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Happy meetings, and the problem of a ball on a sphere

March 16, 2013

I had a good day yesterday. Perhaps it is no coincidence that nice things happen if you help others.

I was in our Liverpool office, giving a talk about my career, and lessons from it, to the tax department, and taking questions at the end. I enjoy doing such things, and believe it is part of my role now at Deloitte to 'Pay it forward'

Anyway, one of the questions, which I received in advance, and I covered at the end, was to ask me which three or four people, dead or alive, I would have as my ideal dinner party guests. Thinking this through over the last week or so gave me a lot of pleasure, in a similar way I am sure to choosing your Desert Island Discs records (though, what gave me less pleasure was that hardly any of the audience, mainly comprising 20 year olds, had heard of it).

After much fun, I narrowed my choices down, and one of my guests would be Sir Isaac Newton, so I could try to understand the basics of physics, hoping that he would be a good teacher of the principles. I do believe that a lot of my O level, A level and degree level science was sometimes cursory: give a concept a name, use it a bit, and then you believe you understand it. A big part of my view of science is that often we tease ourselves into conning ourselves into believing we know more than we do: and in this, I speak as someone who got a top first in engineering from Cambridge: there was lots I could do in practice without deeply, deeply, understanding matters.

Back now to my main point. One of my colleagues, Paul Farr, saw me afterwards, and asked me why I had chosen Sir Isaac over Richard Feynman: I had though of Richard, but ruled him out, but in reflection, Paul was right, Richard was an excellent teacher and his books are well worth reading [ note to self: read more of his work]. I could have added Eric Rogers, the writer of Physics for the Inquiring Mind, the best book on physics that I have ever read.

It turns out that Paul, now a VAT specialist, is also deeply interested in science, maths and engineering. He gave me a puzzle to solve, which I think is pretty neat.

Rolling ball

Imagine a small ball, perched at the apex of a large sphere. The gentlest of forces pushes the ball, so it rolls down the sphere: at what angle does it leave the surface?

I am pleased to say I solved it fairly rapidly, but only by good fortune. Only recently had I been helping Sophie with her homework, and afterwards spent a lot of time trying to bolster my understanding of centripetal forces and why things really go round in circles. So I had one equation instantaneously to mind, working out the take off condition. The speed was slightly harder, but in our chat, Paul had given me the clue of looking at conservation of energy [Paul- that was a big clue] so all I had to do was some not too difficult geometry (though, rustiness meant it took me some while) following which there were three equations which could be solved easily. My solution is below.


1. The angle is somewhat less than I would have intuitively guessed- my a priori guess would have been 60 degrees; perhaps this is real world intuition, taking into account friction?

2. On (1), whilst I suggest friction is a reason for my intuition being wrong, I actually can't see how friction would increase the takeoff theta. Something else to mull over whilst walking Charlie. The real world effect might in fact be air resistance.

3. The initial nudge was incy-wincy-tiny: but were it not, and if instead it gave even the slightest initial velocity, then the departure angle reduces: the maximum velocity which the centripetal force can sustain is exceeded earlier in the slope.

4. Thank you, Paul, for taking an interest in what I had to say, for our discussion, and for giving me an interesting problem to solve.


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