Mathematical Tools for Physics

2—Infinite Series 62

d

ay

2.34 A massm 1 hangs from a string that is wrapped around a pulley of massM. As the massm 1 falls with
accelerationay, the pulley rotates. An anonymous source claims that the acceleration ofm 1 is one of the following
answers. Examine them to determine if any is plausible. That is, examine each and show why it could not be
correct. NOTE: solving the problem and then seeing if any of these agree isnotwhat I want.

(1) (2) (3)

ay=

Mg

m 1 −M

ay=

Mg

m 1 +M

ay=

m 1 g

M

2.35 Combine two other series to get the series forln(cosθ).

2.36 Subtract the series forln(1−x)andln(1 +x). For what range of arguments of the logarithm does this
converge?

R q

p q

2.37 Light travels from a point on the left (p) to a point on the right (q),
and on the left it is in vacuum while on the right of the spherical surface it
is in glass with an index of refractionn. The radius of the spherical surface
isRand you can parametrize the point on the surface by the angleθfrom
the center of the sphere. Compute the time it takes light to travel on the
indicated path (two straight line segments) as a function of the angleθ.
Expand the time through second order in a power series inθand show that
the functionT(θ)has a minimum if the distanceqis small enough, but that
it switches to a maximum whenqexceeds a particular value.
This is the focus.