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 :