Archive for potential energy

The Mechanics of Nursery Rhymes

Posted in Cute Problems, The Universe and Stuff with tags , , , , on December 30, 2020 by telescoper

I’ve always been fascinated by Nursery Rhymes. Some people think these are little more than nonsense but in fact they are full of interesting historical insights and offer important advice for the time in which they were written. One such story, for example, delivers a stern warning against the consequences of placing sleeping babies in the upper branches of trees during windy weather.

Another important role for nursery rhymes arises in physics education. Here are some examples that students of elementary mechanics may find useful in preparation for their forthcoming examinations.

1. The Grand Old Duke of York marched 10,000 men up to the top of a hill and marched them down again. The average mass of his men is 65 kg and the height of the hill is 500m.

(a) Estimate the total work done in marching the Duke of York’s men up to the top of the hill.

(b) If, instead of marching down again, the men take turns sliding down a frictionless slide back to where they started, estimate the average speed of a man when he reaches the bottom of the hill.

(You may assume without proof that when they were up they were up, and when they were down they were down and, moreover, when they were only half way up they were neither up nor down.)

2. By calculating the combined rest-mass energy of half a pound of tuppenny rice and half a pound of treacle, and assuming a conversion efficiency of 10%, estimate the energy released when the weasel goes pop. (Give your answer in SI units.)

3. The Moon’s orbit around the Earth can be assumed to be a circle of radius r. A cow of mass m is standing on the Earth (which has mass M, and radius R). Derive a formula in terms of r, R, M, m and Newton’s Gravitational Constant G for the energy the cow needs in order to jump over the Moon.

(The Earth, Moon and cow may be assumed spherical. You may neglect air resistance and udder frictional effects. )

Feel free to contribute similar problems through the Comments Box.

The Physics of the Pole Vault

Posted in Education, The Universe and Stuff with tags , , , , , on October 17, 2011 by telescoper

At the RAS Club Dinner last Friday I chatted for a while with my former DPhil supervisor, John Barrow. I’m not sure how, but the topic came up about how helpful it is to use sports to teach physics. By coincidence he chose the same example as I have used in the past during first-year tutorials,  the pole vault.

Years ago I went to watch an athletics meeting at Gateshead Stadium and sat right next to the pole vault area. I can tell you that the height the vaulters reach is truly spectacular, especially when you’re close to the action. The current world record for the pole vault is 6.14m, in fact, set by the legendary Sergey Bubka in 1994, so the record hasn’t been broken for 17 years. Here’s a clip of him a few years earlier clearing a mere 6.10 metres (pretty comfortably, by the look of it)…

One might infer, from the fact that the record has not been broken for such a long time, that pole vaulters are working pretty much at the limit of what the human body can achieve. And a bit of physics will convince you of the same.

Basically, the pole is a device that converts the horizontal kinetic energy of the vaulter \frac{1}{2} m v^2,  as he/she runs in, to the gravitational potential energy m g h acquired at the apex of his/her  vertical motion, i.e. at the top of the vault.

Now assume that the approach is at the speed of a sprinter, i.e. about 10 ms^{-1}, and work out the height h = v^2/2g that the vaulter can gain if the kinetic energy is converted with 100% efficiency. Since g = 9.8 ms^{-2} the answer turns out to be about 5 metres.

This suggests that  6.15 metres should not just be at, but beyond, the limit of a human vaulter,  unless the pole were super-elastic. However, there are two things that help. The first is that the centre of mass of the combined vaulter-plus-pole does not start at ground level; it is at a height of a bit less than 1m for an an average-sized person.  Nor does the centre of mass of the vaulter-pole combination reach 6.15 metres. The pole does not go over the bar, but it’s pretty light so that probably doesn’t make much difference. However, it’s not  obvious that the centre of mass of the vaulter actually passes over the bar.  That certainly doesn’t happen in the high jump – owing to the flexibility of the jumper’s back the arc is such that the centre of mass remains under the bar while the different parts of the jumper’s body go over it.

Moreover, it’s not just the kinetic energy of the vaulter that’s involved. A human can in fact jump vertically from a standing position, using elastic energy stored in muscles. One can’t jump very high like that, but it seems likely to me that this accounts for a few tens of centimetres.

Anyway, it is clear that pole vaulters are remarkably efficient athletes. And not a little brave either – as someone who is scared of heights I can tell you that I’d be absolutely terrifed being shot up to 6.15 metres on the end of  a bendy stick, even with something soft to land on!