The following Physics Puzzles are 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.

The purpose of these brain teasers is to evoke serious thoughts about the nature around us. A person aspiring to get into a prestigious university will be helped by this presentation. But more than that, this is to evoke a lasting interest in this subject. Feel free to contact us personally or write e mail to us for your queries.

**Puzzle 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 thin 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?

**Puzzle**** 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.

**Puzzle**** 3 Hanging
Bricks**

What is the maximum overhang you can create with an infinite supply of bricks?

**Puzzle**** 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? 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.

**Puzzle**** 5 Equilibration of
Two Birthday Balloons **

Consider two identical 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?

**Puzzle**** 6 Balloon
tracing**

A helium-filled balloon is tied to the floor of a car that makes a sharp right turn. Does the balloon tilt while the turn is made? If so, which way? The windows are closed so there is no connection with the outside air.

**Puzzle**** 7
Distinguishing two
nearly identical spheres **

You are given two spheres that are
identical in size, weight, appearance, and touch but one sphere is hollow while
the other is solid. (As an example, the solid sphere could be made out of a
light wood and the hollow sphere made out of a denser wood, then both spheres
carefully painted to look and feel the same.) Using only simple items that you
might find at home (no fancy equipment, no drills), determine which sphere is
hollow.

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

As a science undergraduate 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.

**Puzzle**** 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?

**Puzzle**** 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.

**Puzzle**** 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 place where this is possible.

**Puzzle**** 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.

**Puzzle**** 13 Leaning Tower of Pizza
**

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

**Puzzle**** 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?

**Puzzle**** 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
10^{30} kg and the relative speed of stars in a galaxy is of order
10 km/sec.

**Puzzle 16 Where does the water go?**

A person in a boat drops a cannonball overboard; does the water level change?

**Puzzle**** 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.

**Puzzle**** 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. Some other future Puzzles will also have a mathematical flavor.

**Puzzle 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.

**Puzzle 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?

**Puzzle 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.)

**Puzzle 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.

- Using the approximate formula "T = 2 Pi Sqrt[L/g] ( 1 + (1/16)
A
^{2}+ ... )" for the period T of a pendulum of amplitude A measured in radians (with L its length and g=10 meter^{2}/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.)

- 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?

**Puzzle 23 Magnets**

You have two bars of iron. One is magnetized along its length, the other is not. Without using any other instrument (thread, filings, other magnets, etc.), find out which is which.

**Puzzle 24 Hot milk**

You are just served a hot cup of coffee and want it to be as hot as possible later. If you like milk in your coffee, should you add it when you get the cup or just before you drink it?

**Puzzle**** 25
Monkey climbs**

Hanging over a pulley there is a rope, with a weight at one end. At the other end hangs a monkey of equal weight. What happens if the monkey starts to ascend the rope? Assume that the mass of the rope and pulley are negligible, and the pulley is frictionless.

**Puzzle**** 26
Firing at the sun **

If you are standing at the equator at sunrise, where must you point a laser cannon to hit the Sun dead center? Assume that the Sun is stationary and that the Earth's orbit around it is circular.

**Puzzle**** 27 Maximize
the reflections**

Assume two planar mirrors containing angle *α*. How to point a beam
(reflecting from them) to reach as many reflection as possible?

**Puzzle 28 Bat on the hunt**

The bat is on hunt and travels towards the fly at speed of 3.14 m/s, fly
flies ten times slower in opposite direction. The bat sends ultrasound of
frequency f_{0}, which bounce from fly and comes back to the hunter.
Bat's ears are most sensitive to the frequency around 61.3 kHz. Calculate
f_{0} to achieve the best performance in this situation. What frequency
of sound would fly heard if it has ears?

**Puzzle**** 29 Where
did the petrol go? **

Assume a small car with engine pulls at constant force *F* at speed
*v*. Its power is then *P* = *Fv*. But cyclist riding
at constant speed *u* observes power
*P* = *F *(*v*−*u*). Petrol consumption, which
corresponds to the power, is the same as viewed by cyclist, car and pedestrian.
Explain this paradox. Air friction is negligible.

**Puzzle**** 30 Save
our Soul!**

The duck is floating in the middle of the round pond. It wants to join others, unfortunately there is a fox on the bank. The duck cannot take of from water. It can take of only from the ground. Calculate the minimum ratio of the speeds of the duck and the fox for duck to be able to reach the bank and take off and not be eaten by fox. Suggest suitable strategy to reach this goal.

**Puzzle**** 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).

**Puzzle**** 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.

Note: This problem was suggested by Adam Berman, a high school student.

**Puzzle**** 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

for bodies not in relative motion and

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 five 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. As of 1998, the theory of granular flow is lagging badly behind the experiments.

**Puzzle**** 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:

- If both knife switches are closed, then both lights are on.
- If both knife switches are open, then both lights are off.
- 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.
- 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.

**Puzzle**** 35 Crashing
Car**

The driver of the car moving at the speed *v* suddenly recognizes that
it is heading towards the middle of a concrete wall of width *2d* and is at
a distance of *l* from the wall. The coefficient of the friction between
the tires and the surface of road is *f*. What is the best way to do to
avoid the inevitable accident. Decide, what is the maximum velocity with which
he can drive and still avoid the
crash.

**Puzzle**** 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?

**Puzzle**** 37 Rolling
along**

A small cylinder of radius *r* and mass *m* is
rolling on the inclined plane. At the end of the plane it follows smoothly to
horizontal motion and starts to wind onto itself string of the linear
density *ρ*. At what distance from the end of inclined plane will the
cylinder stop? You know the height of the inclined plane *h* and it
slope *α*. The friction is negligible.

**Puzzle**** 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.)

**Puzzle**** 39 ****Find the North!
**

It is possible to find North
with the help of a watch. How? Prove this experimentally. How big is the
deviation between your solution and the correct direction? Can you justify it
with calculation?

**Puzzle**** 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?

**Puzzle**** 41
****World Record
**

Two air-conditioned athletic fields were built
on the equator and on the north pole. World championships were held in the polar
one. The winner of shot-put threw 23 meters and created a new world record.
Could he make it on the equator? How long would be the throw there?

Suppose that the athletes perform the same physical power in both places.

**Puzzle**** 42
****On the
Road **

If you are traveling by car and watching the
road in front of you with binoculars, you can notice that the road is moving
slower than expected. Express how much slower this looks depending on parameters
of your binoculars.

**Puzzle**** 43
**

A shot from an action film: The
hero with a weight of 80 kg is tied to the rope with rigidity 40 N/m.
He jumps from a bridge that is 100 meters high.

The shot is to be made with a
puppet of the hero and the bridge will be only two meters high. What must the
rope rigidity, the weight of the puppet and the slow down of the filming be in
order to make the jump look real in the movie?

**Puzzle**** 44
****Color of the
Moon **

If you look at the Moon at night, it seems to be yellow. If you look at it during the day, it seems to be white. Why is it so?

**Puzzle**** 45
****Hearing is Seeing**

In the house of a person who is weak of hearing, a light bulb is also lit when somebody rings the door-bell. The ring can be operated both from the garden gate and from the door of the house. Draw such a circuit.

**Puzzle**** 46
An insect an hour
**

An insect sets off upwards from the shaft of the minute hand of a church-clock exactly at 12 o'clock. Moving uniformly along the hand, it reaches the end of the hand in a quarter of an hour. When was it at the highest position?

**Puzzle**** 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?

**Puzzle**** 48 Synchronization
problems**

Why do symphonic orchestras tune on the stage of the concert hall and not in the rehearsal room before the performance? List physical reasons that put different instruments (wind-instruments, stringed instruments) out of tune.

**Puzzle**** 49 The
Flying bird **

We put a closed glass-case with air and a small bird inside on a scale. It is obvious that the scale indicates the weight of the bird too, when it is sitting on the bottom of the case. What is the combined weight if the bird is hovering in the mid-air inside the case? What is the weight if we use a cage instead of the closed glass-case?

**Puzzle**** 50 Electron dilemma
**

Electrons inside a solenoid revolve along a circular path. In what direction: according or opposite to that of the electrons moving inside the wire of the solenoid?

**Puzzle**** 51 Jumping
to Conclusions! **

A flea goes up a flight of stairs jumping from edge to edge on the stairs. When can he get up with less work: when he takes the stairs one by one or two by two? In which case does he arrive upstairs sooner?

**Puzzle**** 52 Which
Way?**

Two metal rods of identical cross section but different length and material properties are glued together. The rod obtained this way is hung up by two long and vertical ropes fixed to its end points. In which direction will the point of contact of the two rods be displaced upon increasing the temperature?

**Puzzle**** 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.

**Puzzle**** 54 Changing Seasons
**

A child asks: `Why does winter come when the Earth gets nearest to the Sun?' Answer his question.

**Puzzle**** 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.

**Puzzle**** 56 Get
Back soon **

An astronaut on a space trip moves away from his space shuttle. He has two light, identical spring guns, capable of shooting two projectiles of identical mass. How can he obtain a higher speed: by shooting the two guns simultaneously or consecutively in the same direction?

**Puzzle**** 57
Capillary and the
blade**

A certain liquid in a capillary tube rises half as high as water. What happens when some of this liquid is dripped next to one side of a razor blade floating on top of water?

**Puzzle**** 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 arrangment of lenses and reflectors of any arbitrary size and shape, how hot can you make the beam of light by focusing it onto a single spot?

The sunlight comes from the sun's surface whose temperature is about 6000 K. If you collect and focus enough sunlight, could you create a spot hotter than the temperature of the sun's surface?

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.

**Puzzle**** 59 An inverse rocket
and an inverse sprinkler. **

- 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?

- 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 sprinkler (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?

**Puzzle**** 60
Bank shots on an
elliptical billiard table. **

Consider two point balls B_{1} and B_{2} 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 B_{1} so
that it bounces once off the elliptical side wall and hits ball B_{2}?
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
B_{1} to a side of the table and then extend the perpendicular an equal
distance beyond the table to obtain point P_{1}. Draw the line between
point P_{1} and ball B_{2} and identify the point P_{2}
where this line intersects the side of the table. You then want to point your
cue stick at point P_{2} to hit a bank shot that will connect with ball
B_{2}.

**Puzzle**** 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/m^{3}.

**Puzzle**** 62 Current v/s the
electron beam**

Does the same force act between two conductors with current flowing through them, and a conductor with current flowing through it and an electron beam parallel to it if the other data are identical?

**Puzzle**** 63 Unusual lenses of
air and of iron. **

- 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?
- 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 crystal rock. For an additional Puzzle, 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.

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

Consider a 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.

**Puzzle**** 65
Air or no air
**

The weight of an air-balloon inflated with air is measured using a very sensitive scale. The air is let out from the balloon and the weight of the balloon (and of the thread) is measured. What is the result? Why?

**Puzzle**** 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).

**Puzzle**** 67 Cleaving the
Log**

What is the correct explanation of the phenomenon that if we put the sharp edge of an axe on a wooden clog even with an additional weight put on top of it, the surface of the clog is hardly scratched, but if we lift the axe and strike down, the clog cleaves asunder?

**Puzzle**** 68 Who is
bigger**

In a homogeneous magnetic field there is an electron moving in a circular orbit. Can the magnetic induction vector generated by the moving electron be greater in the middle of the circle than the magnetic induction vector of the homogeneous field?

**Puzzle**** 69
Capacity
**

At the edge of a plate capacitor there is an inhomogeneous electric field, which is usually neglected. Taking this into account as well, would we get a smaller or a greater capacity value?

**Puzzle**** 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).

**Puzzle**** 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 while 10-pin bowling balls start with a mass of about 3.6 kg. The fastest measured baseball pitch had a speed of about 45 m/s.

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

A loudspeaker is attached to the bottom end of a 3 m vertical heavy 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?

**Puzzle**** 73
Compare the
masses**

The elements of a peculiar bead string are made of the same material. All the beads are of spherical shape and have a cylindrical borehole in their middle. The peculiarity is that though the radii of the spheres are different, the borehole length (the height of the cylinder) is exactly the same in all beads. What can we say about the masses of the beads?

**Puzzle**** 74
Dissolving the
energy**

What will happen to the energy of a compressed spring if it dissolves in hydrochloric acid in its compressed state?

**Puzzle**** 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, JerryIt seems that Einstein did not reply to Jerry but this provides an opportunity to do some detective work using physics.

- 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/s^{2}, deduce how far from Earth heaven must lie and also deduce the speed with which Satan struck the surface of the Earth.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=GM*, where_{E}/d^{2}*G*is the universal gravitational constant,*M*is the mass of the Earth, and_{E}*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?

**Puzzle**** 76 Pendulum's time
period**