Note: These problems are designed to require more thinking than traditional problems. Some problems may require techniques from more than one chapter. Some problems may have more than one reasonable solution. Some problems may have no solution, in which case it is your job to explain why there is no solution, the data you would need to obtain a solution, and how you would use this data to obtain a solution. Some problems may include irrelevant information. Some problems may include non-SI units, but your solutions must be given in SI. Some problems may require reasonable assumptions such as free-fall, minimal friction, etc. All solutions must show work to receive full credit. All solutions should be presented symbolically before plugging in numbers. All solutions must include acknowledgements. All problems using Newton's laws must include free- body diagrams to receive full credit. Each force must have a type, direction, approximate magnitude, source, and receiver clearly identified.

Units
Imagine that you are a dictator who would like to impose a new system of units on your nation as an exercise in your arbitrary power. Invent and describe this system of units, including its base units (skip current and luminous intensity) and a few derived ones. Discuss the merits of this system (precision, universality, etc.). You should be creative, comprehensive, and critical. You should not worry about being practical. You are not required to have the same base dimensions (distance, time, etc.) as SI. For example, you could have a base acceleration and distance and then derive the time unit.

Flower Pot 
A flower pot falls from a balcony of an apartment building in Salem, Oregon and injures a person. Your job is to determine the apartment from which it originated. Video footage from the second floor indicates that it traveled past a 4.000 meter high window in 0.2600 seconds. All floors in the building are 4.000 meters high. From which floor was the flower pot dropped? Hint: don't even think about using v = d/t.

Shot Put
The shot put is a track and field event involving "putting" (throwing in a pushing motion) a heavy metal ball (called the shot) as far as possible. A shot putter starts from rest, accelerates the shot uniformly in a straight line, and releases the shot at 15 m/s. How far did his hand move?

Shot Put 2
The shot put is a track and field event involving "putting" (throwing in a pushing motion) a heavy metal ball (called the shot) as far as possible. A shot putter starts from rest, accelerates the shot uniformly in a straight line at 60 m/s² for 0.20 seconds, and releases the shot at 15 m/s. How far did his hand move?

Sledding
A rare "snow day" in Salem, Oregon gives you the opportunity to go sledding. As you drag your sled across level ground at constant speed, you wonder what the coefficient of friction of the snow on the sled is. You estimate the following:

• You pull on the rope with a 1.2 pound force

• The sled weighs 10 pounds

• The sled moves with a speed of 1.5 m/s

• The rope makes an angle of 25 degrees to the level ground

Concrete Block
You are passing a construction site near building 8 on the Chemeketa campus. You see the construction workers leave for a break, but they have left a large concrete block resting at the top of a wooden ramp. As soon as their backs are turned, the block begins to slide down the ramp. Estimate the length of the ramp based on the following information:

• It takes 9 seconds for the block to reach the bottom of the ramp

• The ramp appears to be straight and at an angle of about 20 degrees above horizontal

• Your physics book states that the coefficient of kinetic friction between concrete and wood is 0.35

Hill Driving 
You are driving an 1100-kg car at 40 mph over the top of a hill that is shaped in a circular arc (radius = 60 feet). Calculate the magnitudes of each of the forces on the car when you reach the top. You may assume drag and friction are negligible.

Exercise Energy
It has been proposed that an athletic club use the energy from its customers (on stationary bikes, treadmills, etc.) to generate some of its electricity. You have been hired by the athletic club to calculate the monthly value of harvesting this energy based on the following:

• The average customer on an exercise machine can generate 150 W of electrical power

• There are 10 customers using exercise machines on average throughout the day

• The athletic club is open 18 hours per day

• Electricity is worth $0.12 per kWh

Train Spring
Some train stations have large horizontal springs at the ends of the tracks. These are safety devices to stop the train so that it will not go plowing through the station in the event an engineer is too busy texting. You would like to know how fast a train could be going and still be safely stopped by the spring. You would also like to know the spring constant of the spring. It looks like the spring can be compressed about 3.0 meters. The maximum safe acceleration for passengers while stopping is about half of g. You know from years of reading books about trains that a typical train has a mass of 5.0E5 kg.

Coconut
While getting hit in the head with a falling coconut is not a significant cause of death, it is reasonable to assume that it is an unpleasant experience. In order to estimate the damage from a coconut, it is desirable to know the force of impact (among other things). Use the following data to calculate a force of impact for a coconut on a person's head:

• The coconut tree is at the equator near sea level

• The coconut has a mass of 12 kg

• The coconut falls from a height of 20 m

• The coconut hits a person of height 2 m and mass of 80 kg

• The coconut is in contact with the person's head for 0.040 seconds

• The coconut has a negligible velocity after the collision

Airplane Launcher
You are a Navy officer ordered to design an effective airplane launcher. The launcher consists of a large spring that pushes the plane for the first 6 meters of a 22 meter long runway. The plane's engines can supply a constant thrust of 5.4E4 N for the entire length of the runway. In order for the two (metric) ton plane to successfully take off, it must have a speed of at least 44 m/s by the end of the runway. What is the minimum spring constant for the launcher? Using this spring, what will be the maximum acceleration of the plane (and its pilot)? Do you think this is a good design? Why or why not? If not, what would you suggest changing (thrust, mass, spring, runway, etc.)?

Skateboarder
You have a friend who is entering a skateboarding competition in Los Angeles. She wants to take a running start and then jump onto her 5 kg stationary skateboard. She will then roll along a 2 meter section of level track and go up a sloped concrete ramp. She wants to reach a maximum height of at least 3 meters above where she started. She can run with a maximum speed of 8 m/s and has a mass of 50 kg. Will this trick work?

Tarzan and Jane
As a good physicist and a helicopter parent, you would like to check if your children's cartoons present correct physics. In one cartoon, Tarzan is in peril from a stampede of wildebeests. Lucky for him, Jane is nearby. She grabs a vine and swings down to a stationary Tarzan to save him. At the bottom of the swing, she grabs him. They both continue to swing to safety up to a height which looks to be about one-third that of Jane's original height. It appears that Tarzan has twice the mass of Jane. Decide if you are going to let your children watch this cartoon based on whether or not it is correct physics.

Swing
A ball is attached to a string (extentionless and massless, of course) of length L. The string is attached to a frictionless pivot point. When the string is vertical, the ball is a distance h above the horizontal ground. The ball and string are released from rest at an angle θ₁ from vertical (θ₁ < 90°) and the string is straight. You may place a sharp knife somewhere so that it cuts the string and allows the ball to launch from a chosen angle, θ₂. Derive a symbolic expression (in terms of the given variables and possibly other constants) for the following:

1. Launch speed

2. Landing speed

3. Height attained after launch as measured from the ground

4. Height attained after launch as measured from the launch point

5. Time of flight

6. Horizontal distance as measured from a fixed location on the ground which is directly below the ball and string when they hang vertically

7. Horizontal distance as measured from the launch point

Calculate the maximum possible value and the θ₂ necessary for each of the above quantities when L = 2.00 m, h = 0.50 m, and θ₁ = 40°.

Traffic Accident
Your friend is going to court to determine fault in an automobile accident in Paris, Texas. She thinks that the other car was speeding and she was not. She is looking to you for help to prove her assertion. During discovery, the following evidence is presented, none of which is disputed by the other driver:

• Weather records indicate that there was no rain the day of the collision

• Your friend was traveling north just before the collision and the other car was traveling east

• The two cars got stuck together and remained so after the collision while skidding to a stop

• The speed limit on both roads is 45 mph

• The accident occurred on level ground

• Measurements of skid marks created after the collision indicate that the cars skidded 50 feet at an angle of 30 degrees north of east before stopping

• According to driver's manuals, your friend's car weighs 2400 lbs and the other car weighs 2000 lbs. Each driver weighs 150 lbs and there was no significant cargo

• A physics textbook indicates that the coefficient of kinetic friction for rubber on dry pavement is 0.80

What can an analysis of the physics of this collision and aftermath tell the court regarding the speeds of the vehicles before they collided?

Joe

Joey Joey took a stone
And knocked
Down
The
Sun!
And whoosh! it swizzled
Down so hard,
And bloomp! it bounced
In his backyard,
And glunk! it landed
On his toe!
And the world was dark,
And the corn wouldn't grow,
And the wind wouldn't blow,
And the cock wouldn't crow,
And it always was
Night,
Night,
Night.
All because
Of a stone
And Joe.

-Shel Silverstein

How fast did Joey have to throw a stone to hit the sun? Ignore air resistance. The stone need not make the sun land in his backyard.

Rotating Rod
A rod of length L and mass m is attached to a pivot at one end. The rod is rotated to an angle of nearly 90 degrees above horizontal and released from rest. It is observed to rotate to horizontal. At the instant of release…

• find the moment of inertia of the rod rotating about the fixed end

• find the magnitude of the torque on the rod

• find the magnitude of the angular velocity of the rod

• find the magnitude of the angular acceleration of the rod

• find the magnitude of the velocity of the tip of the rod

• find the magnitude of the tangential acceleration of the tip of the rod

• find the magnitude of the centripetal acceleration of the tip of the rod

• find the vertical acceleration of the tip of the rod

• find the horizontal acceleration of the tip of the rod

• find the vertical acceleration of the center of the rod

• find the horizontal acceleration of the of the center of the rod

• find the x and y components of the force exerted by the pivot point

At the instant just before the rod hits the ground…

• find the moment of inertia of the rod rotating about the fixed end

• find the magnitude of the torque on the rod

• find the magnitude of the angular velocity of the rod (hint: use conservation of energy)

• find the magnitude of the angular acceleration of the rod

• find the magnitude of the velocity of the tip of the rod

• find the magnitude of the tangential acceleration of the tip of the rod

• find the magnitude of the centripetal acceleration of the tip of the rod

• find the vertical acceleration of the tip of the rod

• find the horizontal acceleration of the tip of the rod

• find the vertical acceleration of the center of the rod

• find the horizontal acceleration of the of the center of the rod

• find the x and y components of the force exerted by the pivot point

• compare the above value to g and thoroughly explain any difference

At some angle θ above horizontal…

• find the moment of inertia of the rod rotating about the fixed end

• find the magnitude of the torque on the rod

• find the magnitude of the angular velocity of the rod (hint: use conservation of energy)

• find the magnitude of the angular acceleration of the rod

• find the magnitude of the velocity of the tip of the rod

• find the magnitude of the tangential acceleration of the tip of the rod

• find the magnitude of the centripetal acceleration of the tip of the rod

• find the vertical acceleration of the tip of the rod

• find the horizontal acceleration of the tip of the rod

• find the vertical acceleration of the center of the rod

• find the horizontal acceleration of the of the center of the rod

• find the x and y components of the force exerted by the pivot point

All solutions must be in terms of g, L, m, and θ. Show your work, but place the results of your solution in a table like the following:

• At what angle above horizontal will the magnitude of the vertical acceleration of the tip of the rod be equal to g?

• At what angle above horizontal will the magnitude of the vertical acceleration of the tip be a maximum? What is the maximum acceleration? How can it be greater than g?

Rotating Rod 2
A rod of length L and mass m is attached to a pivot at one end. The rod is attached to a point mass (mass M) at some location on the rod. The rod is rotated to an angle of θ degrees above horizontal and released from rest.

• Calculate the position of the point mass (in terms of L) so that the angular acceleration is identical to that without the ball attached. Hint: there is another solution besides zero.

• Calculate the position of the ball (in terms of L, m, and M) so that the angular acceleration is a maximum. Hint: the answer is not zero.

Boxed Pendulum
Design a pendulum with a period of 5.00 s that will function inside a 1 m³ cube. It must work in an inertial reference frame on Earth. Do not use any springs.

Pendulum Altimeter
You are given a stopwatch and a simple pendulum of length 2.00000 m. The pendulum can undergo 100 oscillations before they become too small to notice. You know that your stopwatch error is about 0.10 s. How high must you climb in order to detect a significant difference in the period of the pendulum?

Dropped Rock
Suppose you drill a hole through the center of the Earth all the way to the other side and then drop a rock.

• Calculate the speed of the rock at the center of the Earth.
• Calculate the time it would take a rock to go all the way through.

Hints: Assume that the Earth has uniform density and that the hole doesn't have any air. Only the matter in the sphere closer to the center than the rock exerts a net gravitational force on the rock. Ignore the rotation of the Earth.

Bobbing in Mercury
A cube of iron is released just below the surface of a large vat of mercury. The top face of the cube is parallel to the surface of the mercury. Calculate the time it takes the cube to return to the point of release. Assume that the drag force of the mercury and the buoyant force of the air are negligible.

Sound and Speed
You've been hired as a consultant to the police department to design a new device that measures the speed of vehicles. The idea is to make a small, rugged device to be placed in the road that will pick out a specific frequency emitted by the car as it approaches and then measure the frequency as it recedes. The device will then use this data to calculate the speed of the car. You are currently in the process of writing a program for the chip in your new device. To complete the program, you need a formula that determines the speed of the car using the measured frequencies. You may also include in your formula any physical constants that you might need.

1. Derive this formula. Note that the source frequency is not known.
2. What could you do (other than improve the quality or quantity of frequency data gathered) to improve the accuracy of this device?

Temperature
Define temperature. The definition should not be an ordinal definition based on direction of heat flow or a measure of "hotness" or "coldness". It should be something that can be calculated based on other measurable quantities (at least in principle). Do not use the ideal gas law (or variations thereof) as it is only approximate.

Entropy
Define entropy. Do not simply say it is a measure of disorder which is not a precise mathematical statement. It should be something that can be calculated based on other measurable quantities (at least in principle).

Maxwell-Boltzmann
The Maxwell-Boltzmann speed probability distribution is the following:

f(v) = √[(2/π)(m/(kT))^3]v^2*e^[-mv^2/(2kT)]
m = mass
k = Boltzmann’s constant = 1.381E-23 J/K
v = speed
T = temperature

Calculate the average speed (v_ave) and most probable speed (v_p). Hints:

Bonus: Find the root-mean-square speed (v_rms) in terms of m, k, and T.

Bonus: Find the fraction of protons predicted by the Maxwell-Boltzmann distribution to have a speed greater than the speed of light (integrate from c to infinity), given a very large temperature, say 7.3E10 K. Find a temperature for which the Maxwell-Boltzmann distribution predicts a significant fraction of the particles (you decide the threshold) will have a speed above that of light.

Charged Spheres
Two small spheres are suspended by strings of length L = 1.20 m from a ceiling at points d = 3.20 m apart. They have equal mass of m = 8.00 kg and equal and opposite charge of q = 1.40E-4 C. Calculate the angle or angles the strings make from vertical at equilibrium.

Harvesting Lightning
It has been proposed that lightning be harvested for energy. Measurements of lightning strikes indicate that they have the following average properties (note, these are reasonable approximations):

• Potential difference between each end of 100 million V
• Current of 30000 A
• Time of 1.67E-4 seconds
• 20 strikes per square kilometer per year in lightning prone regions of the United States and 200 strikes per square kilometer per year in the most lightning prone regions in the world (Congo).

“Lightning collectors” have the following properties (note, I’m just making these up, though I think they are wildly optimistic):

• They can only collect lightning that would strike within a 100 m x 100 m area
• They each cost $10,000
• They last 40 years
• Maintenance costs and scrap value are negligible

The following economic data are also known (note, these are very rough estimations):

• Electrical energy is worth about $0.12 per kWh and is not expected to increase in value
• Investors in such projects expect to get back triple their initial investment in total after 40 years

As an independent energy consultant, it is your job to check if this project is a good investment in either the United States or Congo. Provide a report detailing your assertions and your reasoning. Assume that your audience is investors who have perhaps taken high school physics or college physics many years ago. You will be graded on both the correctness of your physics (and accounting) and your ability to explain the physics to such an audience.

Electron Orbit
Given that the angular momentum of the lowest energy orbit of an electron in orbit in a hydrogen atom is the reduced Planck constant, ħ = 1.056E-34 J*s, calculate the radius of its (assumed circular) orbit. Solve symbolically in terms of ħ and other known constants (mass, etc.). Plug in your numbers to obtain the radius.

Simple DC Circuit
Given a 9 V battery with internal resistance of 0.80 ohms hooked to a single resistor of variable resistance, calculate and graph the current, potential difference, and power for the variable resistor over the range 0 to 5 ohms. What is the resistance of maximum power? Prove your assertion based on your graph and calculus.

Potential and Field
Given the following graph of potential as a function of x, graph the electric field as a function of x. Provide an exact recipe (dimensions of objects, amount of charge, locations) in order to create such electric potentials and fields.

Electromagnetic Induction
A long wire on the x-axis carries 25 A of current to the right. A rectangular loop (0.70 m wide and 0.80 m height) of 36 gauge copper wire in the x-y plane begins with its center at a location of 0.45 meters directly above the wire. The loop moves in the positive x direction at a constant speed of 0.20 m/s for 5.0 seconds. It then moves in the positive y direction at a constant speed of 0.20 m/s for 5.0 seconds.

• Calculate and graph the magnetic flux through the loop as a function of vertical position of the center of the loop. The area vector should be assumed to be on a positive axis.

• Calculate and graph the magnetic flux through the loop as a function of time.

• Calculate and graph the induced emf in the loop as a function of time.

• Calculate and graph the induced current in the loop as a function of time. Use the convention that clockwise currents are negative and counter-clockwise currents are positive.

• Calculate and graph the electric power of the loop as a function of time.

• Calculate and graph the force on the loop as a function of time.

Hint: the magnetic field is not constant over the entire area of the loop. If you treat the magnetic field as fixed over the entire area, you will certainly lose a great deal of credit on this problem.

AC Circuit
Given figure P35.50 with the values as shown, (a) Calculate the resonant frequency in rad/s. (b) Calculate Z, I, V_R, V_c, V_L,  V_RL, and V_LC at this frequency. (c) Redo part b for double and half the resonant frequency.