Archive for the Cute Problems Category

Marginalia

Posted in Cute Problems, Education, The Universe and Stuff with tags , , , , , on August 12, 2020 by telescoper

While this morning’s repeat exams were going on I was leafing through an old second-hand text book, one of many I have acquired over the years looking for nice problems and worked examples. The good thing about old books is that solutions to the problems are usually not available on the internet, unlike modern ones. The book concerned this morning is a classic: Statics by Horace Lamb, which you can still get via Cambridge University Press. I have the first edition, published in 1912.

Looking through I was somewhat alarmed to see what had been pencilled in some of the margins:

Of course anyone who has been to India knows that the swastika isn’t necessarily a Nazi symbol: you find it all over the place in the Indian sub-continent, where it is used as a symbol for good luck. I remember being given a very nice conference bag in Pune many years ago with a swastika on it. I didn’t use it back home, of course.

The first owner of the copy of Statics that I have was acquired in 1913 by a J.H.C (or G) Lindesay of Sidney Sussex College, Cambridge. I know because he/she inscribed their name in the front. That doesn’t look to me like an Indian name, but I think it’s a fair bet that the book passed through many hands before reaching me and that one of the past owners was Indian. I haven’t tried any of the problems marked with the swastika, but perhaps they are difficult – hence the `good luck’ symbol? I notice though that the symbol at the bottom of the page has a chirality different from the others. Is this significant, I wonder?

All of which irrelevance reminded me of an discussion I’ve had with a number of people about whether they like to scribble in the margins of their books, or whether they believe this practice to be a form of sacrilege.

I’ll put my cards on the table  straightaway. I like to annotate my books – especially the technical ones – and some of them have extensive commentaries written in them. I also like to mark up poems that I read; that helps me greatly to understand the structure. I don’t have a problem with scribbling in margins because I think that’s what margins are for. Why else would they be there?

This is a famous example – a page from Newton’s Principia, annotated by Leibniz:

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Some of my friends and fellow academics, however, regard such actions as scandalous and seem to think books should be venerated in their pristine state.  Others probably find little use for printed books given the plethora of digitial resources now available online or via Kindles etc so this is not an issue..

I’m interested to see what the divergence of opinions is in with regard to the practice of writing in books, so here’s a poll for you to express your opinion:

The Geostationary Orbit

Posted in Cute Problems, The Universe and Stuff with tags , , , on May 16, 2020 by telescoper

I’m was mucking out this blog’s blocked comments folder and unsurprisingly found a few from Mr Hine, a regular if sadly deranged correspondent.

One of his blocked comments begins

In the forlorn hope that Mr Hine might some day learn something scientifically correct I thought I’d repost this problem, which is very easy if you have a high school education in physics or applied mathematics but no doubt very difficult if you’re Mr Hine.

Verify that the radius of a circular geostationary orbit around the Earth is about 42,000 km, i.e. find the radius of a circular orbit around the Earth which has a period of 24 hours so that its orbital period matches the Earth’s rotation period, thus ensuring that an object travelling in such an orbit in the same direction as the Earth’s rotation is always above the same point on the Earth’s surface.

(You will need to look up the mass of the Earth.)

How to Solve Physics Problems

Posted in Cute Problems, Education, Maynooth, The Universe and Stuff, YouTube with tags , , on May 14, 2020 by telescoper

Since the examination period here at Maynooth University begins tomorrow I thought I would use the opportunity provided by my brand new YouTube channel to present a video version of a post I did a few years ago about how to solve Physics problems. These are intended for the type of problems students might encounter at high school or undergraduate level either in examinations or in homework. I’ve tried to keep the advice as general as possible though so hopefully students in other fields might find this useful too.

A Virus Testing Probability Puzzle

Posted in Cute Problems, mathematics with tags , on April 13, 2020 by telescoper

Here is a topical puzzle for you.

A test is designed to show whether or not a person is carrying a particular virus.

The test has only two possible outcomes, positive or negative.

If the person is carrying the virus the test has a 95% probability of giving a positive result.

If the person is not carrying the virus the test has a 95% probability of giving a negative result.

A given individual, selected at random, is tested and obtains a positive result. What is the probability that they are carrying the virus?

Update 1: the comments so far have correctly established that the answer is not what you might naively think (ie 95%) and that it depends on the fraction of people in the population actually carrying the virus. Suppose this is f. Now what is the answer?

Update 2: OK so we now have the probability for a fixed value of f. Suppose we know nothing about f in advance. Can we still answer the question?

Answers and/or comments through the comments box please.

A Problem of Snooker

Posted in Cute Problems, The Universe and Stuff with tags , , , on February 25, 2020 by telescoper

I came across the following question in a first-year physics examination from Cambridge (Part 1A Natural Sciences) and, since I have posted anything in the Cute Problems folder for a while I thought I would share it here:

Answers through the comments box please! And please show your working!

P.S. The preamble does not say whether you can also assume irrelevant formulae without proof…

 

Yet another easy physics problem…

Posted in Cute Problems with tags , , , on December 2, 2019 by telescoper

Last week I posted a little physics problem that generated a large amount of traffic (at least by the standards of this blog), so I thought I’d try another one.

The examination comprised two papers in those days (and a practical exam); one paper had long questions, similar to the questions we set in university examinations these days, and the other consisted of short questions in a multiple-choice format. This question is one of the latter. Incientally, for those of you who have asked, the multiple-choice examination contained 50 such questions to be answered in 2½ hours, which is three minutes per question.

(You can assume that the acceleration due to gravity is 9.8 ms-2.)

And here is a poll in which you may select your answer:

Comments on or criticisms of the question are welcome through the comments box…

Another easy physics problem…

Posted in Cute Problems, The Universe and Stuff with tags , , , on November 26, 2019 by telescoper

Many moons ago I posted an `easy’ physics problem from the Physics A-level paper I took in 1981. The examination comprised two papers in those days (and a practical exam); one paper had long questions, similar to the questions we set in university examinations these days, and the other consisted of short questions in a multiple-choice format. The question I posted was one of the latter type. I was reminded about it recently because, years on, it appears people are still trying it (and getting it wrong).

Anyway, since I’m teaching similar things to my first-year Mathematical Physics class I thought I’d put up another question from the same paper.

And here is a poll in which you may select your answer:

Comments on or criticisms of the question are welcome through the comments box…

 

 

 

The Problem of the Disintegrating Asteroid

Posted in Cute Problems, The Universe and Stuff with tags , , on September 30, 2019 by telescoper

I thought you might enjoy this entry in the Cute Problems folder.

An asteroid is moving on a circular orbit around the Sun with an orbital radius of 3AU when it spontaneously splits into two fragments, which initially move apart along the direction of the original orbit. One fragment has a speed which is a fraction 0.65 of the original speed, the other has a speed of 1.35 times the original speed. The original orbit (solid line) is shown above, along with the two new orbits (dashed and dotted).

  1. Which orbit does the fast fragment follow, and which the slow fragment?
  2.  Calculate the original orbital speed in AU/year.
  3. Calculate the angular momentum per unit mass, h, of the original asteroid and of each of the two fragments in units of AU2 per year. [HINT: Show that in these units, for a general orbit of eccentricity e and semi-major axis a, h2=4π2 a (1-e2).]
  4.  Calculate the eccentricities of the orbits of the two fragments.
  5.  Calculate the orbital periods of the two fragments in years.

Answers please through the Comments box. First complete set of answers wins a trip to the Moon on gossamer wings.

 

 

A Problem of Dimensions

Posted in Cute Problems, Maynooth, The Universe and Stuff with tags , , , on August 21, 2019 by telescoper

We’ve more-or-less sorted out who will be teaching what next term in the Department of Theoretical Physics at Maynooth University next term (starting a month from now) and I’ll be taking over the Mathematical Physics module MP110, which is basically about Mechanics with a bit of of special relativity thrown in for fun. Being in the first semester of the first year, these is the first module in Theoretical Physics students get to take here at Maynooth so it’s quite a responsibility but I’m very much looking forward to it.

I am planning to start the lectures with some things about units and dimensional analysis. Thinking about this reminded me that I posted a dimensional analysis problem (too hard for first-year students) on here a while ago which seemed to pose a challenge so I thought I would post another here for your amusement.

 

The period P for an elliptical orbit of semi-major axis a of  a moon of mass m around a planet of mass M, depends only on the quantities  a, m, M and G (Newton’s Gravitational Constant).

(a). Using dimensional analysis only, determine as completely as possible the relationship between P and these four quantities.

(b). How would the period P compare with the period P′ of a system consisting of a moon of mass 2m orbiting a planet of mass 2M in an ellipse with the same semi-major axis a?

Please submit your efforts through the comments box below.

 

More Order-of-Magnitude Physics

Posted in Cute Problems with tags , , , on April 25, 2019 by telescoper

A very busy day today so I thought I’d just do a quick post to give you a chance to test your brains with some more order-of-magnitude physics problems. I like using these in classes because they get people thinking about the physics behind problems without getting too bogged down in or turned off by complicated mathematics. If there’s any information missing that you need to solve the problem, make an order-of-magnitude estimate!

Give  order of magnitude answers to the following questions:

  1. What is the maximum distance at which it could be possible for a car’s headlights to be resolved by the human eye?
  2. How much would a pendulum clock gain or lose (say which) in a week if moved from a warm room into a cold basement?
  3. What area would be needed for a terrestrial solar power station capable of producing 1GW of power?
  4. What mass of cold water could be brought to the boil using the energy dissipated when a motor car is brought to rest from 100 km/h?
  5. How many visible photons are emitted by a 100W light bulb during its lifetime?

There’s no prize involved, but feel free to post answers through the comments box. It would be helpful if you explained a  bit about how you arrived at your answer!