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6 S. The amusement park ride shown in the figure consists of a cylindri-
cal room that rotates about its vertical axis. When the rotation is fast
enough, a person against the wall can pick his or her feet up off the floor
and remain “stuck” to the wall without falling.
(a) Suppose the rotation results in the person having a speed v. The radius
of the cylinder is r, the person’s mass is m, the downward acceleration of
gravity is g, and the coefficient of static friction between the person and
the wall is
µ
s
. Find an equation for the speed, v, required, in terms of the
other variables. (You will find that one of the variables cancels out.)
(b) Now suppose two people are riding the ride. Huy is wearing denim,
and Gina is wearing polyester, so Huy’s coefficient of static friction is three
times greater. The ride starts from rest, and as it begins rotating faster and
faster, Gina must wait longer before being able to lift her feet without
sliding to the floor. Based on your equation from part a, how many times
greater must the speed be before Gina can lift her feet without sliding
down.
7 S. An engineer is designing a curved off-ramp for a freeway. Since the
off-ramp is curved, she wants to bank it to make it less likely that motor-
ists going too fast will wipe out. If the radius of the curve is r, how great
should the banking angle,
.
, be so that for a car going at a speed v, no
static friction force whatsoever is required to allow the car to make the
curve. State your answer in terms of v, r, and g, and show that the mass of
the car is irrelevant.
8 . Lionel brand toy trains come with sections of track in standard
lengths and shapes. For circular arcs, the most commonly used sections
have diameters of 662 and 1067 mm at the inside of the outer rail. The
maximum speed at which a train can take the broader curve without flying
off the tracks is 0.95 m/s. At what speed must the train be operated to
avoid derailing on the tighter curve.
9. The figure shows a ball on the end of a string of length L attached to a
vertical rod which is spun about its vertical axis by a motor. The period
(time for one rotation) is P.
(a) Analyze the forces in which the ball participates.
(b) Find how the angle
.
depends on P, g, and L. [Hints: (1) Write down
Newton’s second law for the vertical and horizontal components of force
and acceleration. This gives two equations, which can be solved for the
two unknowns,
.
and the tension in the string. (2) If you introduce
variables like v and r, relate them to the variables your solution is supposed
to contain, and eliminate them.]
(c) What happens mathematically to your solution if the motor is run very
slowly (very large values of P). Physically, what do you think would
actually happen in this case.
v
r
Problem 6.
.
Problem 7.
Problem 9.
.
L
Chapter 9Circular Motion