Some more Physics/Mathematics puzzles with an unusual structure, tailored for the Google-in-hand generation.

In most cases, you don’t have to derive a numerical value for the solution, to remember the value of the physical constants required, or even to remember the correct formulae to use in the solution. It suffices to describe how you will find an answer (using your cellphone, if needed). Of course, it would be more satisfying, and have a greater chance of being correct, to work through a complete solution. Simplifying assumptions are given in notes. Some helpful (or perhaps misleading) references may be given.

7. Two Slit Experiment

Water waves, light, even electrons show a diffraction pattern in the well-known double slit experiment. This is said to be an interference effect; waves from the two slits travel slightly different distances in slightly differing directions so that their peaks and troughs sometimes align and sometimes cancel each other. The wavy line at the right in the figure below indicates the varying intensity of the diffraction pattern. The pattern is calculated by summing the amplitudes of the two waves at each point on the screen, assuming that the two waves were in phase when they started at the slits.

So now, some questions regarding this phenomenon:

(a) If white sunlight is used as the source, will there be a pattern of white lines on the screen?

(b) If the intensity of the light source or electron source is decreased so that only one photon or electron is in flight at a time, will the pattern change (after sufficient particles have passed through to build up a pattern)?

(c) If one of the slits is blocked off (i.e. a single slit experiment), will the multiple peaks of intensity change to a single peak?

(d) If an extra slit is added (i.e. a three-slit experiment), do we sum up the wave amplitudes coming from the three slits to calculate the intensity pattern on the screen?

Discussion (hidden in blue block; select the text to make it visible)

(a) No. sunlight is a mixture of light of several wavelengths, from red to violet. The diffraction pattern size depends on the wavelength, so the white light separates into rainbows, a bit like how a prism creates a rainbow.
(b) No. One still gets the same pattern. A discussion about this can be readily found on the internet or textbooks.
(c) No. There is still a diffraction pattern, although the height of the peaks falls off rapidly away from the central peak.
(d) No, in the case of photons or electrons. Apparently the Born rule must be applied pair-wise: calculate the interference separately for each pair of slits (i.e. 1,2 – 2,3 – 3,1) and sum up the results. I am not sure about the case of non-quantum waves, but I think the answer would be a ‘yes’. Note: Here is an interesting discussion of Born rule in Quanta Magazine.

8. Flight of Fixed Wing Aircraft

If you search for an explanation of how airplanes fly, you typically get a explanation in terms of Bernoulli’s equation, and how the shape of the airfoil causes low pressure on the upper surface of the wing, as in this MIT web page.

The figure above, from a different source, captures the essence of this explanation.

It is natural to conclude, therefore, that it is impossible for a plane to fly upside-down in steady constant-altitude flight, for the negative lift from the wings plus the weight of the airplane itself would cause the airplane to descend rapidly. But is this really true? Could an airplane continue to fly upside-down?

Discussion (hidden in gray block; select the text to make it visible):

It turns out that the explanation based on airfoil shape is relevant, but not the full story. It is more fruitful to consider the airplane’s flight in terms of momentum transfer, and the law of conservation of momentum. Consider how a rocket travels upward. The momentum imparted to the exhaust gases being shot backward is balanced by an equal and opposite momentum imparted to the rocket. The same principle explains a gun’s recoil when a bullet is fired, or, when a helicopter’s rotors push air down, lift is exerted on the craft. Similarly, the secret of the airplane’s flight is that the slightly tilted wing pushes air flowing over its surface downward. The air carries vertical momentum downward, thereby imparting upward momentum to the airplane. Were the wing to be tilted in the opposite direction (front edge lower than the back edge), it would push the air upward and thus generate negative lift. Now consider a plane flying upside-down. Provided the wings have sufficient tilt (in the picture above, the nose is higher than the tail) they will still push the air downward, generating positive lift that keeps the airplane at constant height above ground. However the airfoil shape decreases lift somewhat in this orientation, so the airplane has to tilt more (or go faster) to compensate, compared to right side up flight.

9. Solar Eclipse

The figure above (not to scale) shows the general geometry of the Sun – Earth – Moon system during solar eclipses.
Now consider a moment during a particular solar eclipse where the Moon’s disc precisely covers the Sun. At this particular moment,
(a) A total eclipse will be seen from some parts of the Earth, what is the area of this region?
(b) A partial eclipse will be seen from some parts of the Earth, what is the area of this region?
(c) An uneclipsed disc of the Sun will be seen from some parts of the Earth, what is the area of this region?

Discussion (hidden in gray block; select the text to make it visible):

First of all, the figure above is somewhat misleading. If the Moon’s disc precisely covers the Sun, the umbra (full shadow) of the Moon will just touch at a point on the Earth. Hence the answer to (a) is zero. Had the Moon’s disc been a little larger than the Sun’s disk the umbra would have had a positive area. On the other hand, had the Moon’s disc been a little smaller than the Sun’s disc, one would get an annular eclipse in the region of the Moon’s antumbra. See this NASA website for details. Remember, the moon’s orbit is slightly elongated so the Earth-Moon distance changes with time.
Now, knowing the sizes of the three bodies and the Sun-Earth distance, one can calculate the exact position of the Moon.
Next, some geometrical calculations are required to find the radius r of the penumbra. If r is small, a good approximation for (b) is simply pi*r^2. Otherwise, one should calculate the area of a spherical cap with this value of r.
To a first approximation, the answer to (c) is half the area of the Earth less the answer to (b). For a more accurate calculation, some geometrical calculation is required to find the area of the slightly less than hemispherical cap that is illuminated by the Sun.
Note: the fact that a total eclipse is generally visible only over a band a few hundred kilometers wide is a direct consequence of the Sun being much larger than the Moon.

10. Acceleration and Accelerators

(a) According to the theory of relativity, nothing material can travel faster than the speed of light. But a scientist claims to have run an experiment in which several objects were accelerated from rest to a speed faster than light. Could this claim be correct?

(b) You must have heard of the Large Hadron Collider at CERN (a machine that accelerates hadrons, such as protons, to high speeds), which has cost several billion US dollars to build and operate so far. But do accelerators have to be so big and costly? Suppose you wanted to own a particle accelerator, and a scientist says that they can set up an elementary particle accelerator in your garage for a mere 1000 US dollars, do you think they can do it?

(c) Acceleration seems to be a simple concept, but in some contexts it may not be so simple. A physics professor asks the class to calculate the acceleration of a piece of rock lying motionless on the top of Mount Everest. Now, three students hand in quite different answers, all of them non-zero, and the professor accepts all three answers as correct. What’s going on?

Discussion (hidden in gray block; select the text to make it visible):

(a) The trick is in the fine print – the ultimate speed limit is c, the speed of light in a vacuum. However, light travels slower than c through material substances. Particle accelerators can regularly produce beams that travel quite close to c, and when these beams hit a tank of water, for example, the particles are traveling faster than the speed of light in water. Look up Cherenkov radiation.
(b) It is not difficult to accelerate electrons in a vacuum under a voltage difference of a few thousand volts. In fact old style TVs have a cathode ray tube (CRT) where the TV picture is formed by high speed electrons hitting a substance that would give off light. Even if CRT TVs are not readily available today, CRT based instruments are. It is a different matter that my idea of a particle accelerator may differ from a CRT TV, although technically the latter is an elementary particle accelerator.
(c) The first student answered that since Earth spins on its axis, the rock undergoes an acceleration. The second student added the additional component of acceleration due to Earth’s orbit around the Sun. The third student took a different track: according to general relativity, the apparent force of gravity is really the rock being in a region of spacetime that is curved due to the mass of the Earth, and the acceleration comes from the mountain pushing the rock up.

4 Pulleys and Plank

A horizontal square plank is held up by one rope at each corner. Each rope passes over a simple pulley and ends in a 10 kg weight that itself rests on the plank. All ropes are vertical. The plank itself is 10 kg. Ignore mass of ropes, and friction,
(a) Is this possible, in the first place?
(b) find the tension in each rope.
(c) Is there a range of masses of the plank for which this configuration is stable? If yes, what is the range?

Solution (hidden in gray block):

The main difficulty in understanding this setup is that each weight is partially lifted up by its rope, so the tension in the rope is somewhere between 0 and 10 kg-force. A good way to understand this is by using a free-body diagram: isolate the system by cutting across the eight vertical ropes. Now there are eight tensile forces T, and gravity pulls on the mass of the system.

The configuration is symmetrical so we can assume equal load at each pulley. Further, since the pulleys have no friction, the tension in the two ropes of each pulley must be equal. Let it be T. The total tension in the eight ropes supports the plank and weights. Thus 8*T upward tension is balanced by the total weight of 50 kg — the four 10 kg weights and the 10 kg plank
Thus T = 50/8 kg-force = 6.25 kg-force.
The maximum sized weights that can be held up are when each weight barely rests on the plank, almost all of its 10 Kg weight is balanced by an almost 10 kg tension in each rope. Thus a maximum of almost 80 kg can be supported in all, of which 40 kg is the weights themselves. That leaves about 40 kg for the plank.

5 Attraction

(a) Two uniform identical spheres of diameter D are placed with a gap of D between them They feel a gravitational attraction of F. If the gap is increased to 2D, what’s the force of attraction in units of F?
(b) Will the answer be same/more/less than above, if the spheres are made of copper, their gravitational attraction is negligible, but each carries an equal and opposite electric charge, and their electrical attraction is F’?

Solution (hidden in gray block):

(a) By Newton’s law of gravitation, the force of gravitational attraction is inversely proportional to the square of the distance between two particles. In the case of uniform spheres, the distance is counted from their centers.

in the first case, the center to center distance r is D/2 + D + D/2 = 2D
in the second case, the center to center distance r’ is D/2 + 2D + D/2 = 3D
That is, r increases by a factor of 1.5, so the attractive force decreases by a factor of 1/1.5^2 = 1/2.25
(b) The force for electrical charge also follows an inverse square law. The main difference here is that the electric charge can move on a conducting surface. So the charge will be concentrated on surfaces that are opposite each other. Hence the calculation of r is roughly D versus 2D, hence the force will decrease by a factor of 1/4 approximately.

6. Rolling cylinders

Two cylindrical objects of the same diameter and mass roll down an inclined plane from a state of rest (without slipping). One object has a uniform density while the second has its mass mostly near the periphery (i.e. like a very thin walled tube). What is the ratio of linear acceleration along the inclined plane, of the first object relative to the second?

Solution (hidden in gray block)

When an object of mass m and moment of inertia I rolls a distance s down a slope of angle theta, its gravitational energy decreases by mgs* sin(theta) and its kinetic energy increases.

Let v be the linear velocity along the plane, and omega be the angular velocity.
Total Kinetic Energy = (1/2){mv^2 + I omega^2). .
When there is no slippage, omega = v/R.

Assuming no dissipation of energy, the drop in potential energy equals gain in kinetic energy, since initial K. E. was zero.
2 mg sin(theta) * s = m v^2 + I/R^2 v^2 = (m + I/R^2)* v^2
Differentiate both sides with respect to time:
2 mg sin(theta) * v = (m + I/R^2) * 2 v a
Now 2v cancels, hence
a = mg sin(theta) / (m + I/R^2)
Now, what is the difference between the solid cylinder and the tube?
The moment of inertia I is mR^2/2 for the cylinder, and mR^2 for the tube.
So the denominator in the formula for a differs for the two objects.
It is {m + m R^2/2 / R^2} = 1.5m for the cylinder and {m + mR^2/R^2} = 2m for the tube. The tube will thus have a slower
acceleration than the cylinder by a factor of 1.5 / 2 = 0.75