Homework for PHY 309K
Welcome to homework assignments for the Elementary Physics (I)
course, PHY 309 K.
The homeworks on this page are for the section taught by
Professor
Vadim Kaplunovsky in Fall of 2006 (unique #61680).
Other sections assign different homeworks.
Many homework problems are taken from the Trefil & Hazen
textbook; such problems are listed by problem numbers.
All other problems are written in full as plain (HTML) text.
Please note that the physical laws and the formulae you use
and the way you put them together are more important for your grade
than the numbers you calculate.
If you understand the physics governing a problem, use correct formulae,
and properly put them together, you would get a high partial score even
if your arithmetic is faulty.
But you would get a low score for a numerical answer without a clear explanation of
where it came from, even if the number happens to be correct.
Assignment List
 Set I, due September 13.
This time, there are only nontextbook problems:
 Enrico Fermi once gave a class which lasted precisely 1 microcentury.
How long was his class in minutes?
 A British naturalist with a penchant for archaic units once reported the
average speed of a snail as 1 furlong per fortnight.
(A furlong is one eight of a mile or 220 yards; a fortnight is 14 days&nights.)
How far would a snail crawl in one minute?
Please state your answer in metric units.
 A lake has area one square kilometer and average depth one meter.
Calculate the volume of water in the lake in cubic meters and in liters.
Note: In consistent units, Volume = Area × Average_Depth
 Planet Earth has mass M=6×10^{24} kg and radius
R=6.37×10^{6} m.
Calculate its average density and compare it to the density of water
(ρ_{water} = 1 g/cm^{3} = 1000 kg/m^{3}).
For your information, the average density of a body is the ratio of its net mass
to its net volume, ρ = M/V.
The volume of a sphere of radius R is
V = (4π/3) R^{3}.
Solutions.
 Set II, due September 22.
Textbook problems 17, 19, and 21 at the end of chapter 3.
Also, one nontextbook problem:

Distances between planets are often measured in astronomical units
(au) where 1 au = 149.6×10^{6} km
is the average radius (semimajor axis) of the Earth's orbit around the Sun.
(a) Mercury's orbit around the Sun has average radius (semimajor axis)
0.387 au and eccentricity approximately 20%
(i.e., ε=0.2).
How close does Mercury get to the Sun at the closest point of its orbit
(the perihelion) and how far is it at its most distant point (the aphelion)?
(b) How long is Mercurian year?
Solutions.
 Set III, due September 29.
Nontextbook problems:
 Consider a steamboat in a river. The river current flows at speed 6 m/s.
The boat moves at speed 8 m/s relative to the water; an observer
standing on the bank sees the boat steaming faster or slower than that, depending
on the boat's direction.
(a) When the boat steams downstream, what is its speed relative to the river's banks?
(b) When the boat steams upstream, what is its speed relative to the river's banks?
(c) When the boat heads directly across the river, it motion relative to the banks is diagonally
across and downstream (cf. figure 2.9 of the textbook). What is the speed of this motion?
(Hint: Use Pythagoras's theorem for adding perpendicular vectors.)
 As the Earth spins, every point on its surface moves through a latitude circle in 24 hours.
In particular, the UT Austin campus moves in a circle of radius approximately 5400 km.
(a) What is the speed of this motion?
(b) What is the centripetal acceleration of the campus?
 How much force does it take to accelerate a 1000 kg car from zero to 60 miles/hour in 6 seconds?
 Bob and Alice try to play tugofwar while sitting on ice. The ice is very slippery (no friction),
so Alice and Bob are both sliding towards each other. Bob's mass is 75 kg, Alice's mass is 50 kg.
Bob accelerates towards Alice at 2 m/s^{2}. What is Alice's acceleration?
Solutions.
 Set IV, due October 11.
Textbook problem 5 at the end of chapter 5.
Also, three nontextbook problems:
 Last year
Huygens probe landed on the Saturn's moon Titan and reported a surprisingly Earthlike
world: dense atmosphere, seas, rivers, etc. Titan is bigger than Luna but smaller than
Earth, and less dense. To be precise, its radius is 0.468 of R_Earth but its mass is only
0.0226 M_Earth.
If you go to Titan, how much would you weigh there compared to your weight on Earth?
 Consider a planet on a circular orbit of radius 2 au
around a star of mass 3 M_Sun.
How long is the year on that planet?
 Communication satellites have geostationary orbit: a circular orbit in equatorial
plane with period equal to 24 hours, so that the satellite appear to hang stationary above
some point on Earth's surface.
The radius of this orbit is 42,240 km.
Consider a similar "geostationary" or rather arestationary orbit around Mars.
What is the radius of such orbit?
For your information, Mars has mass 6.42×10^{23} kg
and diurnal period (day+night) of 88775 seconds
(24 hours, 39 minutes, and 35 seconds).
Solutions.
 Set V, due October 18.
Textbook problems 5 and 8 at the end of chapter 6.
Also, two nontextbook problems:
 A 2000 pound car moving at 60 MPH speed hits a 1000 pound cow standing on the road.
The collision is inelastic: the cow ends up on the hood of the car, and both are
badly damaged.
What is the speed of the car (with the cow on its hood)
immediately after the collision?
 In a perfectly elastic collision of two bodies, their relative velocity
changes its direction but not magnitude.
For example, in one elastic collision between two steel balls of different masses,
their respective velocities were v_{1}=+10 m/s and v_{2}=0
before the collision, and v_{1}′=−5m/s and
v_{2}′=+5m/s after the collision.
In the process, the relative velocity v_{rel}=
v_{2}v_{1} changed from −10 m/s to +10 m/s:
same magnitude but opposite direction.
What was the mass ratio m_{1}/m_{2} of the two
balls involved in this collision?
Hint: Use momentum conservation.
Solutions.
 Set VI, due October 25.
Four nontextbook problems:
 A CD disk has radius R=6 cm; in a 48x CD drive it
spins at frequency 10080 RPM.
(a) What is the period of this rotation?
(b) What is its angular velocity?
(c) What is the linear speed of a point at the rim of the disk?
(d) What is the centripetal acceleration of that point?
(e) What are the linear speed and the centripetal acceleration
of another point located 3 cm from the disk's center?
 The mass of that CD is 15 grams.
(a) What is the disk's moment of inertia?
(For simplicity, ignore the hole in the CD and treat it as
a solid disk. The formila for such disk's moment of inertia
is given in figure 711 of the textbook.)
(b) What is the angular momentum of the spinning CD?
(c) How much torque should the CD drive's motor supply to spin
the disk from zero to full speed in just one second?
 When you step on the platform of medical scales, your weight is
transmitted through a cable to an asymmetric balance;
the cable is attached 1 cm left of the pivot.
On the right side of the pivot, there is a 1 kg mass that
can slide along the balance; you adjust its position to balance
the scales.
A person steps on the scales and they balance when the sliding
weight is 80 cm right from the pivot.
What is that person's mass?
 Two astronauts on a space walk are connected to each other by
a 20 meter line.
The mass of each astronaut is 100 kg, including the spacesuit.
The astronauts spin around their common center of mass —
located in the middle of the line — at frequency=3
revolutions per minute.
(a) What is the net angulat momentum of the two astronauts?
(b) The astronauts pull on the line connecting them until
they are 10 meters from each other.
What happens to their angular momentum?
(c) What happens to the rotation frequency?
Solutions.

Training exercise for the second midterm.
It's not graded, but you should do it anyway before October 30.
 Ariel is a moon of Uranus. Comparing its orbit around Uranus
to Luna's orbit around the Earth, one finds that Ariel's
orbit has radius one half of Luna's but Ariel's period is
only 1/11 of Luna's period:
R_{Ariel}=(1/2)R_{Luna} but
T_{Ariel}=(1/11)T_{Luna}.
Use these data to calculate Uranus's mass in units of Earth's masses
(that is, calculate the ratio
M_{Uranus}/M_{Earth}).
 Two friends are speedboating on Lake Travis.
Andy's boat has gross mass 1000~lb while Bob's boat has gross mass 2000~lb.
Being somewhat drunk, Andy and Bob play chicken, and eventually collide head on
with each other.
Just before the collision Andy's boat speeds North at 35~MPH while Bob's boat
speeds South at 25~MPH.
The collision is inelastic: the two boats smash each other and become entangled
together.
What is the common velocity of the two boat wrecks immediatly
after the collision?
(`Immediately' means before the wreckage slows down (due to water resistence) and
eventually sinks.)
 Charlie and David carry a 90~lb beam on their shoulders.
Charlie's shoulder is 2 feet ahead of the beam's center of mass
while David's shoulder is 4 feet behind the center of mass.
How much weight is carried by Charlie and how much by David?
 Consider a 60~kg student walking from one campus building
to another for her next class.
The second building is 1~km away from the first and has
10~m higher elevation.
Assuming perfect efficiency, how much mechanical work does
the student perform while walking from one classroom to another?
Solutions.
 Set VII, originally due November 8, extended to November 10.
Textbook problems 13, 16, and 19 at the end of chapter 8 (page 183).
Also a nontextbook problem:

A big horse working hard has mechanical power
of about one horsepower, hence the name of the unit.
In English units 1 hp = 550 feetpounds per second, and in metric
1 hp = 746 W.
Consider a 2000 pound (900 kg) horse walking up a 1000 ft (300 m) high hill.
If his mechanical power is 1 hp, how much time does he need to get
to the top of the hill?
Note: homework 19 is badly phrased. What is should say is:
«Approximate the continent as a square slab of rock, 5000 km by 5000 km by
30 km deep, of average density ρ=2800 kg/m^{3}».
Solutions.
 Set VIII, due November 15.
Nontextbook problems:
 Riding a bicycle at 20 km/h (12.5 MPH) takes about 60 Watts of mechanical power.
(Assuming level ground, no wind, an average bike, and a 150 lb rider.)
How many calories does a rider burn in an hour of such ride?
Note that human muscles' efficiency is about 25%.
That is, for each calorie of food converted to mechanical energy, another 3 calories
are converted to body heat.
Also note that food calories are actually kilocalories, 1 kcal=4186 J.
 The main ingredient of sugar is sucrose (chemical formula
C_{12}H_{22}O_{11}), the rest is water and impurities.
Assuming pure sugar is pure sucrose, how many
C_{12}H_{22}O_{11} molecules
are there in one pound of pure sugar?
 Copper (Cu) has two common isotopes, ^{63}Cu
(29 protons, 34 neutrons, atomic mass 62.93) and ^{65}Cu
(29 protons, 36 neutrons, atomic mass 64.93).
Natural copper is a mixture of the two isotopes;
when the chemists measure its atomic mass they find an average value of 63.546.
What are fractions of each isotope in the natural copper?
 Deuterium — the ^{2}H_{1} isotope of hydrogen —
has atomic mass μ[^{2}H_{1}]=2.0141.
In nuclear fusion, two atoms of deuterium fuse into one atom of helium4 (isotope
^{4}He_{2}), which has atomic mass
μ[^{4}He_{2}]=4.0026.
Note that μ[^{4}He_{2}] < 2×μ[^{2}H_{1}]
because a small fraction (about 0.64%) of deuterium mass is converted to energy
according to Einstein's formula
E=ΔM×c^{2}.
(a) How much energy is released when one kilogram of deuterium is fused to helium?
(b) One liter of seawater contains about 35 milligram of deuterium.
If you could fuse that deuterium in a controlled manner, with the same efficiency
as burning gasoline, how many liters of gasoline would you be able to replace with one
liter of seawater?
FYI, Burning 1 liter of premium gasoline releazes about 35 MJ of energy.
Solutions.
 Set IX, due November 27.
Nontextbook problems:
 The Challenger Deep (named after research ship Challenger II)
is the deepest place in all the oceans of Earth, 10923 meters deep.
(35840 feet, or almost 7 miles below the surface).
What is the water pressure at the bottom of the Challenger deep?
Note that seawater is a bit denser than fresh water;
for the purpose of this exercise, use
ρ(seawater)=1050 kg/m^{3}.
 The average air pressure at the sea level is 760 Torr or 101300 Pascal.
For simplicity, let's ignore mountains and assume the atmosphere
starts at the sea level all over the planet's surface of total area
A=4πR^{2}=511×10^{6} km^{2}.
Calculate the total mass of the Earth's atmosphere.
 A glass bowl weighs 5 pounds in air, but when weighed underwater,
its apparent weight is only 3 pounds.
What is the density of glass this bowl is made of?
 An ice cube floats in a glass of warm water.
What happens to the water level when the ice melts?
Does it go up, down, or stays the same?
Please explain your answer.
Solutions.
 Set X, due December 6, extended to December 8 (last class day).
If you turn it in by 12/6, the grader will return it on the last class day.
If you turn it in on 12/8, there is no penalty, but if you want you homework
back, you would need to come to my office.
This set has six problems. You can do any four problems for full credit,
or you can do all six problems and get extra credit.
All problems are nontextbook:
 Consider a horizontal water pipe of inner diameter 2 cm carrying water
at speed 10 m/s. At some point, the pipe narrows down to 1 cm in diameter,
and then the pipe ends and the water jets out.
(a) What is the speed of water in the narrow part of the pipe?
(b) The narrow part of the pipe opens to the air, so the water pressure there
is equal to the atmospheric pressure P_{atm}=100 kPa.
What is the water pressure in the wide part of the pipe?
 Most solid materials expand with temperature according to
L(T)L_{0}=L_{0}×α×(TT_{0})
where L(T) is length measured at temperature T
and L_{0} is L measured at some standard
temperature T_{0}, usually 22°C
(72°F).
As the temperature changes, the length changes according to
ΔL=L_{0}×α×ΔT.
The thermal expansion coefficient α differs from material
to material.
For most types of steel, α=6.5×10^{6}/°F.
Consider a 150 meter (500 foot) steel rail suffering weather extremes
between 0°F in winter and 100°F in summer.
How much does its length changes between winter and summer?
 A heliumfilled balloon is launched underinflated to allow for the
gas expansion at higher altitude.
On the ground, the air pressure was 1000 mbar and the temperature
77°F, but when the balloon reached altitude of
10,000 ft, the air pressure dropped to 690 mbar
and the temperature to 23°F.
On the ground, the balloon's volume was V_{0}=100 m^{3}.
What was the volume of the balloon when it reached the 10,000 ft altitude?
Hint: everyday temperature in degrees Fahrenheit or Celsius is not the same as absolute
temperature in Kelvins (also known as degrees Kelvin).
 Planet Venus has a thick atmosphere whose main constituent is
carbon dioxide (CO_{2}, molecular weight μ=44 g/mol).
Near the surface, the atmospheric pressure on Venus reaches 92 bar
(9.2 MPa) and the temperature 740 Kelvin (872°F).
What is the density of CO_{2} under such conditions?
FYI, according to the universal gas law, 1 mol of any gas
satisfies PV=RT, where T is absolute
temperature (in Kelvins) and R=8.314 Pa·m^{3}/mol/K
is the universal gas constant.
 Steinbier
is an ancient beermaking technology which involves heating wort in
wooden pots by dropping hot rocks into the pot.
Granite rock has specific heat c_{r}=0.19 cal/g/°C
while wort has specific heat c_{w}=1.04 cal/g/°C
(similar to water).
Suppose you drop a 20 kg rock (or rather ten 2 kg rocks) heated to 360°C (680°F)
into 20 kg of 35°C (95°F) wort.
When the rocks and the wort reach thermal equilibrium, what would be their temperature?
 An experimental gun shoots big bullets made of ice at high speed.
When such a bullet (initially at 0°C) hits a target, its kinetic energy is converted
to heat, which melts some of the ice.
(a) If a 100 g bullet impacts the target at speed 400 m/s, how much ice would melt?
(b) At much higher impact speeds, there is enough energy to melt the whole bullet, heat the
resulting water to 100°C, and vaporise it all.
What is the minimal impact speed that would get the whole bullet vaporized?
For your information, water has specific heat c=1 cal/g/°C,
latent heat of fusion (melting) L_{f}=80 cal/g,
and latent heat of vaporization (boiling) L_{v}=540 cal/g;
1 calorie of heat is equivalent to 4.186 Joules of energy.
Solutions.
Exams
For your records, here are the exams we had thus far:
Last Modified: December 9, 2006.
Vadim Kaplunovsky
vadim@physics.utexas.edu