Archive for the Cute Problems Category

A Cube of Resistance

Posted in Cute Problems with tags , , , on September 14, 2017 by telescoper

It has been brought to my attention that I haven’t posted any cute physics problems recently, so here’s one (which involves applying Kirchoff’s laws) that’s a bit harder than A-level standard which might be of interest to students about to begin a degree in physics this month!


The above image, produced using the advanced computer graphics facilities available at Cardiff University’s Data Innovation Research Institute, represents a cube formed of 12 wires each of which has resistance 1Ω.

What is the electrical resistance between: (i) A and G; (ii) A and H; and (iii) A and D?

As usual, answers through the comments box please!

Advertisements

A Problem with Spitfires

Posted in Cute Problems, History, The Universe and Stuff with tags , , , , , on July 25, 2017 by telescoper

This problem stems from an interesting exchange on Twitter last night, prompted by a tweet from the Reverend Richard Coles:

I think his clerical vocation may be responsible for the spelling mistake. The answer to his question doesn’t require any physics beyond GCSE but it does require data that I didn’t have access to last night.

Here’s a version for you to try at home with all the necessary numbers (though not necessarily in the right units):

A model of a Mark VI Spitfire showing its two 20mm cannons.

A Supermarine  Mark VI (Type 350) Spitfire fighter aircraft weighing 6740 lb is initially travelling at its top speed of 354 mph. The aircraft is armed with two Hispano-Suiza HS.404 20mm cannons, one on each wing, each of which is fed by a drum magazine containing 60 rounds. Each projectile  fired from  the cannon weighs 130 grams, the rate fire of each cannon is 700 rounds per minute and the muzzle velocity of each shell is 860 m/s.

(a) Calculate the reduction in the aircraft’s speed if the pilot fires both cannon simultaneously until the magazines are empty, if the pilot does nothing to compensate for the recoil. Express your answer in kilometres per hour.

(b) Calculate the average deceleration of the aircraft while the cannons are being fired, and express your result as a fraction of g, the acceleration due to gravity at the Earth’s surface which you can take to be 9.8 ms-2.

(c) A Mark 24 Spitfire – which is somewhat heavier than the Mark VI, at 9,900 lb (4,490 kg) – is armed with 4×20mm cannons, two on each wing. The inboard cannon on each wing has a magazine containing 175 rounds; the outboard one has 150 rounds to fire. Repeat the above  analysis for these new parameters and comment on your  answer.

Answers through the comments box please!

 

 

 

A Problem of Brightness

Posted in Cute Problems with tags on July 20, 2017 by telescoper

It’s been a while since I posted anything in the Cute Problems folder, so here’s a nice one:

Two physics students are studying at desks in the same room. The two outlets of the wall socket to which their desk lamps are attached have inadvertently been wired in series rather than in parallel. The only bulbs available for the lamps are designed for the nominal mains voltage (240V) : Student A chooses a 200W bulb for his lamp; Student B opts for a 50W bulb for hers.

Which is the brighter student?

Hint: Assume that the filament in each bulb obeys Ohm’s Law.

Answers through the comments box please, preferably with working…..

A Problem of Gravity

Posted in Cute Problems with tags , , on May 9, 2017 by telescoper

Here’s a nice one for the cute problems folder.

Two spherically symmetric stars A and B of equal mass M and radius r have centres separated by a distance 6r. Ignoring any effects due to the orbital motion of the stars, determine a formula (in terms of G, M and r) for the minimum velocity with which material can be ejected from the surface of A so as to be captured by B.

Answers through the comments box please. First correct answer receives 7 points.

 

A Question of Equilibrium

Posted in Cute Problems with tags , on March 30, 2017 by telescoper

It’s been quite a while since I’ve been able to find time to post any items in the cute problems folder, and I don’t have much time today either, but here’s a quickie. You may well find this a lot harder than it looks at first sight. At least I did!

An isolated system consists of two identical components, each of constant heat capacity C, initially held at temperatures T1 and T2 respectively. What is the maximum amount of work that can be extracted from the system by allowing the  two components to reach equilibrium with each other?

As usual, answers through the comments box please. There is no prize, even if you’re right.

 

 

The Neyman-Scott ‘Paradox’

Posted in Bad Statistics, Cute Problems with tags , , , , on November 25, 2016 by telescoper

I just came across this interesting little problem recently and thought I’d share it here. It’s usually called the ‘Neyman-Scott’ paradox. Before going on it’s worth mentioning that Elizabeth Scott (the second half of Neyman-Scott) was an astronomer by background. Her co-author was Jerzy Neyman. As has been the case for many astronomers, she contributed greatly to the development of the field of statistics. Anyway, I think this example provides another good illustration of the superiority of Bayesian methods for estimating parameters, but I’ll let you make your own mind up about what’s going on.

The problem is fairly technical so I’ve done done a quick version in latex that you can download

here, but I’ve also copied into this post so you can read it below:

 

neyman-scott1

neyman-scott2

I look forward to receiving Frequentist Flak or Bayesian Benevolence through the comments box below!

Reflections on Quantum Backflow

Posted in Cute Problems, The Universe and Stuff with tags , , , , on November 10, 2016 by telescoper

Yesterday afternoon I attended a very interesting physics seminar by the splendidly-named Gandalf Lechner of the School of Mathematics here at Cardiff University. The topic was one I’d never thought about before, called quantum backflow. I went to the talk because I was intrigued by the abstract which had been circulated previously by email, the first part of which reads:

Suppose you are standing at a bus stop in the hope of catching a bus, but are unsure if the bus has passed the stop already. In that situation, common sense tells you that the longer you have to wait, the more likely it is that the bus has not passed the stop already. While this common sense intuition is perfectly accurate if you are waiting for a classical bus, waiting for a quantum bus is quite different: For a quantum bus, the probability of finding it to your left on measuring its position may increase with time, although the bus is moving from left to right with certainty. This peculiar quantum effect is known as backflow.

To be a little more precise about this, imagine you are standing at the origin (x=0). In the classical version of the situation you know that the bus is moving with some constant definite (but unknown) positive velocity v. In other words you know that it is moving from left to right, but you don’t know with what speed v or at what time t0 or from what position (x0<0) it set out. A little thought, (perhaps with the aid of some toy examples where you assign a probability distribution to v, t0 and x0) will convince you that the resulting probability distribution for moves from left to right with time in such a way that the probability of the bus still being to the left of the observer, L(t), represented by the proportion of the overall distribution that lies at x<0 generally decreases with time. Note that this is not what it says in the second sentence of the abstract; no doubt a deliberate mistake was put in to test the reader!

If we then stretch our imagination and suppose that the bus is not described by classical mechanics but by quantum mechanics then things change a bit.  If we insist that it is travelling from left to right then that means that the momentum-space representation of the wave function must be cut off for p<0 (corresponding to negative velocities). Assume that the bus is  a “free particle” described by the relevant Schrödinger equation.One can then calculate the evolution of the position-space wave function. Remember that these two representations of the wave function are just related by a Fourier transform. Solving the Schrödinger equation for the time evolution of the spatial wave function (with appropriately-chosen initial conditions) allows one to calculate how the probability of finding the particle at a given value of evolves with time. In contrast to the classical case, it is possible for the corresponding L(t) does not always decrease with time.

To put all this another way, the probability current in the classical case is always directed from left to right, but in the quantum case that isn’t necessarily true. One can see how this happens by thinking about what the wave function actually looks like: an imposed cutoff in momentum can imply a spatial wave function that is rather wiggly which means the probability distribution is wiggly too, but the detailed shape changes with time. As these wiggles pass the origin the area under the probability distribution to the left of the observer can go up as well as down. The particle may be going from left to right, but the associated probability flux can behave in a more complicated fashion, sometimes going in the opposite direction.

Another other way of thinking about it is that the particle velocity corresponds to the phase velocity of the wave function but the probability flux is controlled by the group velocity

For a more technical discussion of this phenomenon see this review article. The exact nature of the effect is dependent on the precise form of the initial conditions chosen and there are some quantum systems for which no backflow happens at all. The effect has never been detected experimentally, but a recent paper has suggested that it might be measured. Here is the abstract:

Quantum backflow is a classically forbidden effect consisting in a negative flux for states with negligible negative momentum components. It has never been observed experimentally so far. We derive a general relation that connects backflow with a critical value of the particle density, paving the way for the detection of backflow by a density measurement. To this end, we propose an explicit scheme with Bose-Einstein condensates, at reach with current experimental technologies. Remarkably, the application of a positive momentum kick, via a Bragg pulse, to a condensate with a positive velocity may cause a current flow in the negative direction.

Fascinating!