q08.gravity. 1.suppose you have a pendulum clock which keeps correct time on earth (acceleration due...
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Q08. Gravity
1. Suppose you have a pendulum clock which keeps correct
time on Earth (acceleration due to gravity = 9.8 m/s2).
Without changing the clock, you take it to the Moon
(acceleration due to gravity = 1.6 m/s2). For every hour
interval (on Earth) the Moon clock will record:
1. (9.8/1.6) h
2. 1 h
3.
4. (1.6/9.8) h
5.
9.8 / 1.6 h
1.6 / 9.8 h
2. The mass density of a certain planet has spherical symmetry
but varies in such a way that the mass inside every spherical
surface with center at the center of the planet is proportional to
the radius of the surface. If r is the distance from the center
of the planet to a point mass inside the planet, the gravitational
force on the mass is :
1. not dependent on r
2. proportional to r2
3. proportional to r
4. proportional to 1/ r
5. proportional to 1/ r2
Mass inside radius R :
2
04
RM R r r d r R 2
1
r
2
m M rf G
r 2
r
r
1
r
Force due to mass shell beyond m vanishes.
Force due to mass sphere beneath m is the same as if it’s a
point mass M(r) placed at the center.
3. Each of the four corners of a square with edge a is occupied by a
point mass m. There is a fifth mass, also m, at the center of the
square. To remove the mass from the center to a point far away
the work that must be done by an external agent is given by :
1. 4Gm2/a
2. –4Gm2/a
3. (42) Gm2/a
4. –(42) Gm2/a
5. 4Gm2/a2
Potential energy of m at center of square is :
2
0 42
2
mU G
a
28
2
Gm
a
To bring it to infinity, where 0U requires work
0 0extW W
2
4 2Gm
a
0U
0U U
2
4 2Gm
a
4. A projectile is fired straight upward from Earth's surface with
a speed that is half the escape speed. If R is the radius of
Earth, the highest altitude reached, measured from the surface,
is
1. R/4
2. R/3
3. R/2
4. R
5. 2R
21
2 2escv
K m
By definition:
Maximum height H is given by:
21
2E
esc
Mv G
R
E K U R
21
2 2escv
K m
1 1 1
4R R R H
1 3
4R H R
3
RH
U R H
1 1EGmM
R H R
( Energy conservation )
5. An object is dropped from an altitude of one Earth radius
above Earth's surface. If M is the mass of Earth and R is its
radius the speed of the object just before it hits Earth is given
by :
1.
2.
3.
4.
5.
/GM R
/ 2GM R
2 /GM R
2/GM R
2/ 2GM R
2E U R K U R
21 1 1
2 2mv GmM
R R
1
2
GmM
R
GMv
R
Energy conservation :
6. A planet in another solar system orbits a star with a mass of 4.0
1030 kg. At one point in its orbit it is 250 106 km from the
star and is moving at 35 km/s. Take the universal gravitational
constant to be 6.67 10–11 m2/s2 kg and calculate the
semimajor axis of the planet's orbit. The result is :
1. 79 106 km
2. 160 106 km
3. 290 106 km
4. 320 106 km
5. 590 106 km
E K U
84.547 10 M ( M = mass of planet )
Using2
SM ME G
a ( MS = mass of star, a = semimajor axis ) , we have
11 30
8
6.67 10 4.0 10
2 4.547 10a
112.93 10 m 6290 10 km
3023 11
9
4.0 10135 10 6.67 10
2 250 10M
Energy conservation :
7. A spaceship is returning to Earth with its engine turned off.
Consider only the gravitational field of Earth and let M be the
mass of Earth, m be the mass of the spaceship, and R be the
distance from the center of Earth. In moving from position 1
to position 2 the kinetic energy of the spaceship increases by :
1.
2.
3.
4.
5.
2 22 1
1 1GMm
R R
2 21 2
1 1GMm
R R
1 22 21 2
R RGMm
R R
1 2
1 2
R RGMm
R R
2 1
1 2
R RGMm
R R
1 1 2 2E K U K U
2 1 1 2K K U U 1 2
1 1GMm
R R
1 2
1 2
R RGMm
R R
Energy conservation :
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