183
10. Psychology professor R.O. Dent requests funding for an experiment
on compulsive thrill-seeking behavior in hamsters, in which the subject is
to be attached to the end of a spring and whirled around in a horizontal
circle. The spring has equilibrium length b, and obeys Hooke’s law with
spring constant k. It is stiff enough to keep from bending significantly
under the hamster’s weight.
(a) Calculate the length of the spring when it is undergoing steady
circular motion in which one rotation takes a time T. Express your result
in terms of k, b, and T.
(b) The ethics committee somehow fails to veto the experiment, but the
safety committee expresses concern. Why. Does your equation do any-
thing unusual, or even spectacular, for any particular value of T. What do
you think is the physical significance of this mathematical behavior.
11. The figure shows an old-fashioned device called a flyball governor,
used for keeping an engine running at the correct speed. The whole thing
rotates about the vertical shaft, and the mass M is free to slide up and
down. This mass would have a connection (not shown) to a valve that
controlled the engine. If, for instance, the engine ran too fast, the mass
would rise, causing the engine to slow back down.
(a) Show that in the special case of a=0, the angle
.
is given by
. =
cos
–1
g(m+M)P
2
4
p
2
mL
,
where P is the period of rotation (time required for one complete rota-
tion).
(b) There is no closed-form solution for
.
in the general case where a is
not zero. However, explain how the undesirable low-speed behavior of the
a=0 device would be improved by making a nonzero.
[Based on an example by J.P. den Hartog.]
12. The figure shows two blocks of masses m
1
and m
2
sliding in circles
on a frictionless table. Find the tension in the strings if the period of
rotation (time required for one complete rotation) is P.
13. The acceleration of an object in uniform circular motion can be given
either by |a|=|v|
2
/r or, equivalently, by |a|=4
p
2
r/T
2
, where T is the time
required for one cycle. (The latter expression comes simply from substitut-
ing |v|=distance/time=circumference/T=2
p
r/T into the first expression.)
Person A says based on the first equation that the acceleration in circular
motion is greater when the circle is smaller. Person B, arguing from the
second equation, says that the acceleration is smaller when the circle is
smaller. Rewrite the two statements so that they are less misleading,
eliminating the supposed paradox. [Based on a problem by Arnold Arons.]
Problem 10.
Problem 11.
L
1
L
2
m
1
m
2
Problem 12.
m
m
M
L
a
.
L
Homework Problems