Duke Physics Challenges

The following Physics Challenges are for Duke undergraduates and others to enjoy. The motivation for these problems comes from Mark Twain's famous quip that "I have never let my schooling interfere with my education." and similarly I think physics is too much fun to restrict just to classrooms, labs, and homework assignments.

If you would like to propose a solution to a Challenge, talk about one of the Challenges, or get a hint, please send me e-mail at hsg@phy.duke.edu or drop by my office in the Physics building, Room 097. Please also forward to me any neat problems that you find or invent that would be fun to present as a challenge.

References and links to other collections of physics problems and challenges can be found here.

Professor Henry Greenside
Physics  097     919-660-2548

Challenge 1    Heating of Two Identical Balls

You are given two identical steel balls of radius 5 cm. One ball is resting on a table, the other ball is hanging from a string. Both balls are heated (e.g., with a blow torch) until their radii have increased to the same value of 5.01 cm. Which ball absorbed more heat and why?

Challenge 2    Let Go or Hang On?

A painter is high up on a ladder, painting a house, when unfortunately the ladder starts to fall over from the vertical. Determine which is the less harmful action for the painter: to let go of the ladder right away and fall to the ground, or to hang on to the ladder all the way to the ground.

Challenge 3    How to knock a bottle over with a sandbag and drinking straw.

A heavy 300 kg sandbag one meter tall is hung from a playground swing with a rope 3 meters long so that the bottom of the sandbag just clears the ground. A bottle is then placed on the ground a meter away from the sandbag as shown.

Explain how to to knock the bottle over with the sandbag if you are given a paper drinking straw but are not allowed to touch anything (sandbag, rope, bottle, swing) with your body or with the straw.

Challenge 4    Running away from killer bees.

While walking through an open field on a windy day, you accidentally step on a nest of killer bees. In which direction should you run to save your life? Will you be able to run fast enough to escape?

For this Challenge, assume that the wind is blowing from the east at 4.5 meters/sec (10 miles/hour) and use the fact that bees have an experimentally measured maximum speed of about 8 meters/sec (18 miles/hour). The fastest runners can attain 10 meters/sec (23 miles/hour), most people much less than that.

Challenge 5    Equilibration of Two Birthday Balloons

Consider two identical spherical birthday balloons, one of which is inflated to 2/3 its maximum diameter and the other inflated to 1/3 its maximum diameter.  What happens when the openings of the two balloons are connected to each other by a straw so that air can flow back and forth between the two balloons?

Note: This experiment is simple enough that you should try it before making your mind up about what the "obvious" answer is.

Challenge 6    Temperature of a kilogram of ice and a kilogram of boiling water?

A kilogram of ice at 0oC and a kilogram (liter) of boiling water at 100oC are mixed together in a thermally insulated tank. What is the temperature of the water in the tank after the contents have reached equilibrium?

Challenge 7    Distinguishing three nearly identical spheres

You are given three spheres that are identical in size, weight, appearance, and touch but one sphere has a spherical hollow core, one sphere is completely solid, and one sphere is the same as the solid sphere except that a hollow spherical core was created and then filled with a liquid with the same mass density as the solid part of the solid sphere. (For example, the solid sphere could be made out of a light wood and the hollow sphere made out of a denser wood, then all three spheres are carefully painted to look and feel the same.) Using only simple inexpensive items that you might find at home (and I assume that your home is not a physics laboratory so you don't have access to something like an X-ray machine) and without damaging the spheres in any way (so no drilling), explain how to figure out which sphere is which.

Challenge 8    Why don't clouds fall like a rock to the ground?

As someone living near the beginning of the 21st century, can you explain a problem that badly perplexed the ancient Greeks and Romans (and also people throughout the medieval ages): how come clouds don't come crashing down to the ground? After all, clouds are made of water droplets and ice crystals which are about 800 times more dense than air, comparable in density to rocks. So why don't clouds fall like a rock to the ground? To give this problem focus, propose some specific experiments that you could carry out that would help you to discover the answer.

Challenge 9    A bird flying between colliding trains

Two trains each traveling at 30 km/hour are approaching each other on the same straight railroad track. When the trains are 30 km apart, a bird resting at the front of one train takes off and flies at a constant speed of 50 km/hour to the other train. As soon as it reaches the other train, it instantly turns around and flies back to the original train, and keeps repeating this back and forth at the same constant speed until the trains collide. How far will the bird have flown at the time of the collision?

Challenge 10    Which switch controls the desk lamp?

A light bulb in a desk lamp is turned on and off by exactly one of three simple switches which are located in a remote room such that one can not see the desk lamp from the location of the three switches. Explain how to determine which switch controls the desk lamp if you are allowed to flip the switches any number of times but are allowed to visit the room with the desk lamp only once. You can assume that the on and off positions of each switch are correctly labeled.

Challenge 11    Spherical Thinking

Assuming that the earth is a sphere, where on the earth's surface is it possible for a person to walk one kilometer south, one kilometer east, and one kilometer north and end up in the exact same place?

A hint: there is more than one such place.

Challenge 12    Atomic thickness of your signature

When you write your name on paper using a pencil, you create a thin layer of graphite. Invent and carry out an elementary experiment to estimate how many atoms thick is your signature.

Note: The graphite in a pencil is a pure form of carbon consisting of many planar sheets of carbon atoms stacked one above the other. The carbon atoms have strong bonds within a planar sheet and much weaker bonds between the sheets and so one sheet can slide rather easily with respect to an adjacent sheet, which explains why graphite is so useful as pencil lead. The spacing between sheets has been measured by X-ray crystallography to be 0.34 nanometers from which you can then determine from your experiment how many atomic sheets thick is your signature.

Challenge 13    Leaning Tower of Pizza

Assume that you have a large supply of identical strong square pizza boxes of dimension one inch deep by 18 inches wide. By stacking these pizza boxes on a sturdy table, one on top of the other and one box per layer, how far out into space can you extend this stack beyond the edge of a table?

Challenge 14    Wrongly rotating wagon wheels of a stagecoach

In watching a cowboy movie, you may have noticed that the wagon wheels of a stagecoach sometimes rotate the wrong way: the stagecoach may be moving left to right across the movie screen while the spokes of the wheel rotate backwards (counterclockwise). If a movie displays 24 frames per second, if the wagon wheels are 5 feet tall and have eight spokes each, and if the horses pull the stagecoach left to right at 20 miles/hour (32 km/hour), will the wagon spokes be rotating forward or backwards compared to the direction of the stagecoach? What will the movie audience perceive as the angular velocity of the wheels?

Challenge 15    Galactic Pinball

An indestructible sphere of mass 100 kg is launched by rocket into space. What will its speed be after a sufficiently long time?

Note: It may be useful for you to know that the mass of a star is of order 1030 kg and the relative speed of stars in a galaxy is of order 10 km/sec.

Challenge 16    A Universal Reflector

Consider three identical square planar mirrors that are glued together to form three adjacent sides of a cube meeting at a corner, with the mirrored sides all facing towards each other. Show that these mirrors act as a universal reflector that sends light back to its source: any light beam entering this arrangement of mirrors will leave parallel and opposite to its original direction.

If you look into such a corner reflector, what kind of image will you see of your face?

Note: Such corner reflectors were left on the surface of the moon by Apollo astronauts and were used in ranging experiments, in which laser beams from an observatory on Earth were bounced off the surface of the Moon and returned to the observatory, with the time of transit being measured. This enabled the distance from the Earth to the Moon to be measured with high accuracy which has been useful in testing the theory of general relativity and also for investigating the geology and origin of the Moon.

Challenge 17    Candy-Bar Powered Marathon Runner

A candy bar provides about 300 calories of energy. By thinking about the physics of running, estimate how many candy bars a person would have to eat to obtain enough energy to run a Boston Marathon of 26 miles and 385 yards (42.2 kilometers), if that person weighs 65 kg (143 lb) and is 1.7  meters tall (5 feet 7 inches).

Note 1: A food calorie is a so-called "large calorie", the amount of energy needed to raise one kilogram of water one degree Celsius at atmospheric pressure, and is equal to about 4.2 kilojoules.

Note 2: Estimating orders of magnitudes of phenomena is a fun and important skill and often provides surprisingly useful insights into some problems. A famous historical example was the order-of-magnitude estimate by Lord Rayleigh (1842-1919) of the lifetime of the sun if it obtained its heat from chemical means (e.g., burning coal). His estimated lifetime was orders of magnitude shorter than known geological and evolutionary times and so strongly suggested that the sun obtained its energy by some unknown non-chemical mechanism, which we now know to be nuclear fusion.

Challenge 18     Volume of a Holey Cube

Consider a solid silver cube whose side has length L=4 cm. If three holes of diameter D=3 cm are drilled completely through, and perpendicular to, the centers of all the faces of the cube, what is the volume V of the remaining metal in the cube?

Note: This is not strictly a physics problem but does require the kind of practical mathematical knowledge that an undergraduate science student should have.

Challenge 19    Resistance is Futile

Consider an electrical circuit consisting of a cube of 12 identical resistors such that each edge of the cube is a 1 ohm resistor, and each group of three resistors meeting at a vertex are soldered together. Calculate the resistances between nearest neighbor, second-nearest neighbor, and third-nearest neighbor pairs of vertices.

Challenge 20    Broken Symmetry Game

Consider a circular table (e.g., a bridge table) and a large supply of identical circular disks that are much smaller than the table (e.g., checker pieces). Now consider the following simple game: each of two players take turns choosing a disk and putting it down on the surface of the table so that the disk lies flat and no disk rests on top of another disk.

If the first person who is unable to put a disk down loses (because of lack of space), should you go first or second to win this game?

Challenge 21    Deducing the size of the Earth from a lovely sunset

You are enjoying a Caribbean vacation and happen to have a stopwatch with you at the beach. As you watch the sun set over the ocean, you carry out the following eccentric sequence of events: (1), you lie down on your stomach in the sand and wait until the top of the sun just disappears below the horizon; (2), you then quickly stand up and simultaneously start your stopwatch. By standing up, a bit of the sun is now visible again and (3), you wait until the top of the sun again dips below the horizon, at which point you stop the stopwatch. Knowing this elapsed time, your height, and that a day lasts 24 hours, explain how you can deduce the radius of the Earth. (And next time you find yourself watching a sunset at the beach, give this a try and compare your answer with the known value of 6400 km.)

Challenge 22    Can you trust your heart?

According to Daniel Boorstein in his interesting book "The Discoverers" (Random House, 1983), Galileo was nineteen years old in 1583 when he made the apparent discovery that the period T of a pendulum seemed to be independent of the amplitude A of its swing (measured in radians, with zero radians corresponding to the pendulum being directly underneath its support). He was supposedly attending prayers in the baptistery of the Cathedral of Pisa and was distracted by the swinging of an altar lamp, whose period did not seem to change as its amplitude slowly diminished.

In fact, as you hopefully know, the period of a pendulum does depend on the amplitude of the swing, becoming longer as the amplitude becomes larger. So here is an interesting historical question: could Galileo have discovered this while in the baptistery? (He did discover this later on in his life.) The only clock he would have had available in the baptistery would have been his heartbeat. Since this is an unreliable clock (one's heartbeat can speed up or slow down), this raises an interesting physics question: given an unreliable clock and some knowledge of what makes it unreliable, how accurately can one measure a time interval or difference in time intervals?

Try to do some history of science and determine whether Galileo could have detected the nonlinear dependence of period on amplitude by just using his heartbeat as a clock. Let's guess that the length L of the lamp's support was L=10 meters and that the amplitude of motion was moderate, say A=20 degrees from the vertical.

1. Using the approximate formula "T = 2 Pi Sqrt[L/g] ( 1 + (1/16) A2 + ... )" for the period T of a pendulum of amplitude A measured in radians (with L its length and g=10 meter2/sec the acceleration of gravity), estimate in seconds the difference in period between an amplitude of 20 degrees and zero degrees. Galileo would have had to detect this small difference in time to realize that the period depended on amplitude.

Note: This formula is derived in many textbooks and is surprisingly accurate, even for amplitudes as large as 45 degrees (see "Mechanics, 3rd Ed." by L. D. Landau and E. M. Lifshitz (Pergamon Press,1976), Section 11.)

2. Assume that Galileo's heart beat on average 60 beats per minute but could beat as fast as 70 beats per minute or as slow as 50 beats a minute at any given time. By counting heart beats, could Galileo have detected the small difference in period between a nearly vertical lamp (A=0) and a swinging lamp with A=20 degrees?

Challenge 23    North, South, East, or West, and When and Where?

The above picture shows the Sun over eleven successive hours as photographed from a certain fixed location. What time was it when the Sun reached its minimum height above the horizon? In what compass direction (north, south, east, or west) was the camera facing when the Sun reached its minimum height? Is the Sun moving right to left or left or right? Was it spring, summer, fall or winter when this picture was taken? And where on the surface of the Earth could this picture have been taken?

Challenge 24    How sensitive is the human eye?

Some books say that the human eye is so sensitive that it can perceive the light of a match two miles (3.2 km) away on a dark night. Other books say that the human eye is so sensitive that it can detect as few as five photons (quantized light particles). By using an appropriate order of magnitude estimate, determine whether these two statements are consistent with each other.

Challenge 25    Law of reflection for a moving mirror.

An elementary fact that people learn about mirrors is the law of reflection, that the angle of incidence of a light beam striking the mirror (as measured with respect to a normal) equals the angle of reflection.

Does this law also hold for a mirror that is moving? Consider a square mirror that is moving at speed v in a direction perpendicular to the mirror. (You can think of the mirror as starting in the xy-plane and moving in the positive z direction of a Cartesian coordinate system.) As the mirror approaches a certain observation point, a friend shines a laser beam of frequency w at the mirror so that the beam makes an angle A with the normal to the mirror. (You can think of the beam as lying in the yz-plane.)

What angle and frequency will you measure for the reflected light beam? Does the law of reflection still hold?

Do your conclusions change if the mirror moves parallel, rather than perpendicular, to its plane (say in the y direction if it starts in the xy-plane)?

Note: The large mirror of the Hubble space telescope is an example of a mirror in motion as it orbits the earth. From your analysis, do you think the users of the Hubble have to take into account the motion of the mirror when measuring properties of its images?

Challenge 26    First-order single-variable dynamics are asymptotically boring

Consider some quantity y(t) that varies with time t, e.g., the pressure, temperature, mass, voltage, or chemical concentration of some system. Show that if this quantity evolves according to a first-order ordinary differential equation of the form dy/dt=f(y) with f(y) some differentiable function, then the asymptotic (nontransient) dynamics of y(t) are boring: y(t) either diverges to infinity (which is unphysical) or y(t) approaches a constant time-independent behavior. In particular, no matter how complicated the function f(y), the asymptotic behavior can never be oscillatory.

Note 1: In thinking about this problem, try to use a qualitative approach based on the possible signs (negative, zero, or positive) of the function f(y) rather than on any detailed properties of this function.

Note 2: This elementary and neat result suggests that one needs at least two coupled variables or a higher-order time derivative to get nontransient non-constant behavior, e.g., sustained oscillations. A famous and rather difficult theorem from the turn of the century, the Poincare-Bendixson theorem, generalizes your single-variable analysis to two coupled first-order equations with arbitrary smooth functions of two variables: the only nontransient bounded behavior is either constant or periodic. For three or more coupled first-order equations, new kinds of nontransient dynamics can occur such as quasiperiodic behavior (multiple oscillations present with frequencies whose ratios are irrational numbers) or chaos which is nontransient bounded dynamics that is neither periodic nor quasiperiodic.

Challenge 27    Ultimately wrong theory of everything

A scientist Dr. X of Country Y excitedly holds a news conference and says: "I have finally succeeded in finding the ultimate complete theory of the universe which will allow the detailed explanation and prediction of all phenomena from elementary particles to condensed matter to galactic structure to black holes and beyond. It consists of the following 23 coupled partial differential equations involving space and time derivatives." At this point Dr. X holds up a 2 meter by 4 meter panel covered with the exceedingly complex 23 equations so that the world television audience can be properly intimidated.

Why is this claim obviously wrong?

Challenge 28    Vector or not a vector?

Consider a smooth vector function ${\bf F} = \bigl( F_x(x,y,z), \, F_y(x,y,z), \, F_z(x,y,z) \bigr) ,$ with three components. If $$\partial_x$$, $$\partial_y$$, and $$\partial_z$$, denote partial differentiation with respect to $$x$$, $$y$$, and $$z$$ respectively, is the triplet $\bigl( \partial_x F_x , \, \partial_y F_y , \, \partial_z F_z \bigr)$ a vector quantity?

Challenge 29    Fission of charged rain drops

Small water drops can sometimes merge into a single bigger drop (one sees this when rain falls on the windshield of a car). Fortunately, this mechanism seems to have some upper limit else we might get hit by rain drops a meter or more in size which would be most painful (and probably fatal for small creatures). This raises an interesting physics question: what determines the size of rain drops in a storm?

Using an elementary knowledge of electrostatics, you can determine whether electric charge might play a role in determining the characteristic size of a rain drop. (That electrical charges might play a role is suggested by the occurrence of lightning in many storms.) Consider a spherical water drop of radius R carrying an electrical charge Q that is uniformly distributed over the surface of the drop. If this drop splits into two smaller spherical drops of equal size with each drop having charge Q/2 uniformly distributed over their surfaces, show that electrostatics favors such a splitting by calculating the decrease in electrostatic energy caused by fission.

The total surface area of the two equal smaller drops turns out to be larger than the surface area of the original drop and so, because of surface tension which holds a drop together, it costs energy to split the original drop. If S denotes the surface tension (which has units of energy per unit area or Newton/meter), how much energy is needed to split a drop into two equal smaller drops?

Now put your two observations together. For a water rain drop of size R=1 mm and approximate surface tension of S=0.07 Newton/meter at room temperature, how many electrons N would have to be deposited on the drop in order for the electrostatic energy gained by fission to offset the energy lost by creating more surface area? Do you think this would be a reasonable amount of charge to accumulate by friction as one rain drop bumps against another in a rain storm? Is the resulting electric field at the surface of the drop large in the sense of being close to the value 30,000 volts/meter at which air breaks down?

Note: Find a copy of the Guiness Book of World Records and look up the weight of the heaviest hailstone ever found (it was heavy enough to easily kill an elephant!). Evidently powerful convection currents in storms can suspend large weights so that the distribution of rain drop sizes is more likely determined by stability arguments of the above sort rather than by the largest mass that can be supported in a storm.

Challenge 30    Time for a vertical pencil to fall over

Estimate the time for a motionless vertical pencil to fall over (1), because of quantum mechanics and (2), because of thermal fluctuations.

Challenge 31    Length of a helical string

Consider a cylindrical rod of length 12 cm and circumference 4 cm. Starting at one end of the rod and ending up at the other end, a string is wound evenly and exactly four times around the cylinder. What is the length of the string?

Note: With an appropriate insight, only elementary high school mathematics is needed to solve this (no calculus, no differential geometry).

Challenge 32    Time for a marble to roll down and up a kitchen bowl.

Consider a hemispherical kitchen bowl of radius R. If a marble is released with zero velocity at one edge of the bowl (a distance R above the kitchen table), how long will it take for the marble to roll down and then up to the opposite side of the bowl? For simplicity, assume that the marble rolls without slipping.

Challenge 33   Critical angle of rolling for two adjacent cylinders on a tilted board.

Consider two cylinders of equal radius and uniform mass density that are placed on a board so that they are touching each other and such that their axes are parallel to the bottom of the board:

Now for a single cylinder, as soon as the board is tilted up from the horizontal, the cylinder will start to roll. But for two cylinders, the tendency for the bottom cylinder to roll is opposed by a friction force arising from the contact with the upper cylinder. Calculate the critical angle A above the horizontal at which the two cylinders will start to roll down the incline.

Note 1: Assume that the complex friction forces can be modeled by the usual simplified rules given in an introductory physics course. If f and N denote the friction and normal forces respectively at a contact and if µ denotes the coefficient of friction, then

0   <=   f   <=   µ N,

for bodies not in relative motion and

f   =   µN,

for bodies in relative motion. For this simple model, sliding at a contact begins when f first attains its maximum value of µN. For simplicity, you can also assume that the friction forces of the cylinders with the board and with each other are all described by the same friction coefficient.

Note 2: This kind of problem arises when trying to understand the dynamics of granular flow, e.g., how grain flows down a pipe in a silo or sand in an hourglass, or why sand forms a conical heap with a characteristic angle of repose. Over the last ten years, granular flow has become a hot topic in the physics community and many extraordinary discoveries have been made as researchers started carrying out careful experiments.

Note 3: As a harder challenge, see if you can work out the critical angle for rolling if a third cylinder is stacked on top of the first two cylinders.

Challenge 34    Crazy series circuit

Consider the following series circuit:

consisting of two 15 watt light bulbs L1 and L2 and two knife switches K1 and K2 connected to black boxes that themselves are wired in series by simple single-strand copper wires. The entire circuit is then connected to an AC source as shown, e.g., the usual 120-volt, 60-cycle American voltage source.

Explain how to connect at most a few passive electrical components (e.g., capacitors, diodes, inductors, or resistors) in each black box so that the following is achieved:

1. If both knife switches are closed, then both lights are on.
2. If both knife switches are open, then both lights are off.
3. If knife switch K1 is closed and switch K2 is open, then light L1 goes on but light L2 goes off (this is crazy). This is the situation shown in the figure.
4. If knife switch K2 is closed and switch K1 is open, then light L2 goes on but light L1 goes off (this is also crazy).

Note: This circuit makes a great demo for people just learning physics or electronics. The parts in the black boxes are sufficiently few and small that they can easily be concealed inside the bases of the knife switches and of the light bulbs, leading to a truly paradoxical circuit for the uninitiated.

Challenge 35    Formula for the derivative of a determinant

Science students need to have a solid understanding of linear algebra so here is a Challenge to test that solid understanding: Consider a N×N matrix M(x) whose matrix elements Mij are differentiable functions of some variable x and that is nonsingular for all values x of interest. If a prime ' denotes differentiation with respect to x, derive the following formula:

det( M )'   =   det( M )  Tr( M-1 M' ) .

In words, this formula says that the derivative of the determinant of the matrix M is equal to the determinant of M times the trace of the matrix given by multiplying the inverse of M (the matrix M-1) with the derivative of M (the matrix M').

Challenge 36    Does a pendulum violate conservation of momentum and angular momentum?

Consider a pendulum consisting of a heavy mass attached to a thin rigid metal rod (of negligible weight compared to the mass). The top end of the rod is attached to some pivot so that the pendulum can swing freely back and forth from left to right. This familiar innocent pendulum seems to have the alarming property of violating the fundamental conservation laws of momentum and angular momentum! For the momentum of the pendulum oscillates in time (is not conserved), going from zero (when the mass is at its highest say on the left), increasing to a large positive value as the pendulum swings tot the right, decreasing to zero as the pendulum reaches its maximum height on the right, then changing sign and becoming negative as the pendulum swings to the left. Similarly, the angular momentum of the mass with respect to the pivot also oscillates in time. Explain this paradox: how is it possible that these fundamental conservation laws are violated?

Challenge 37    The Human Basilisk

The basilisk is a small iguana-like reptile that has the remarkable ability to run over water (e.g., when threatened):

You know from your own experience that you can not run over water with your bare feet. By making appropriate order-of-magnitude estimates (how much power a person can produce, how fast one can pump one's legs up and down, how much force you apply when you slap the water with your feet), estimate whether a person could run on top of water if large oval flat pads of some area A were attached to the soles of each foot. (These pads are not balloons that provide buoyancy, just a way to spread the impact over a larger area.)

Challenge 38    A simple weight-loss program: visit Ecuador

Assuming that the earth is spherical with radius R = 6400 km and taking into account that it rotates once per day, calculate how much less you weigh at the earth's equator than if you were standing at the north or south pole.

Note 1: Your analysis partially explains why countries place their rocket launching pads as close to the equator as possible: the rockets weigh less and so it costs less to launch them. A more complete analysis requires taking into account that the earth is not a sphere but an oblate spheroid, bulging a bit at the equator and being flattened at the poles. This means that someone on the equator is further from the center of the earth than at the poles and so weighs a bit less even in the absence of rotation. This makes the equator even more favorable for launching rockets.

Note 2: Assuming a spherical rigid earth, see if you can work out the more general case of the effect of the earth's rotation on gravity: if your latitude is T degrees (measured from the equator), how much less do you weigh than if the earth were not rotating? By what angle would a hanging plumb bob deviate from the normal to the surface? (It is the fact that gravity no longer points along the normal that distorts a sphere into an oblate spheroid.)

Challenge 39    A Pre-20th Century Derivation of the Radius of a Black Hole

By the time one has finished a high school or first-year college course on physics, one has learned about Newton's laws of motion, about gravitational attraction, that the kinetic energy of a particle of mass $$m$$ and speed $$v$$ is given by $$(1/2)mv^2$$, and that light travels in vacuum with speed $$c$$. Use this pre-20th century knowledge to show that if a sphere of mass $$M$$ has a radius $$R$$ given by $R = {2 M G \over c^2 }$

where $$G$$ is the gravitational constant, then a particle on the surface of this sphere can never attain escape velocity since it speed would have to exceed $$c$$, the maximum speed possible. The sphere is then effectively an invisible "black hole", since even light can not escape from its surface.

Look up the appropriate masses in an astronomy textbook and calculate this radius for the sun and for the earth and compare with their actual radii. Also compare the density (kg per cubic meter) for a solar black hole and compare with the density of water, 1000 kg/m3.

Note 1: A particle on the surface of some object is said to attain "escape velocity" if its kinetic energy exceeds the gravitational potential energy gained by bringing the particle from infinity to the surface of the attracting object.

Note 2: This formula for the radius (known as the Schwarzschild radius) is "correct" in the sense that an identical formula is found when using Einstein's general theory of relativity. However, there are considerable subtleties about the meaning of space, time, and a radius in the vicinity of the enormous gravitational field of a black hole and so it is somewhat of a coincidence that the non-relativistic formula for kinetic energy and a Newtonian treatment of gravity yield this particular formula.

Challenge 40    How does a battery lose its power?

Flashlights and radios depend on familiar D, C, and AA-type batteries. If these batteries are in constant use (e.g., a flashlight is left on), investigate and plot how the voltage and current vary with time t. Does the battery maintain a constant voltage until close to the end of its life? Do you get different answers for different kinds of batteries with the same voltage, say alkaline and carbon?

Challenge 41    A relativity paradox: flashlights pointing in opposite directions?

An astronaut in space turns on two flashlights simultaneously and pointing in opposite directions, sending out two beams that both move away from the astronaut at the speed of light. How fast does the front of one beam move away from the front of the other beam? Is this a contradiction of Einstein's theory of special relativity?

Challenge 42    How does a sailboat move upwind?

A sailboat has the somewhat paradoxical ability to move upwind, i.e., to move towards the source of the wind that is blowing on its sail. Explain how this is possible.

As a more quantitative challenge, determine how to choose the angle u of the sail with respect to the wind and the angle v of the boat with respect to the sail so as to move upwind as fast as possible.

Comment: If you make some idealizations, e.g., that the boat has a keel that keeps the boat moving in a fixed direction along the axis of the boat, that the sail is a vertical planar sheet, that the pressure on the sail is sin(u) times the maximum wind pressure, and that the speed of the boat is proportional to the wind pressure, you should be able to show that the maximum speed upwind is 1/8 the maximum speed downwind, and that this maximum upwind speed is achieved when the sail is turned to an angle of u=30 degrees with respect to the wind and when the boat is turned to an angle of v=30 degrees with respect to the sail.

Challenge 43    Exploring the solar system with a light sail.

Visionaries have proposed exploring the solar system with "light sails", small vessels attached to huge thin flat reflecting sheets that would be pushed around by the pressure of sun light.

Work out some of the details of a light sail, which involves understanding the balance of gravitational and light pressure forces. A simple way to think of a light sail is a huge flat sheet of area A and mass M which can be steered by pointing its normal vector in various directions. How would you choose the orientation of the light sail to move in a particular direction? If the total light intensity on the sail is I (in kilowatts/meter2), how fast could you accelerate in some given direction? Since light sails don't have keels like water-based sailboats (which keeps them moving in a straight line), is it possible to sail upwind towards the sun?

To get a sense of the orders of magnitude involved, assume that you have a solar sail that is perfectly reflective and perfectly flat, that you want to accelerate a payload of 50 kg (representing say a few friends, food, a few suitcases, and the weight of the solar sail itself), and that the solar light intensity at Earth's distance from the sun is 1.4 kilowatt/meter2. How large would the area of the sail have to be if you wanted to sail from from the Earth to the Moon in a week's time? From the Earth to Mars in a year's time?

Challenge 44    Three problems about the single-particle one-dimensional time-independent Schrodinger equation.

When learning quantum mechanics, one first learns about the single-particle one-dimensional time-independent Schrodinger equation for a particle of mass m in a potential V(x). Determine whether the following three statements are true or false about the wave functions satisfying this equation:

1. Unbound (scattering) states always lie above bound states, i.e., the lower bound of energies for unbound states is always greater than the upper bound of energies for bound states.

2. For bound states, the variance <(x - <x>)2 > of the particle's position always increases as its energy E increases (becomes less negative).

3. For any two bound states with energies E1 and E2, their separation |E1-E2| decreases as the particle mass m increases.

More generally: the energy of a photon needed to excite a molecule from one bound state to another decreases as the nuclear masses of the molecule are increased (leaving the potential unchanged, e.g., by replacing atoms by heavier isotopes).

Challenge 45    Transmission of Light Through Three Polarized Sunglasses

Many sunglasses have the property of being "polaroid" which means that they filter the light in a special way (they produce what is called "linearly polarized light"). Now a single pair of polaroid sunglasses is a strong filter and substantially reduces the intensity of light passing through it. If you look through two pairs of polaroid sunglasses simultaneously, you get a further reduction in light intensity but something funny happens: if you rotate the second pair of sunglasses while holding the first fixed, you can reduce the amount of light that gets through almost to nothing; one sees nearly total darkness.

So here is something strange to think about: if you hold two pairs of sunglasses so that almost no light gets through, and then you insert a third pair of polaroid sunglasses between the first two pairs, you will now find that, for certain orientations of the middle (third) pair of sunglasses, light gets through again. Explain this: how is it possible that one can add a filter that, by itself, reduces light intensity but here increases the light intensity? Can you identify and explain the angle at which you should hold the middle pair of sunglasses to maximize the amount of light that gets through?

Challenge 46    Laws of Friction for Boats

Introductory physics courses often talk about some simple empirical "laws" of friction for one solid surface rubbing against another solid surface. One surprising law is that the friction force is independent of the surface area of contact. Another law is that the friction force is proportional to the normal force that presses one surface against the other. (See Challenge 33, "Critical angle of rolling for two adjacent cylinders on a ramp", which gives you a chance to test your understanding of these friction laws.)

So here is an experimental Challenge for the next time you take a bath in a big bathtub: what are the "laws of friction" for a solid surface rubbing against a liquid surface, e.g., a flat-bottom boat pulled along on the surface of a deep body of water? More specifically:

• Is the "friction force" needed to pull a boat through the water at constant speed proportional to the area of the boat's bottom?
• Is the friction force for a boat proportional to the normal force, i.e., to the weight of the boat? You can test this by taking a toy flat-bottom boat and loading it with varying amounts of nails.
• Does the friction force needed to pull a boat through the water depend on the shape of the boat's bottom? If so what shape will minimize the force for a given area? You could explore this by cutting out various simple shapes of equal area out of water-proof cardboard.
• Does the friction force depend on the speed of the boat?
For the purposes of this Challenge, qualitative answers are fine although a quantitative study would certainly be more interesting and satisfying.

A historical comment: at my undergraduate University (Harvard), there was a spring-time "Adam's House Raft Race" in which students were invited to create their own rafts and race against each other in Boston's Charles River. This race stimulated many theoretical and experimental questions about what kind of raft would be the fastest for 2-4 students who would sit in or on the raft and paddle like crazy. By the way, some students went for creativity rather than speed, e.g. one raft was a two-level bunk bed on foam pillows.

Challenge 47    Two-Slit Interference Pattern With Polarized Light

Consider the following variation of the two-slit interference experiment that is often discussed in introductory physics courses to illustrate the fact that light and sound are wave-like phenomena. Take a monochromatic but unpolarized beam of light (e.g., using sunlight, a prism and some lenses) and focus the beam on an opaque wall with two vertical slits. Also assume that the widths of the slits and the distance between the slits are chosen so that one gets a nice interference pattern of light intensity on some screen beyond the wall. (Basically, the slits and their separation should be comparable to the wavelength of the monochromatic light).

Now consider putting in front of each slit a high quality linear polarizing filter than can each be rotated around an axis perpendicular to the direction of light. Explain what happens to the interference pattern on the screen as one linear polarizer is rotated through an angle of 360 degrees while the other polarizer is fixed in its orientation. In other words, what is the effect of polarization on the interference pattern of the beams?

Challenge 48    Ant On a Loudspeaker

A loudspeaker is placed on its back so that the speaker cone is open to the air and faces up towards the ceiling. As a small ant starts to walk across the speaker cone, the cone is set into vibration up and down by playing a pure sinusoidal tone of amplitude A and of frequency f. Assuming that the ant rests on the surface of the speaker without any adhesion (ignore its sticky feet!), determine the values of A and f for which the ant will be thrown clear of the speaker, i.e., for which the ant will no longer be in physical contact with the speaker cone.

Comment: Scientists have recently explored what happens when a shallow layer of tiny brass balls is put in a cup and this cup is shaken up and down sinusoidally, just like the poor ant above. Much to everyone's surprise, the scientists found a remarkable richness of spatial patterns and dynamics as the amplitude and frequency of the shaking were varied. You can see some pictures and learn more from the web page http://chaos.ph.utexas.edu/research/granular/granular.html . Your analysis of the ant on the speaker cone is actually a valuable first step towards understanding the origin of the patterns in the thin layer of brass balls.

Challenge 49    Is X-Ray Vision Possible?

Animals, insects, and reptiles can see light with wavelengths varying from 10,000 nm in the infrared to about 100 nm in the ultraviolet. (Humans see from deep red at 700 nm to violet at 400 nm.) Given that vision depends on the absorption of light by some molecule, is it possible in principle for a biological organism to develop X-ray vision, i.e., the ability to detect light with wavelengths of 10 nm or less?

Challenge 50    The Nano-nut-Dropping Nano-Squirrel

A nano-squirrel on a nano-tree is trying to drop nano-nuts into a small nano-hole directly under itself. If the nano-squirrel is at height $$H$$ above the ground, if the gravitational acceleration is $$g$$, if each nano-nut is a point particle of mass $$m$$, and each nano-nut is released with nearly zero initial velocity using the best reproducible nanotechnology, use the position-momentum uncertainty principle to show that, despite the best efforts of the nano-squirrel, the nano-nuts will end up spread over the ground in a region of at least radius $$R$$ satisfying the inequality $R \ge \left[ { 8 \hbar^2 \over g} { H \over m^2 } \right]^{1/4} ,$ where $$\hbar$$ is Planck's reduced constant.

Challenge 51    How Quickly Does a Scent Travel By Collisions Only?

Consider a straight glass tube of length L=2 cm connected to a closed vial of a strong perfume at one end and to your nose at the other end. Assuming that you can detect a scent the first time at least 10 perfume molecules enter your nose simultaneously, estimate how long it will take for you to smell the perfume after the vial is opened.

Some comments: The purpose of the glass tube is to eliminate possible air currents so that the perfume spreads from the vial to your nose only by molecular collisions. You can assume that the vial is connected to the tube by a sealed joint so that the tube is the only way that the perfume can reach your nose. Also assume that you can hold your breath long enough to carry out this experiment, so that you don't have to take into account the strong wind currents associated with your breathing. Finally, don't be too quick to use a diffusion equation (if you know what that is) since this problem concerns specifically a regime in which a continuum description of the perfume concentration is likely not to be a good approximation.

Challenge 52    How high can a tree grow?

The Sequoia pines in California are among the tallest trees in the world and can attain a height of 110 meters. Is this the largest possible height for a tree on earth? This is not just a biological or evolutionary question but involves some interesting physics, much of which is accessible at the level of an introductory undergraduate course in thermodynamics and statistical mechanics.

One constraint on the height of a tree is simply the material strength of the wood, or of the root system that supports the tree in a vertical position. A tree that is too tall may break under its own weight or become unstable to falling over because of strong winds. The latter seems to be more likely from my own observations of when Hurricane Fran swept through Durham, North Carolina, in 1996, knocking over many big trees without breaking them. Evidently the roots were not strong enough to withstand the large forces and either broke or were pulled out of the ground which had been softened by the rain. Can you suggest experiments, say on beams of wood bought at a lumber store or on small trees, that could help to determine the mechanical limits of a tall tree? Is the height of a Sequoia determined by the strength of its wood or of its roots?

When one looks more into the biology of trees, another constraint arises that has nothing to do with mechanical strength, which is the issue of water transport. Trees get water primarily from their roots which implies that trees have to raise the water from their roots through a gravitational potential Mgh to provide water to leaves at the top of the tree. (Here M is the mass of a water molecule in kilograms, g=9.8 m/sec2 is the acceleration of gravity, and h is the height of the tree in meters.) How can a Sequoia lift water so high?

• One possibility is to think of the tree as a tall straw that sucks the water from its roots up to its highest branches. Assuming that the tree can produce a perfect vacuum in the vicinity of its leaves (don't ask me how), determine the maximum height in meters for a tree allowed by suction. You can assume that the surrounding pressure is standard atmospheric pressure, P=105 N/m2.

• Your answer should be nonsensical, trees would be limited to unreasonably small heights if they transported water by suction. An alternative possibility comes from knowing that tree leaves "transpire", i.e., give off water moisture through their leaves. As water evaporates from the leaves, it effectively pulls up a little bit an entire chain of water leading from the roots to the leaves, as other water molecules move into the place of the recently evaporated ones. More carefully, the difference in concentration between the saturated water vapor just at the surface of the leaf and the less saturated water vapor in the atmosphere beyond the leaf creates a difference in chemical potential that could provide the energy to lift a water molecule through the height of a tree. Given this, calculate the height h of the largest possible tree and determine whether trees are limited in height by this mechanism of water transport.

Some suggestions: First, let's assume that the entire tree and its surrounding are in thermal equilibrium at room temperature (T=293oK) so we can try to use ideas from thermodynamics. The energy Mgh needed to lift a water molecule through the height of the tree could then be obtained by the change in chemical potential when a water molecule in the liquid phase inside the leaf evaporates to become part of the water vapor away from the leaf. Thermodynamics books show that the change in chemical potential for a gaseous molecule that moves from a region of one concentration to a region of a different concentration is given by -kTln(r) where r is the ratio of the concentrations and where k is the Boltzmann constant. A plausible value for the ratio r might be the relative humidity, which is found empirically to be about 90% in the vicinity of tree leaves.

Note that a mechanism based on the change in chemical potential leads to the testable prediction that trees can not transport water if the surrounding air is completely saturated with water vapor.

• Breaking a chemical bond can release of order one tenth to one electron volt of energy (which is also the range of energy available by the absorption of a single photon for photosynthesis). About how many chemical bonds would have to be broken in a biochemical reaction to raise a single water molecule to the top of a Sequoia tree?

Note: Using energy released by chemical reactions for biological transport is called "active transport", as opposed to "passive transport" mechanisms like suction or transpiration which involve no chemical reactions. Is active transport a practical possibility for explaining water transport? It may help to appreciate that a 15-meter tall maple tree with about 180,000 leaves and total leaf area of about 700 meter2 evaporates of order 200 liters of water per hour on average, from which you can estimate a lower bound on the daily energy needed for active water transport.

• Another possibility for raising water is by capillary action, the tendency of water in a thin vertical tube to climb up along the walls because of surface tension and wetting. By balancing the force of surface tension pulling water up along the walls of a tube against the force of gravity pulling the mass of a cylindrical water column down, one can derive the following relation for the height h to which water will climb up a tube from its base:

h = 2 S cos(alpha) / (r rho g) ,

where h is the height in meters, S is the surface tension (about 0.07 Newton/meter at room temperature), alpha is the "contact angle" in radians which the water makes with the tube at the edge of the miniscus, r is the radius of the column in meters, rho is the density of water (about 1000 kg/m3 at room temperature), and g is the gravitational acceleration (9.8 m/sec2). For glass capillaries and in small plant channels, the contact angle alpha is about 0 so cos(alpha) can be approximated by 1.

From this relation and these data, estimate the radius of an internal tube in the tree that could raise water to a height of 100 meters. Is this radius biologically plausible?

Challenge 53    The Great Snowplow Chase

On a certain winter day, snow starts to fall at a heavy and steady rate. Three identical snowplows start plowing the same road, the first leaving at 12 noon, the second leaving at 1 pm, and the third leaving at 2 pm. At some time later, they all collide. At what time did the snow start to fall?

Note: Assume that the speed of a snowplow is inversely proportional to the depth of the snow.

Challenge 54    A telescope made from a rotating mercury mirror.

Before recent breakthroughs in telescope design (which allow images from many small reflecting mirrors to be combined by computer into an image corresponding to a single effective mirror with total area equal to that of the smaller mirrors), the largest possible ground-based reflecting telescope was limited by how large a single plate of glass could be made and ground into a mirror. However, scientists had thought of and tried an ingenious alternative, which was to fill a large cylindrical tank with liquid mercury and rotate the tank around its axis. The shiny surface of the mercury then deforms into a shape that can be used as a large and inexpensive telescope mirror.

Work out the design of such a rotating mercury mirror. First show that the surface of the mercury will take on the shape of a paraboloid of revolution. Next, determine the focal length of this paraboloid as a function of the angular rotation frequency w of the tank. Finally, determine the angular frequency needed to achieve a focal length of one meter.

Is it possible to avoid the drawback that such a mercury-based mirror will always point directly overhead?

Challenge 55    Sinking Submarines Versus Floating Balloons

Explain why an inflated balloon (made of a rigid plastic material) will rise to a definite height once it starts to rise, while a submarine will always sink to the bottom of the ocean once it starts to sink.

Challenge 56    The optimal shape for a snowman to reduce melting.

A snowman is traditionally made of three balls of snow stacked one above the other. Explain why a sphere is also the optimal shape for a snowman body part in the sense that, for a given volume of snow, it will melt the least rapidly as the weather becomes warm. For example, a snowman made of cubes, boxes, cylinders, triangular prisms, pyramids, or tetrahedra will melt more rapidly than a traditional snowman.

For simplicity, assume that the air is uniformly the same warm temperature, ignore the effects of wind and sunlight, and ignore the fact that the shape will eventually sag because of melting and gravity.

Challenge 57    Average distance between two random points in a sphere.

Show that the average distance between two points that are chosen randomly and uniformly in a sphere of radius 1 is 36/35, about 1.029.

Most computer languages have a function rand() for generating random numbers uniformly in the interval [0,1]. Can you figure out how to use such a generator to create randomly and uniformly distributed points in a sphere? If so, write a short computer program and confirm your analytical result by generating many random uniform pairs of points inside the sphere and by averaging the distances between the pairs of points.

Challenge 58    The temperature of a hot spot made from a magnifying glass.

You have presumably had the fun of focusing sunlight with a magnifying glass to burn a hole in a piece of paper. Now think about this more deeply from a physics point of view: given any arrangement of lenses and reflectors of any arbitrary size and shape, how hot can you make a single spot by focusing light from the Sun on that spot? In particular, could you make a spot of focused sunlight hotter than the surface temperature of the Sun, which is about 6000 K?

This problem has an interesting historical precedent. Archimedes supposedly recommended that Greek warriors try to set fire to Roman ships by focusing sunlight with their shields onto the wood ships. Assuming that the shields were flat and that about 50% of the light is reflected off the shields, estimate how many Greek soldiers would be needed to focus the sunlight and set a Roman ship on fire. A useful piece of data is that a lens of diameter 3 cm and focal length 10 cm is capable of burning wood with sunlight.

Challenge 59    An inverse rocket and an inverse sprinkler.

1. Consider a cylindrical can of highly compressed gas in outer space. You know that if you puncture a hole in the can so that the gas can leak out, the can will start moving like a rocket, in a direction opposite to that of the leaking gas.

Now consider an "inverse" problem in which a cylindrical can is completely empty (has a vacuum) and is inserted into a big tub of water. Also imagine the experiment being done on the space shuttle so that there is no buoyancy force that would push the can to the surface of the tub. The can is now punctured at one end so that a jet of water starts to stream into the can. In what direction will the can move and why?

2. A similar problem: Consider a lawn sprinkler consisting of three arms that rotate in a circle as water sprays out. But now consider putting the sprinkler at the bottom of a big tank of water and using a pump to suck water out of the sprinker (so that water now flows into the arms of the sprinkler and out through the hose). In what direction will the arms turn and why?

Challenge 60    Bank shots on an elliptical billiard table.

Consider two point balls B1 and B2 placed on a mathematical billiard table whose shape is that of an ellipse, rather than the traditional rectangle. In what direction should one shoot ball B1 so that it bounces once off the ellipical side wall and hits ball B2? For this problem, ignore the spinning of the billiard ball.

In case you haven't played billiards before, you should know that a ball bounces off a wall according to the law of reflection, i.e., the angle of incidence equals the angle of reflection as measured with respect to a line normal to the tangent at the point on the wall where the ball bounces. For a rectangular table, the strategy would be this: drop a perpendicular from ball B1 to a side of the table and then extend the perpendicular an equal distance beyond the table to obtain point P1. Draw the line between point P1 and ball B2 and identify the point P2 where this line intersects the side of the table. You then want to point your cue stick at point P2 to hit a bank shot that will connect with ball B2.

Challenge 61    Size of smallest asteroid that a person could jump off of.

In the not so far future, it may be possible to land an astronaut on an asteroid. Based on how high you can jump on earth, determine the maximize size of a spherical asteroid that you could jump completely off of. The typical density of a rocky asteroid is about 3000 kg/m3.

Challenge 62    Why does a rotating positronium atom live longer?

There exists in nature a positively-charged particle called the positron, which is the antiparticle of an electron in that it is completely identical (same mass, same amount of electrical charge, same amount of spin) except that it is positively charged. Experiments show that an electron and positron can combine into a neutral atom called positronium, which is nearly identical in its properties to a hydrogen atom once the difference in relative mass is taken into account. However, while a hydrogen atom can persist forever in its ground state, a positronium atom exists only for a short time, about 10-10 seconds. The reason is that there is a finite probability of finding the electron and positron in the same small region of space in which case the two particles can annihilate one other, the positronium atom disappears, and in its place two high-energy photons (gamma rays) are observed.

Explain why a rotating positronium atom with large orbital quantum number l will, on average, live for a longer time than a positronium atom in its ground state before disintegrating into two gamma rays.

Challenge 63    Unusual lenses of air and of iron.

1. A scuba-diving archeologist would like to create an underwater magnifying glass out of a flexible plastic bag filled with air so that she can read the fine print of a sunken Greek ruin. What shape should the bag have to act as a magnifying glass?

2. Sound waves move with different speeds in different media and so sound can be refracted and focused by lenses just like light waves. As an impressive example, the earth's large spherical iron core acts like a lens that can focus sound waves emitted by an explosion or earthquake at the earth's surface. Using the following information and your elementary knowledge of optics (in particular, the formula that describes how paraxial rays are refracted by a spherical interface between two media of known indexes of refraction), estimate the focal length of the earth's iron core as a spherical lens embedded in less dense crustal rock. For an additional challenge, work out the details of how sound from an explosion in Durham, North Carolina, will appear as an "image" on the opposite side of the world.
Here are the data: the speed of sound (and so the index of refraction) varies with depth in the earth in a complicated way, as summarized in a Encyclopedia Britannica figure. For simplicity, assume that the earth's inner iron core of radius 3,500 km has a constant sound speed of 9 km/sec. Similarly, let's assume that the earth's crust, which extends from 3,500 to 6,400 km has a constant sound speed of 11 km/sec. Let's also ignore the fact that sound waves in a solid medium, unlike light in vacuum, can be longitudinal as well as transverse and these components travel with different speeds. The above speeds are for the faster longitudinal component.

Challenge 64    Does a neutrally buoyant balloon rise or fall as the temperature increases?

Consider a spherical balloon filled with helium gas and then weighted so that it remains motionless in the center of a sealed box of air at room temperature and atmospheric pressure. If the box is slowly and uniformly warmed so that the temperature everywhere inside increases by a small amount, determine whether the balloon will rise, fall, or remain in the same place.

Challenge 65    Origin of Unusual Radio Noise

Challenge 66    Swimming through the air on the International Space Station.

Imagine that you are a future tourist on the International Space Station and, having forgotten to buckle yourself into bed at night, you wake up the next morning floating freely and weightless in the middle of your bedroom chamber. Would it be possible for you to "swim" through the air to get back to your bed? If so, would you use the same kind of swimming strokes as you would to swim underwater? If you can swim through the air, what would be the order of magnitude of your maximum speed?

Note: You could always get back to your bed by taking off your pajamas, wadding them into a ball, and then throwing them in a direction opposite to that of your bed. Conservation of momentum would then give you a small velocity in the direction of your bed (can you estimate the order of magnitude of this speed?). But here the interest lies in the fluid dynamics of a large mass (you) trying to swim through a medium of small viscosity (air).

Challenge 67    Distinguishing a Fast Cooler Star From a Slow Warmer Star

Introductory astronomy and physics courses teach the interesting fact that the color of a remote star can be used to determine how hot the star is. More precisely, these courses teach that the light from a star, when passed through a prism or diffraction grating, produces a special rainbow called a "blackbody spectrum" whose shape (light intensity I plotted as a function of the wavelength λ of light) is "universal" in the sense that the curve I(λ) depends only on the temperature T (in kelvin) of the surface of the star and not at all on the star's chemical composition or size. Further, the blackbody curve I(λ) has a single peak and the wavelength λmax corresponding to that peak can be used to deduce the temperature of the star's surface by something called "Wien's law", namely that T = C / λmax where C is some universal constant that applies to all hot opaque objects.

The discussions in these introductory courses usually assume that the star is sitting still in space, but in fact most stars are moving toward or away from the Earth, some with a high speed. So consider a remote star whose surface temperature is T and assume that the star is moving directly toward the Earth with a speed s. Then something called the Doppler effect will cause the wavelength of each component of the light to become a little shorter ("blueshifted"). In particular, the wavelength λmax corresponding to the peak of the star's spectrum will be blueshifted and and so an astronomer applying Wien's law to the star's spectrum will deduce an incorrect higher temperature for the surface of the star since in the relation T = C / λmax, the wavelength in the denominator is a bit smaller than its actual value.

Your challenge: Is there some way for the astronomer to determine that he or she is looking at a fast cooler star versus a slow warmer star?

Note: The spectra from stars often have dark sharp absorption lines caused by elements in the outer cooler atmosphere of the star absorbing and then reemitting parts of the blackbody radiation coming from the star's surface. It would be straightforward to determine the speed of the star by measuring how much the wavelengths of the absorption lines are Doppler shifted compared to wavelengths of emission lines of the same elements in a laboratory on Earth. Here the question is how the Doppler effect modifies the blackbody spectrum and whether, in principle, observations of just the blackbody spectrum can be used to deduce the speed and temperature of a remote glowing object.

Challenge 68   The ollie: how does a skateboarder get that board off the ground?

A basic move in skateboarding is an "ollie", which gets the skateboarder and skateboard high into the air.

Since the skateboarder's feet are resting on the skateboard without any attachment, use your knowledge of physics (and perhaps of skateboarding) to explain how it is possible to push down on the board and so get the board high into the air. How also is it possible for the skateboard to stay in contact with the feet during a jump like this?

Challenge 69    Will a neutrally buoyant relativistic submarine sink or rise?

Consider some futuristic submarine that can travel close to the speed of light while submerged under water and let us assume that, when the submarine has its propulsion system turned off and is at rest in the water, that its ballast is adjusted so that the submarine is neutrally buoyant and so neither sinks nor rises. Now assume that the submarine speeds up to close to the speed of light in some huge ocean, initially traveling parallel to the surface of the ocean. Will the submarine sink, rise, or continue to travel parallel to the surface of the ocean?

Some thoughts: a fish that is at rest with respect to the ocean and that watches the submarine zoom by will presumably see the submarine relativistically contract along its length and so the fish will conclude that the submarine is denser than the surrounding water and will sink. But sailors in the submarine will presumably see the water coming toward them at high speed and so they will conclude that the relativistically contracted water is denser than the submarine and so the submarine should rise. So who is right, the fish or the sailors?

Challenge 70    Does one have to be quiet in order not to scare the fish away?

Fishermen on the shore of a lake or in a stream often try to be quiet so as not to scare the fish away. Using the fact that the speed of sound is about 340 m/s in air and about 1,500 m/s in water, use Snell's law of refraction to determine how far back from the shore a 1.7 m tall fisherman would have to stand so that the sound of the fisherman's voice could not be heard by any fish in the water. (Assume that the sound does not propagate through the ground.) Show also that if the fisherman stands in the water near the shore, then a fish would be able to the fisherman's voice no matter where the fish is located (although more loudly in some places than others).

Challenge 71    Throwing a baseball versus throwing a bowling ball.

If you can throw a baseball with a certain maximum speed, what would be the maximum speed you can throw a more massive object like a bowling ball?

Some data: a baseball has an official mass of about 0.15 kg (weight of about 5 oz) while 10-pin bowling balls start with a mass of about 3.6 kg (8 lb). The fastest measured baseball pitch had a speed of 100.9 mph (about 45 m/s).

Challenge 72    Doppler shift or not for co-falling source and detector?

A loudspeaker is attached to the bottom of a 3 m vertical rigid rod and a microphone is attached to the top of the same rod. If the loudspeaker emits a pure tone of frequency f = 1000 Hz when the rod is at rest, what frequency does the microphone measure as a function of time if the entire apparatus is dropped from a tall tower?

Challenge 73    High tide on the Moon.

If the Moon were warm and had a water ocean like the Earth, would there be tides on the Moon like there are on Earth? If so, how often would high tide occur?

Challenge 74    The economics of a lunar rocket base.

President Bush has proposed to build a base on the Moon by 2018 from which future rockets could be launched to Mars or other parts of the solar system. Where should such a rocket base be placed on the Moon, in what direction should the rocket be pointed when launched, at what point in the Moon's orbit should the rocket be launched, and with what minimum speed should the rocket be launched so that it can escape the Moon-Earth system?

Challenge 75    Deducing the location of heaven from Satan's fall.

In the book Dear Professor Einstein: Albert Einstein's Letters to and from Children edited by Alice Calaprice (Prometheus Books, 2002), a student Jerry from Richmond, Virginia, wrote the following letter to Einstein in 1952:

                     Dear Sir,

I am a high school student and have a problem.

My teacher and I were talking about Satan. Of course
you know that when he fell from heaven, he fell for
nine days, and nine nights, at 32 feet a second and was
increasing his speed every second.

I was told there was a foluma [formula] to it. I know
you don't have time for such little things, but if
possible please send me the foluma.

Thank you,
Jerry

It seems that Einstein did not reply to Jerry but this provides an opportunity to do some detective work using physics.
1. Assuming that Satan indeed fell from heaven for nine days and nine nights, assuming that Satan's initial speed was zero, and assuming that Satan fell with a constant acceleration of g=9.8 m/s2, deduce how far from Earth heaven must lie and also deduce the speed with which Satan struck the surface of the Earth.

2. Make a more realistic calculation by assuming that the gravitational acceleration g of Satan during his fall was not constant but decreased with increasing distance from Earth according to Newton's universal law of gravity, g=GME/d2, where G is the universal gravitational constant, ME is the mass of the Earth, and d is the distance of Satan to the center of the Earth at a particular moment. (Ignore the fact that Satan would also be acted on by gravitational forces from the Sun and other planets.) What now would be the location of heaven from Earth if Satan fell for nine days and nine nights under these conditions, and with what speed would Satan now strike the Earth's surface?

Challenge 76    Which way was the bicycle moving?

The picture below shows the tracks made by the two wheels of a bicycle as it was traveling through snow. In which direction (left to right or right to left) was the bicycle moving? Which trace corresponds to the rear wheel, which to the front wheel?

Challenge 77    Rendezvous

You have been wandering in a desert, crazed with thirst for several days, when you see a truck traveling on a straight east-west road, where the road is south of your location. In what direction should you run to maximize your chance to arrive at the road in time to be picked up by the truck and be saved?

For this Challenge, assume that you run at a constant speed in a straight direction, and that the truck also travels with a constant speed (whose value you do not know). A hint: you generally do not want to run to the nearest part of the road.

This challenge could also be posed in nautical terms: you are on a small lifeboat when you see off in the distance a large ship moving in a straight line at a constant speed. In what fixed direction should you paddle as fast as you can (at some constant speed) to maximize your chance of intercepting the ship so you can be rescued?

Challenge 78    Tough balancing act

Explain how to arrange ten large identical steel nails so that they are all supported off the ground by just the head of an eleventh identical vertical nail.

Note: The nails can touch only each other and the head of the vertical nail. You can not use any other items in solving this Challenge such as glue or magnets.

Challenge 79    Can One Boil Water With Boiling Water?

A pot of water is brought to a steady boil on a stove and then a thin plastic cup of room temperature water is suspended in the boiling water so that no part of the cup touches the pot (see above figure). Will the water in the cup start to boil if you wait long enough?

After you have convinced yourself what the answer should be, try the experiment (but please be careful, don't burn yourself!).

Challenge 80    How should a lifeguard run and swim to save a drowning person?

Assume that you are a lifeguard at location $$L$$ on a beach when you see someone starting to drown at location $$D$$ in the water. Assume that you can run on sand (orange region) with a speed $$v_s$$ that is substantially larger than the speed $$v_w$$ which you can swim in water (blue region), and assume that to save the person, you run in a straight line from point $$L$$ until you reach some point $$P$$ on the shore and then swim from $$P$$ in a straight line until you reach the drowning person. Determine the location of $$P$$ that would let you reach the drowning person in the shortest amount of time.

To make the problem specific, assume that the Cartesian coordinates of the lifeguard is $$x_L, y_L$$, that the coordinates of the drowning person is $$x_D, y_d$$ and that the shoreline has coordinate $$y = 0$$.

Challenge 81    Maximum electric field from an electrically charged blob of clay?

You are given a volume V of a malleable incompressible non-conducting material like clay that has a uniform charge density ρ throughout its volume. Determine how to shape the material and place the resulting shape to as to produce the largest possible electric field magnitude E at some point P of interest.

Challenge 82    Thickness of a flat Earth.

It was once thought that the Earth was flat, rather than in the shape of a ball. Assuming that the Earth is a large flat slab with a mass density of 5,500 kg/m3 (this is the average mass density of Earth), use Gauss's law to determine how thick the slab would have to be so that the gravitational acceleration at the surface of the slab is 10 m/s2.

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