Physics 106, Spring 2008
K. Kauder
Group Problem #5
–Answer key–
1.
♠
Gravity
A 2000 kg spaceship is initially traveling in a circular orbit around a planet at a distance four times
the radius of the planet as measured from the planet’s center. The radius of the planet,
R
p
is 400 km.
The spaceship goes into a new circular orbit
whose radius is sixteen times the radius of the
planet as measured from the planet’s center. The
planet’s mass is unknown but the acceleration due
to gravity on its surface
g
p
has been measured to
be 2.5
m
s
2
.
(a)
Determine the planet’s mass
M
knowing the
value of the acceleration due to gravity on its
surface.
(b)
Determine the speed of the spaceship when it
is in its initial orbit.
(c)
Determine the work done by the spaceship’s en
gine in traveling from the initial orbit to the
final one.
Solution
:
(a) We can express the force acting on some test object of mass
m
on the surface in two ways:
universal law of gravitation:
F
=
G
m M
R
2
p
and local law of gravitation:
F
=
m g
p
These are two ways to describe the same force
F
so we can equate the two terms:
G
m M
R
2
p
=
m g
p
Conveniently, the test mass
m
can be eliminated and we can rearrange the expression to find the
solution:
M
=
R
2
p
g
p
G
=
(400
×
10
3
m)
2
(2
.
5
m
s
2
)
6
.
67
×
10

11
N
m
2
kg
2
= 6
.
0
×
10
21
kg
(b) Reminder: Paraphrasing the textbook (about
centripetal force
, equation 6–15 or thereabouts):
“A force must be applied to an object [with tangential speed v] to give it circular motion. For an
object of mass
m
the net force acting on it must have a magnitude given by
f
cp
=
m a
cp
=
m v
2
r
and must be directed toward the center of the circle.”
Luckily, we are concerned with circular motion (treating a general elliptic orbit would be a bit
nastier). Here, the centripetal force is supplied by gravity so we can equate the two and solve for
v
(note that
r
= 4
R
p
):
G
m M
r
2
=
m v
2
r
⇔
v
2
=
G
M
r
⇒
v
=
radicalBig
G
M
r
=
radicalBig
6
.
67
×
10
−
11
N
m
2
kg
2
6
.
0
×
10
21
kg
4(400
×
10
3
m)
= 500
m
s
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Physics 106/GP5
– Page 2 of 2 –
–Answer key–
(c) In order to determine the amount of work done by the rocket’s engine we must write down the
most general form of the workenergy theorem which is
W
= Δ
K
+ Δ
U
where Δ
U
is the change in the gravitational potential energy of the astronaut. We can write this
equation explicitly as
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 Spring '08
 KOJA
 Physics, Force, Gravity, Mass, kg

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