9. (6 pts.)
At a circus, a clown on a motorcycle with a mass
M
travels along a horizontal track
and enters a vertical circle of radius
r
.
Which one of the following expressions determines the
minimum speed that the motorcycle must have at the top of the track to remain in contact with
the track?
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A.
B.
C.
v
=
gR
D.
v
= 2
gR
E.
v
=
MgR
v
gr
=
10. (6 pts.)
Note the following three cases. In which case will the magnitude of the normal force
on the block be equal to (
Mg
+
F
sin
θ
)?
11. (6 pts.)
The figure gives the angular momentum L of a wheel versus time t. Which of the
following is true for the
magnitudes
of the torques in the intervals shown?
B
C
L
t
A
A
A
12. (6 pts.):
A bullet is aimed at a target on the wall a distance
L
away from the firing position.
Because of gravity, the bullet strikes the wall a distance
Δ
y
below the mark as suggested in the
figure.
Note: The drawing is not to scale.
If the distance
L
were half as large, and the bullet had
the same initial velocity, how would
Δ
y
be affected?
A
B
C
L
t
D
2
v
gr
=
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Problem 1 (25 pts.):
A bowling ball of mass
M
and radius
R
(
treated as a uniform solid sphere
is thrown along a
level
floor with kinetic energy
K
0
. It is thrown so that initially (at
t
= 0) it slides
with a linear speed
ν
0
but does not rotate. As
the ball slides, kinetic friction causes it to
begin spinning. Eventually the ball begins to
roll without slipping along the floor, with
linear speed
ν
and angular speed
ω
, as shown
below. The coefficient of kinetic friction
between the ball and the floor is
μ
k
a) (
15 pts.
) At what time
t > 0
does the bowling ball begin rolling smoothly along the floor?
What is the linear speed
ν
at that time?
Express your answers in terms of
ν
0
, g, and μ
k
, as needed.
)
.

b) (
10 pts.
) What is the change
Δ
K
in the bowling ball’s kinetic energy during this entire process
(
i.e
., from sliding motion to rolling without slipping)?
Express your answer in terms of
K
0
.