefunction

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).

15.3 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

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

(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?

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 ( )21

TT P

cδ γ δ

ρ= − .

(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 × 10-7

m 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? On the chart in your text on page 521, find a

comparably loud sound.

(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?

2.00 m

(16.3.40) (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.

17.1 Superposition and Standing Waves

(17.1.10) 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?

(17.1.20) (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?

θ

speakers

M

80 cm

(17.1.30) The following functions represent waves traveling on the same string in opposite

directions:

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.