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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