Homework for PHY 302K

Welcome to homework assignments for the General Physics (I) course, PHY 302 K. The homeworks on this page are for the section taught by Professor Vadim Kaplunovsky in Spring 2010 (unique #57965). Other sections assign different homeworks.

Many homework problems are taken from the Giancoli textbook; such problems are listed by problem numbers. (The numbers are according to the 6^th edition; if you use an older or a newer edition, the number in your book will be different.) 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.

Schedule of Homeworks and Exams.

Each homework set should be posted by the "assigned" date in this schedule. If it is not, let me know ASAP.

The textbook chapters in the schedule are tentative. They may change if the class goes faster or slower than I expect.

Skipped Sections

When I decide to skip some sections of the textbook, I shall list them here and announce them in class. The exams will not involve the skipped material.

Sections skipped:

1.8, 7.9, 8.9, 9.5–7, 10.13, 13.5, 13.14, 15.6 15.9–12.

Chapters I did not get to:

11 and 12.

Supplementary Notes and Pictures:

• Significant Figures.
• Kepler Laws (external link).
• Rotation Kinematics, Moment of Inertia, and Torque.
• Laminar vs Turbulent flows.

Exams

• First mid-term, February 19; solutions.
• Second mid-term, March 29; solutions.
• Third mid-term, April 28; solutions.
• Final exam, May 15 (2-5 PM); no solutions.

Homework sets

Set 1

Only non-textbook problems this time:

1. Convert the following distances into kilometers, meters, and centimeters. Use the scietific notations when appropriate, and keep the right numbers of significant digits. (a) The mapped length of the Mammoth cave is 348 miles. (b) The Ribbon Falls drop 1612 ft. (c) Mount Denali's height is 20,320 ft. (d) King's Canyon's depth is 8200 ft.
2. When the Great Pyramide of Cheops was built, it had base area 13.0 acres and height 481 ft. Calculate the volume of the Pyramide in cubic feet and in cubic meters. FYI, 1 acre is 43560 ft², exactly. The volume of a pyramide or a cone is V=Bh/3, but you need consistent units to use this formula.
3. (a) How many liters of paint does it take to paint two yellow lines in the middle of of a 100 kilometer long highway? For the sake of definiteness, assume each line is 10 cm wide and 1 mm thick. (Or rather, when the yellow paint was still wet, it was 1 mm thick.) (b) What's the mass (in kilograms) of all that paint if its density is ρ=1100 kg/m³?
4. The Moon has radius 1737 km and mass 7.35·10²² kg. Calculate the Moon's average density in units of g/cm³ and compare to the density of water.
5. Consider two balls made of solid aluminum. The second ball has twice the radius of the first, R₂=2R₁. The first ball has mass M₁=1 kg. What is the mass M₂ of the second ball? Note that you do not need to know the density of aluminum to solve this problem — the fact that both balls have the same densities should be enough.
6. A student weighs 150 lb, his backpack 17.3 lb. Write down their combined weight with a correct number of significant figures.
7. A circle has radius R=10.5±0.2 m. Calculate the circumference length and the area of this circle and write them down with appropriate numbers of significant figures.
8. Your friend is celebrating his 20^th birthday. What is his age in seconds? Please write your answer with a correct number of significant figures. Note that you do not know the hour and minute of his birth, only the day.
9. Make an order-of-magnitude estimate of the total amount of beer consumed by the UT students during one long semester. Please write down all of your assumptions.
10. Joe has been running Marathons for years, and one day he decides to run all the way across the American continent, from New York to Los Angeles. Make a rought estimate of time Joe will need for this run.

Due January 27; solutions.

Set 2

Non-textbook problems:

1. The velocity and the acceleration of some body can be positive or negative. Explain the meaning and gives examples of all possible combinations of v and a: (a) v>0 and a>0; (b) v>0 and a<0; (c) v<0 and a>0; (d) v<0 and a<0; (e) v>0 and a=0; (f) v<0 and a=0; (g) v=0 and a>0; (h) v=0 and a<0.
2. A driver wishing to go from Austin to San Antonio (80 miles South of Austin) takes a wrong ramp into I-35 and ends up going North instead of South. He does not notice his mistake until he gets to Waco (100 miles North of Austin). At that point he turns around, goes back to Austin and continues on to San Antonio. The whole trip took 4 hours. What was this trip's average speed and what was its average velocity?

Textbook problems 6, 7, 12, 18, 23, 28, 35, 50, 52(a) at the end of chapter 2.

Due February 3; solutions.

Set 3

Textbook problems 42, 47, 63, 77 at the end of chapter 2 and problems 9, 18, 19, 24, 31 at the end of chapter 3.

Non-textbook problems:

1. The figure on the right shows two vectors A and B and their sum A+B. Draw similar pictures for the A-B, B-A, and A-2B.
2. A plane flies due North for 90 miles, then turns and flies 180 miles in the direction 120° (clockwise from North), and then turns again and flies another 90 miles in the direction 210° (clockwise from North). (a) Draw a map of the plane's motion. (b) Find the net displacement of the plane in (x,y) components. (c) Find the magnitude and the direction of the net displacement vector.

Recommended order: First the textbook problems from chapter 2, then first non-textbook problem, the textbook problem 3.9 (i.e problem 9 from chapter 3), then the second non-textbook problem, and finally the rest of textbook problems from chapter 3.

Due February 10; solutions.

Set 4

Textbook problems 38, 66, and 41–42 at the end of chapter 3.

Non-textbook problems:

1. A passenger in an open-top convertible car moving at 30 miles per hour tosses a bottle over his shoulder. In the reference frame of the car, the initial velocity of the bottle is directed 60° above the backward horizontal. But a pedestrian standing on the street where this happens sees the bottle raising straight up and then falling down without any horizontal motion. (a) Explain how is this possible. (b) To what height did the bottle fly up before falling down.
2. A car moving at 45 MPH (20 m/s) hits a tree; while the engine compartment is crashed, the steering column moves through L=1.0 meter before coming to stop. To save the car's driver from a deadly injury, his body must also come to stop over just one meter of distance. If the driver weighs 180 lb (80 kg), how much force should the seat belt and/or the air bag apply to driver's body to keep him alive in this crash?

Textbook problems 5, 15, 23, 31, 32, and 33 at the end of chapter 4.

Due February 17; solutions.

Set 5

Textbook problems 37 and 61 at the end of chapter 4.

Non-textbook problems:

1. Bob is driving a 900 kg (2000 lb) car at speed 27 MPH (12 m/s) on an icy road. Because of the ice, the static friction coefficien between the car's tires and the road is only μ_s=0.12, and the kinetic friction coefficient is even lower, μ_k=0.08. What's the minimal stopping distance of Bob;s car under such condition? For simplicity, ignore Bob's reaction time and assume the road is straight and horizontal.
2. A box is dropped with zero initial velocity onto a conveyor belt moving horizontally with velocity 8.0 m/s. The kinetic friction coefficient between the box and the belt is μ_k=0.40. How much time does it take the box until its motion catches up with the conveyor's?
3. A 2 kg book lays at rest on an inclined surface. The incline angle (between the surface and the horizontal) is slowly increasing; when it reaches θ_c=37°, the book begins to slide down the incline. (a) What is the static friction coefficient μ_s between the book and the surface? (b) Earlier, when the incline angle was only θ=30°, what was the static friction force between the book and the surface?
4. A carton full of old junk is shoved up a steep ramp and released. The carton sides up with deceleration a_up=7 m/s², comes to a stop, and slides back down the ramp with acceleration a_down=3 m/s². (a) Explain why the carton's accelerations on the way up and on the way down are different. (b) Derive formulae for the accelerations a_up and a_down in terms of the the angle θ the ramps makes with the horizontal and the kinetic friction coefficient μ_k between the carton and the ramp. (c) Solve those equations and find θ and μ_k.

Textbook problems 17, 13, 14, 9, 18, 79 at the end of chapter 5.

Due February 26 (Friday); solutions.

Set 6

Textbook problems 27 and 73 at the end of chapter 5.

Non-textbook problems:

1. In 2005 Huygens probe landed on the Saturn's moon Titan and reported a surprisingly Earth-like 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?
2. Distances between planets are often measured in astronomical units (au) where 1 au = 149.6×10⁶ km is the average radius (semi-major axis) of the Earth's orbit around the Sun.
   (a) Mercury's orbit around the Sun has average radius (semi-major axis) 0.387 au and eccentricity approximately 20% (i.e., ε=0.20). 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?
3. 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?
4. Communication satellites are usually put in the geostationary orbit: a circular orbit in equatorial plane with period equal to sidereal day (23 hours, 56 minutes and 4 seconds, slightly shorter than the means solar day of 24 hours), so that the satellite appear to hang stationary above some point on Earth's surface. The radius of this orbit is 42,164 km.
   Consider a similar "geostationary" or rather arestationary orbit around Mars. What is the radius of such orbit?
   For your information, Mars has mass 6.42×10²³ kg and sidereal day 88642 seconds (24 hours, 37 minutes, and 22 seconds).
5. A man pulls a 10.0 kg bucket full of water out of a deep well. Man's mechanical work is 6000 J. How deep is the well?
6. How fast should a 2.45 g ping-pong ball move in order to have the same kinetic energy as a 7.00 kg bowling ball moving at 3.0~m/s?
7. A 9 g bullet hits a tree with impact speed of 250 m/s. The bullet makes a 8 cm long hole in the wood, then stops. Find the average force resisting bullet motion through the wood.
8. A 12.0 kg crate is pulled 5.00 meters up a 20° ramp by a 125 N force parallel to the ramp. (a) How much work was done by this force? (b) How much work was done by the force of gravity acting on the crate? (c) How much work was done by the normal force? (d) The kinetic friction coefficient between the crate and the ramp is μ_k=0.400. How much work was done by the friction force? (e) By how much did the kinetic energy of the crate change during this process? (f) The initial speed of the crate was 5.0~m/s. What was its speed after it had moved through 5.00~m?
9. A big horse working hard has mechanical power of about one horsepower, hence the name of the unit. In English units 1 hp = 550 feet-pounds 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?

Textbook problem 69 at the end of chapter 6.

Due March 5; solutions.

Set 7

Textbook questions 7, 10, 13, 24 and problems 91, 43, 82, 86, 39, 37, 49, 55 at the end of chapter 6. (Note: the questions are on pages 160–161 while the problems are on pages 162–166.)

A non-textbook problem:

1. Consider a bungee jump (illustration). A bungee cord acts like a spring when stretched: its tension T depends on length L as T=k(L-L₀) for some force constant k and un-stretched length L₀. However, for L<L₀ the cord folds instead of becoming compressed, so there is no tension force and no elastic energy.
   (a) To test the reliability of a bungee cord, it's tied to a 120 kg steel ball on one end and to a 61 m high bridge on the other end. The ball is dropped off the bridge (with zero initial velocity) and falls down, but the bungee cord slows down the fall and eventually stops the ball and yanks it back up. The lowest point reached by the ball is 1 m above the water, 60 m below the bridge. The bungee cord has un-stretched length L₀=20 m; what is its force constant k?
   (b) Once the cord is tested, a 60 kg student ties himself to the cord instead of the steel ball and jumps off the same bridge. How close to the water below the bridge does he get before the bungee cord yanks him up?

Due March 12; solutions.

Set 8

Textbook problems 16, 5, 4, 8, 26 at the end of chapter 7.

Non-textbook problems:

1. A 12,000 lb truck is southbound on the Guadalupe street. When it enters the intersection with the MLK boulevard, its velocity is 20 MPH in the direction 30° East from South. (Note that the Guadalupe street bends at the intersection.) A 3000 lb car going West on the MLK runs a red light and collides with the truck. The two vehicles become stuck together in the collision, and the combined wreckage moves due South until it hits the 7/11 gas station at the corner.
   How fast was the car moving before it hit the truck?
2. Pluto is a small planet (radius=1137 km, mass=1.27·10²² kg) with a large moon Charon (radius=606 km, mass=1.9·10²¹ kg); the center-to-center distance between Pluto and Charon is about 19,600 km.
   How far is the center of mass of the Pluto–Charon system from the Pluto's center? Is it above or below the Pluto's surface?
3. A 140 lb man is sitting at the stern of a 70 lb, 12-foot-long boat. The prow of the boat touches the edge of a pier and the boat isn't moving. The man stands up and walks to the prow so he can tie the boat to the pier. But as he walks forward, the boat moves back; when he comes to the prow and stops, the boat also stops moving, but now it's a few feet away from the pier.
   (a) Explain the boat's motion.
   (b) How far has the boat moved back by the time the man reaches the prow?

Textbook problems 12, 18, 68, 48 at the end of chapter 8.

Update: problem 48 at the end of chapter 8 is postponed to the next homework set.

Due March 26 (Friday after the break); solutions.

Set 9

Textbook questions 7, 9, 19, 22 and problems 24, 26, 30, 34, 48, 50, 53, 63 at the end of chapter 8.

A non-textbook problem:

1. A rod of mass M, length L, and uniform density and thickness swings around a frictionless pivot at one end of the rod; the other end is free. The rod is held at angle θ below the horizontal and then released with no initial angular velocity, ω₀=0. Find the linear acceleration a of the rod's free end immediately after the rod is released.

Due April 5 (Monday); solutions.

Set 10

A non-textbook problem:

1. Two paramedics, Bob and Charlie, carry a 200 lb patient on an 8-foot-long stretcher. Bob holds the front end of the stretcher and Charlie holds the back end. The patient lies closer to the front end of the stretcher, so his center of gravity is 3 feet from Bob's hands and 5 feet from Charlie's. How much of the patient's weight is carried by Bob and how much by Charlie?

Textbook problems 12, 6, 57, 18, 27, 36, 68(a) at the end of chapter 9.

Textbook questions 3, 6 and problems 68, 11 at the end of chapter 10.

Another non-textbook problem:

2. According to Boyle's Law, the volume of a fixed amount of gas held at constant temperature is inversely proportional to the absolute pressure of the gas.
   A bubble of gas of volume V=1.0 cm³ forms at the bottom of a lake. The bubble floats up; by the time it reaches the lake's surface, its volume increases to V'=3.0 cm³. Assuming constant gas temperature in the bubble and 0.97 atm (98 kPa) air pressure above the water, find the depth of the lake.

Due April 12; solutions.

Set 11

Textbook question 7 and problems 16, 22, 72, 80, 34, 39, 38, 41, 55, 58 at the end of chapter 10.

Non-textbook problems:

1. A small plane fies horizontally with airspeed 50 m/s (100 knots or 180 km/hr) at altitude where air density is ρ=1.00 kg/m³. Along the top surface of a wing, the air flows at speed v_t=55 m/s relative to the plane. Along the bottom surface of a wing, the air flows at a smaller speed v_b=45 m/s.
   (a) Explain how the difference between these two speeds creates the lift force which keeps the plane from falling down.
   (b) The combined area of the plane's two wings is 12 m². What is the weight of this plane?
2. This video shows a beautiful curve-ball goal in a 1978 Brazil-Italy soccer game. The ball starts parallel to the endline, but then veers right and flies into the goal, right through a crowd of defenders who can't figure out where it's going.
   Which way should the soccer ball spin to veer right? How does one kick a ball to give it such spin?

Suggested order of problems: First textbook problems 16, 22, 72, 80, and 34, then textbook question 7, then textbook problems 39, 38, and 41, then the non-texbook problems, and finally the textbook problems 55 and 58.

Update: textbook problems 55 and 58 are postponed to the next homework set.

Due April 19; solutions.

Set 12

Textbook problems 55 and 58 at the end of chapter 10.

Non-textbook problems:

1. A blood panel analysis done by medical labs includes measuring the erythrocyte sedimentation rate — the terminal velocity (in mm/hr) of the red blood cells sinking through the plasma. A red cell sinks because it's slightly dense than the plasma — ρ(cell)=1125 kg/m³ while ρ(plasma)=1025 kg/m³ — so the buoyant force is less than the cell's weight. But the cell's motion is opposed by the viscous drag force proportional to the cell's velocity, so the cell reaches the terminal velocity v_t and does not accelerate any further.
   Calculate the terminal velocity (in units of mm/hr) of a red blood cell of radius R=6μm (6·10^-6 m) sinking through the plasma of viscosity η=1.5·10^-3 Pa·s. For simplicity, treat the red blood cell as a sphere (actually, it's disk shaped) and use the Stokes formula for the viscous drag force F_D=6πRη×v.
2. Titan — the largest moon of Saturn — has a fairly dense atmosphere (mostly nitrogen and methane); near the Titan's surface it has density ρ=5.5 kg/m³. In 2005, the Huygens probe landed on Titan using aerobraking to kill its orbital velocity and then a sequence of parachutes to slow down its descent. The last parachute had diameter 2R=3.03 m and aerodynamical drag coefficient C=1.50 — meaning, the air drag on the chute is F_D=C×½ρ_airAv²; Huygens mass was 318 kg; Titan;s gravity is 1.35 m/s².
   As Huygens descended, it had plenty of time to reach the terminal velocity. What was its velocity as it landed on Titan?

Textbook problems 7, 90, and 86 at the end of chapter 13.

More non-texbook problems:

3. An alcohol thermometer has a bulb of volume 1.00 cm³ and a cylindrical tube of inner radius 0.0100 cm. The alcohol has volume expansion coefficient β=1.09·10^-3(°C)^-1. In the morning, when the temperature was 50°F (10°C), the alcohol filled the bottom 3.0 cm of the tube. How much would it fill in the afternoon when the temperature raises to 86°F (30°C)?
   For simplicity, ignore the expansion of the glass containing the alcohol.
4. A helium-filled balloon is launched under-inflated 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 V₀=100 m³. What was the volume of the balloon when it reached the 10,000 ft altitude?
   Note: mind the difference between the everyday temperature in degrees Fahrenheit or Celsius and the absolute temperature in Kelvins (also known as degrees Kelvin).
5. Planet Venus has a thick atmosphere whose main constituent is carbon dioxide (CO₂, molecular weight μ=44  g/mol). Near the surface of Venus, the atmospheric pressure reaches 92  bar (9.2  MPa) and the temperature 740  Kelvin (872°F). What is the density of CO₂ under such conditions?
6. To avoid Nitrogen narcosis, a diver is breething a helium-oxygen gas mixture. Inside his lungs, water vapor and carbon dioxide are added to the mix, and the temperature reaches 100°F (311 Kelvins). What are the average (root mean square) velocities of molecules for each component of the gas mix, namely He, O₂, H₂O, and CO₂?

Finally, textbook problem 53 at the end of chapter 13.

Due April 26; solutions.

Set 13

Textbook problems 70 and 99 at the end of chapter 13.

Textbook problems 6, 10, 12, 16, 18, 26, 28, 32, 36, 38 at the end of chapter 14.

Update 5/3: problem 14.38 (problem #38 at the end of chapter 38) is canceled.

Due May 5 (Wednesday); solutions.