Physics 2211 A
Quiz #5
Solutions
Summer 2007
g
Earth
= 9
.
8 m
/
s
2
R
Earth
= 6
.
37
×
10
6
m
G
= 6
.
673
×
10

11
N
·
m
2
/
kg
2
M
Earth
= 5
.
98
×
10
24
kg
Unless otherwise directed, all springs, cords, and pulleys are ideal, and drag should be neglected.
I
. (16 points) A solid sphere of mass
M
and radius
R
is pivoted on a horizontal axle that passes through the
sphere a distance
R/
2 from the center. The sphere is held so the center is an angle
θ
below the axle, and
released. What is the angular acceleration at the instant of release, in terms of any or all of
R
,
M
,
θ
and
physical or mathematical constants?
(On Earth.)
.
.
.
.
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The net torque on the sphere can be related to its angular acceleration by the rotational version of Newton’s
Second Law,
∑
~
τ
=
I~α
.
The only torque about the axle is due to the force of weight, which acts at the
center of mass. So
X
τ
=
Iα
⇒
τ
w
=
Iα
⇒
r
w
Mg
sin
φ
=
Iα
where
φ
is the angle between the
~
r
vector and the weight vector when placed tail to tail, or
φ
= 90
◦

θ
.
The rotational inertia of a solid sphere about its center of mass is
I
cm
=
2
5
MR
2
. This sphere is not rotating
about its center of mass, though, so the parallel axis theorem,
I
=
I
cm
+
Md
2
can be used to find the
rotational inertia about the axle. Since the axle has been displaced a distance
R/
2 from the center of mass,
I
=
I
cm
+
Md
2
=
2
5
MR
2
+
M
R
2
¶
2
=
2
5
+
1
4
¶
MR
2
=
13
20
MR
2
Putting these together
r
w
Mg
sin
φ
=
Iα
⇒
R
2
¶
Mg
sin (90
◦

θ
) =
13
20
MR
2
α
and solving for the angular acceleration (remember that sin (90
◦

θ
) = cos
θ
)
α
=
(
R/
2)
Mg
cos
θ
(13
/
20)
MR
2
=
10
13
‡
g
R
·
cos
θ
Quiz #5 Solutions Page 1 of 5
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II
. (16 points) A 45 kg scientific satellite is to be launched into a circular orbit an altitude two Earthradii above
the Earth’s surface.
The launch will take place from the South Pole, so the satellite will have no kinetic
energy before it is launched. How much work must be done to place this satellite in orbit?
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Use the WorkEnergy Theorem, since it relates the work done by an external force (the answer to the
question) to the change in kinetic and potential energy. In this case, there are no nonconservative forces in
the Earthsatellite system, and the potential energy is that of Universal Gravitation.
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