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Erich Poppitz

PHY254 Lecture 4 Supplementary Notes Motion in plane, polar coordinates Here is a practice problem on motion in plane, polar coordinates: Example (Kleppner and Kolenkov): A bead with unit mass moves outward with constant speed u along the spoke of a wheel that rotates with angular frequency !. The bead starts from the center at t = 0. Find the velocity, acceleration, and the magnitude of the force on the bead, and make a sketch of the motion. Answer: The radial coordinate of the bead (distance from centre) is given by r = ut since u is constant, and the azimuthal velocity is ▯ = !. The velocity is therefore ~v = _^+ r▯▯ = ur^+ ut!▯ ^ (1) The acceleration is _2 _  ^ 2 ^ a = (▯ r▯ )r^+ (2r_▯ + r▯)▯ = ▯ut! r^+ 2u!▯: (2) Therefore, the radial force on the bead is given by ▯ut! (the r ^ compo- nent of the above equation times mass which equals 1), and the azimuthal force is given by 2u! (the ▯ component of the above equption times mass which equals 1). The magnitude of the force is therefore u t ! + 4u ! =2 q 2!u 1 + (!t=2) . As the particle spirals outward, the force required to main- tain its motion increases. For !t ▯ 1, the force increases linearly with time. Here is a sketch of the motion. For additional practice, do the example from Morin on p. 70. 1 Solving F = ma problems numerically In class, we derived the approximate formulas F(t;x(t);v(t)) v(t + ▯t) ▯ v(t) + ▯ta(t) = v(t) + ▯t (3) m and x(t + ▯t) ▯ x(t) + ▯tv(t) (4) These motivated the recursion rule: given x a0d v , w0 have x = x + ▯tv i+1 i i vi+1 = v +i▯tF(x ;v it i; i ▯ 1 To better understand the approximations we have made here, let’s think about the Taylor expansion of a function f(t): df(t) 1 2 d f(t) 1 3 d f(t) f(t + ▯t) = f(t) + ▯t + (▯t) 2 + (▯t) 3 + ::: (5) dt 2 dt 6 dt Now, we expect the terms to get smaller at each step. To demonstrate this, consider the case f(t) = et=▯, ▯ > 0. You might think that since f(t) is exponentially growing, retaining only a couple of terms in a Taylor expansion would somehow not work so well as an approximation. But this is not the case, as long as ▯t
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