Physics 2211
Quiz #3
Solutions
Spring 2008
Unless otherwise directed, all springs, cords, and pulleys are ideal, and drag should be neglected.
I
. (16 points) The Earth is a planet with mass
M
and radius
R
. What would the period of the Earth’s rotation
be if objects at the equator had an apparent weight that was onethird of the gravitational force on them?
Express your answer in terms of any or all of
M
,
R
, and physical or mathematical constants.
Note: Do NOT
use the physical constant “
g
”, as it includes the effect of the Earth’s actual period.
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This is a Newton’s Second Law problem with Uniform
Circular Motion. Sketch a Free Body Diagram of the
object at the equator. Choose the positive direction to
be in the direction of the centripetal acceleration, to
ward the center of the Earth. The only forces acting on
the object are the force of gravity and a normal force.
The normal force is the object’s apparent weight. The
force of gravity on the object will be calculated using
the Law of Universal Gravitation. The acceleration is
a centripetal acceleration. Note that the speed of the
object,
v
, is the circumference of the Earth, 2
πR
, over
the period,
T
. Letting
M
be the mass of the Earth,
and
m
be the mass of the object,
X
F
r
=
F
G

n
=
ma
r
⇒
F
G

1
3
F
G
=
m
v
2
R
⇒
G
Mm
R
2

1
3
G
Mm
R
2
=
m
R
2
πR
T
¶
2
Solve for
T
. Note that the mass of the object cancels.
2
3
G
Mm
R
2
=
4
mπ
2
R
T
2
⇒
T
2
=
3
2
4
π
2
R
3
GM
⇒
T
=
r
6
π
2
R
3
GM
Quiz #3 Solutions Page 1 of 5
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1. (6 points) An object of mass
M
is attached to the end of a string and swings in a vertical circle having a
radius
r
. At an instant when the string is at an angle
θ
above the horizontal, the tension in the string is
T
.
What is the radial component of the weight? (
On Earth.
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 Spring '08
 UZER
 Force, Mass

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