Download Kg - cloudfront.net

2.
4.
The period of a satellite circling planet Nutron is observed
to be 84 seconds when it is in a circular orbit with a
radius of 8.0 x 106 m. What is the mass of planet Nutron?
AP Physics
A satellite circles planet Zeron every
98 min. The mass of
24
this planet is known to be 5 #10 Kg. What is the radius
of the orbit?
98min = 5880sec
4" 2 3
4" 2
r
M = 2 r3
GM
GT
2
3
4"
4" 2
M = 2 r3 =
8.0 #10 6 m)
2
2 (
$11
N%m
GT
6.67 #10 Kg 2 (84 sec)
T2 =
Chapter 13
(
Review
T2 =
)
r=
M = 4.29 #10 28 Kg
!
3
!
4" 2 3
r
GMC
GMT 2
4" 2
r=3
(6.67 #10
!
$11 N %m 2
Kg 2
)(5 #10
24
Kg)(5880 sec)
2
4" 2
r = 6.6 #10 6 m
#2
7
4
!
1.
3.
A satellite circles planet Roton
every 2.8 hours in an orbit
7
having a radius of 1.2 # 10 m . If the radius of Roton
is 5.0 # 106 m , what is the magnitude of the free-fall
acceleration on the surface of Roton?
& 3600sec )
+ =10080sec
' hr *
(2.8hr)(
2
2
T2 =
2
4" 3
4"
T =
r
M = 2 r3
GM
GT
!
3
4" 2
4" 2
M = 2 r3 =
1.2 #10 7 m)
2
2(
$11
N
GT
6.67 #10 Kg%m2 (10080sec)
2
(
A 50-kg satellite circles planet Cruton every 5.6 hours in
an orbit with a radius of 12 x 106 m. What is the
magnitude of the gravitational force on the satellite by
planet Cruton?
)
MC =
!
4" 3
r
GT 2
3
4" 2 3
4" 2
r =
1.2 #107 m)
2
2
2(
$11
GT
6.67 #10 NKg%m2 (20160 sec)
(
D!
!
GMm G(6M
! )m
=
2
x2
D
$
( x)
!2
)
6M
m
(D $ x )
!
= 6x 2
D
=x
1+ 6
x = 0.29D
D $ x = 6x
M = 2.52 #1024 Kg
(
)
D = 1+ 6 x
M = 1.066 #10 25 Kg
2
D$ x
x
M
2
4" 3
r
GMC
MC =
#5
!
5
8
!
!
!
6.
(
g=
(
)
(1.066 #10 Kg)
GM 6.67 #10
=
= 26.9 secm2
2
2
6
r
5.0
#10
m
(
)
$11 N%m 2
Kg 2
)
$11 N% m
24
GMm 6.67 #10 Kg 2 (2.52 #10 Kg)(50Kg)
F= 2 =
2
r
(1.2 #10 7 m)
25
2
What is the magnitude of the free-fall acceleration at a
point that is a distance 2R above the surface of the earth,
where R is the radius of the earth?
gs = ge
re2
r2
= ge e 2 = 19 ge = 1.08 secm 2
2
rs
( 3re )
F = 58.3N
!
!
!
3
6
9
9. A spaceship of mass m circles a planet (mass = M) in
an orbit of radius R. How much energy is required to
transfer the spaceship to a circular orbit of radius 3R?
7. A satellite is in a circular orbit about the earth at an
altitude at which air resistance is negligible. Which of
the following statements is true?
12. An object is released
from rest at a distance
h above the surface of
a planet (mass = M,
radius = R << h). With
what speed will the
object
strike
the
surface of the planet?
Disregard
any
dissipative effects of
the atmosphere of the
planet.
,E = E f $ E i
a. There is only one force acting on the satellite.
& GMm ) & GMm )
,E = ( $
+ $ ($
+
' 2( 3r) * ' 2r *
& GMm ) & GMm ) & 3GMm ) & GMm )
,E = (
+ $(
+=(
+$(
+
' 2r * ' 6r * ' 6r * ' 6r *
& GMm )
,E = (
+
' 3r *
(K + U ) P = ETop
1 mv 2 $ GMm = $ GMm
2
r
R+ h
1 v 2 = GM $ GM
2
R
R+ h
1 v 2 = GM ( R + h ) $ GMR
2
R( R + h)
1 v 2 = GMh
2
R (R + h )
v=
10
2GMh
R( R + h)
13
16
!
!
8.
What is the gravitational force on a 20-Kg satellite circling
the earth (Re = 6.4 x 106 m, Me = 6.0 x 1024Kg) with a
period of 5.0 h?
T2 =
r=3
4" 2 3
r
GMC
r=3
(6.67 #10
2
$11 N % m
Kg 2
GMT 2
4" 2
)(6 #10
4"
24
10. A spacecraft (mass = m) orbits a planet (mass = M) in a
circular orbit (radius = R). What is the minimum energy
required to send this spacecraft to a distant point in space
where the gravitational force on the spacecraft by the planet
is negligible?
13. What is the kinetic energy of a 200-kg satellite as it follows
a circular orbit (radius = 8.0 x 106 m) around the earth?
(mass of earth = 6.0 x 1024 kg)
(
Kg)(18000 sec)
& GMm ) & GMm )
,E = ( $
+ $ ($
+
' 2(-) * ' 2r *
& GMm )
,E = 0 + (
+
' 2r *
& GMm )
,E = (
+
' 2r *
2
2
r = 1.486 #107 m
11
)
$11 N %m
24
GMm 6.67 #10 Kg 2 (6 #10 Kg)(200Kg)
K=
=
2r
2(8 #10 6 m)
,E = E f $ E i
2
K = 5.0025 #10 9 J
14
!
17
!
!
F=
(
11. A projectile is launched from the surface of a planet (mass =
M, radius = R). What minimum launch speed is required if
the projectile is to rise to a height of 2R above the surface of
the planet? Disregard any dissipative effects of the
atmosphere.
)
$11 N % m
24
GMm 6.67 #10 Kg 2 (6 #10 Kg)(20Kg)
=
2
r2
(1.486 #10 7 m)
2
14. An object is released from rest when it is a height h above
the surface of a planet of mass M and radius R. What is the
speed of the object just before striking the surface of the
6
planet?
Let h = 4.0 # 10 m, R = 5.0 #10 6 m and M =
4.0 #10 24 kg.
K + U = ET
1 mv 2 $ GMm = $ GMm
2
r
3r
!
1 mv 2 = GMm $ GMm
2
r
3r
F = 36.2N
v =!
v=
1 v 2 = GM $ GM = 2 GM
2
r
3r 3 r
!
& GM )
4GM
v = 2( 2
+=
'3 r *
3r
2
2 6.67 #10$11 NKg%m2
)(4 #10 Kg)(4 #10 m)
24
6
(5 #10 m)(9 #10 m)
6
6
18
!
!
(
!
m
v = 6887 sec
15
12
2GMh
R( R + h)
15. What is the kinetic energy of a 180-kg satellite which
circles the earth with a period of 8.0 h?
(Re = 6.4 x 106 m, Me = 6.0 x 1024Kg)
19. Three 5.0-kg masses are located at points in the xy plane as
shown in the figure. What is the magnitude of the resultant
force (caused by the other two masses) on the mass at the
origin?
16. What is the escape speed from a planet of mass M and
radius R if M = 3.2 x 1023Kg and R = 2.4 x 106m?
2GM
R
v esc =
GMm
K=
2r
(
2
2 6.67 #10$11 NKg% m2
v esc =
Find r.
y
23
(2.4 #10 m)
6
30cm
!
!
m
4217.4 sec
!
!
!
19
Gm1m 2
= 1.853 #10$8 ˆjN
r122
Fiˆ =
Gm 2 m3
= 1.042 #10$8 iˆ N
r232
m1
)(3.2 #10 Kg)
F ˆj
!
m2
Fˆi
!
v esc =
F ˆj =
!
!
40cm
!
m3
x
FNet = iˆ 2 + ˆj 2 = 2.13 #10$8 N
25
22
!
!
!
!
15. What is the kinetic energy of a 180-kg satellite which
circles the earth with a period of 8.0 h?
(Re = 6.4 x 106 m, Me = 6.0 x 1024Kg)
T2 =
r=3
4" 2 3
r
GMC
(6.67 #10
r=3
$11 N % m 2
Kg 2
GMT 2
4" 2
)(6 #10 Kg)(28800sec)
24
20.
17. Two satellites are placed in geosynchronous orbits,
orbits with a period of 24 hours, where each satellite
hovers over a spot on the Earth's equator. Satellite B
has three times the mass of satellite A. What is the
relationship between the magnitudes of the
gravitational forces of the Earth on the two satellites?
GMm A
R2
GM 3mB
FB =
R2
FA =
2
4" 2
FA 1
=
FB 3
A satellite is placed in a geosynchronous orbit. In
this equatorial orbit with a period of 24 hours, the
satellite “hovers” over one point on the equator.
Which statement is true for a satellite in such an
orbit?
C. The satellite is in a state of free fall toward
the Earth.
FA = 13 FB
FB = 3FA
r = 2.03 #10 7 m
!
!
20
26
23
!
!
K=
(
$11 N%m 2
Kg 2
)
18. A satellite of mass m circles a planet of mass M and radius R
in an orbit at a height 2R above the surface of the planet.
What minimum energy is required to change the orbit to
one for which the height of the satellite is 3R above the
surface of the planet?
(6 #10 Kg)(180Kg)
GMm 6.67 #10
=
2r
2(2.03 #10 7 m)
24
,E = E f $ E i
21. Which of the following quantities is conserved
for a planet orbiting a star in a circular orbit?
Only the planet itself is to be taken as the
system; the star is not included.
B.
Energy and angular momentum.
& GMm ) & GMm )
,E = ($
+ $ ($
+
' 2( 4R) * ' 2( 3R) *
& GMm ) & GMm ) & 4GMm ) & 3GMm )
,E = (
+ $(
+=(
+$(
+
' 6R * ' 8R * ' 24R * ' 24R *
& GMm )
,E = (
+
' 24R *
9
K =1.77 #10 J
!
21
24
!
27
24. The planet Venus requires 225 days to orbit the sun,
which has a mass M = 2.0 x 1030 kg, in an almost
circular orbit. Calculate the radius of the orbit and the
orbital speed of Venus as it circles the sun.
22. The figure below shows a planet traveling in a
counterclockwise direction on an elliptical path
around a star located at one focus of the ellipse.
When the planet is at point A,
T2 =
A. its speed is constant.
B. its speed is increasing.
C. its speed is decreasing.
r=
D. its speed is a maximum.
E. its speed is a minimum.
3
4" 2 3
r
GMC
r=3
(6.67 #10
$11 N % m 2
Kg 2
GMT 2
4" 2
)(2 #10 Kg)(1.944 #10 sec)
30
7
2
4" 2
r = 1.1#1011 m
28
31
!
23.
Rui and Jennifer are arguing about whether or not it is
possible to escape the gravitational field of the Earth. Rui
shows Jennifer the system below where mass m is Re (not
the Earth's radius) distant from Earth and Rp (not planet
P's radius) distant from planet P. Rui states that the mass
m has escaped if FP on m = $FE on M . Which one, if either,
is correct, and why?
E
!
!
Re
!
m
!
!
RP
!
!
!
!
m
!
!
2"r
T
2"(1.1#1011 m )
(1.944 #10
v = 3.56 #104
7
sec)
m
sec
!
A. Rui, because the total gravitational force on m is
zero at that point.
B. Rui, because there is no gravitational force from
Earth on m at that point.
C. Rui, because there is no gravitational force on m
from Earth when r > Re.
D. Jennifer, because there is a gravitational force
on m from Earth no matter how great the
distance from the Earth.
E. Jennifer, because the gravitational force from
the Earth can only be blocked by a body that is
larger than the Earth.
Re
v=
P
29
E
v=
RP
P
30
32