Marble on a turntable
By Minghao Zhang (Yr 12)
Just as everyone has a thousand reasons to love or hate maths, I have my own. I appreciate maths, because it can provide even the most ridiculously counter- intuitive phenomenon with a solid solution. Hardly any other forms of explanations could be as convincing as a mathematical expression. Here in this article, I would like to discuss with you one example of such problems, the bizarre behaviour of a marble ball, which turns out to fall right into the mathematical prediction. Imagine a large, flat turntable, which is rotating at constant a speed. A naughty boy placed a marble ball on it, allowing it to start rotating before letting go. What would you expect to happen? Will the marble be expelled immediately? Or will it start to orbit around the centre resembling a planet? I shall not reveal the answer right away, because that way I would ruin your interests in taking out a record player and carrying out your own ‘experiment’. While you are setting up the gadgets, we will begin our theoretical approach - first by modelling the situation and analysing the dynamics. Let the angular velocity of the turntable be Ω⃗ , and , , 𝑅⃗ and ⃗ denote the mass, moment of inertia, radius (pointing from centre of ball to surface of contact) and angular velocity of the ball bearing respectively. Since the turntable is a plane, we use a 2-D vector to represent the ball bearing’s position. We assume the force is some friction force exerted on the ball by the surface, which should be the only horizontal force that concerns us. Our goal is to work out the equation of motion of the ball bearing.
First, apply Newton’s Second Law,
2 2
= ⃗⃗⃗
Then Newton’s second law for rotation,
= 𝑅⃗ × ⃗⃗⃗ There is no slipping between the ball and the turntable, so 𝛺⃗ × = + ⃗ × 𝑅⃗
Job done for all the mechanics. With three equations and three unknowns, we should be able to acquire our solution. However, there is no division rules for vectors and this will bring us trouble in manipulating the algebra. At this critical point, the flexibility of maths comes onto stage. While Cartesian coordinate system, vector space and Argand diagrams are essentially the same thing, the quantities they carry are very different. Therefore, we are able to make one final facilitation to our equations before solving them - mapping all vector quantities onto an Argand diagram. Let = ∙ ̂ + ∙ ̂ , = ⃗ ∙
25
Made with FlippingBook - professional solution for displaying marketing and sales documents online