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# Gravitation: Important Questions

### Numerical Problems

1. An unknown planet orbits the Sun with distance twice the semi major axis distance of the Earth’s orbit. If the Earth’s time period is T1, what is the time period of this unknown planet?

Solution

The earth’s time period = T1

Semi major axis distance of earth = α1

Time period of unknown planet = T2

Semi major axis of unknown planet = 2α1,

Kepler’s III law T2 =2√2T1

Ans: T2 = 2 2T1

2. Assume that you are in another solar system and provided with the set of data given below consisting of the planets’ semi major axes and time periods. Can you infer the relation connecting semi major axis and time period? Solution

For planet kurinji

Time period T12  α13

T1=2;

α1 = 8

α1 = 2.22 => 2.T12

Similarly for other planets

α2 = 2.32 => 2T22

α3 = 2.42 => 2T32

α4 = 2.52 => 2T42

α  2T2

Ans: a  2T2

3. If the masses and mutual distance between the two objects are doubled, what is the change in the gravitational force between them? Now the masses and distance between two objects is doubled. F2 = F1

No change in gravitational force when masses and distance are doubled.

Ans: No change

4. Two bodies of masses m and 4m are placed at a distance r. Calculate the gravitational potential at a point on the line joining them where the gravitional field is zero. Ans: V = −9Gm/r

5. If the ratio of the orbital distance of two planets d1/d2 =2, what is the ratio of gravitational field experienced by these two planets? Ans: E2 = 4 E1

6. The Moon Io orbits Jupiter once in 1.769 days. The orbital radius of the Moon Io is 421700 km. Calculate the mass of Jupiter?

Solution

Time period of moon Io = 1.769 days

= 1.769 × 24 × 60 × 60s.

Orbital radius of the moon (r) = 4.217 × 108m. M = 1.898 × 1027kg

Ans: 1.898 ×1027 kg

7. If the angular momentum of a planet is given by = 5 t 2iˆ 6tjˆ +3kˆ . What is the torque experienced by the planet? Will the torque be in the same direction as that of the angular momentum?  8. Four particles, each of mass M and equidistant from each other, move along a circle of radius R under the action of their mutual gravitational attraction. Calculate the speed of each particle  9. Suppose unknowingly you wrote the universal gravitational constant value as G = 6.67x1011 instead of the correct value G = 6.67x10-11 ,  what is the acceleration due to gravity g' for this incorrect G? According to this new acceleration due to gravity, what will be your weight W'?

Solution:

Mass of the earth M = 6.024 × 1024 kg

Radius of the earth R = 6.4 × 106m

Gravitational constant G' (Incorrect) = 6.67 × 1011

Gravitational constant G (Correct) = 6.67 × 1011

Acceleration due to gravity g' = ? Ans: g' =1022 g, W' = 1022W

10. Calculate the gravitational field at point O due to three masses m1,m2 and m3 whose positions are given by the following figure. If the masses m1 and m2 are equal what is the change in gravitational field at the point O?  11. What is the gravitational potential energy of the Earth and Sun? The Earth to Sun distance is around 150 million km. The mass of the Earth is 5.9 × 1024 kg and mass of the Sun is 1.9 × 1030 kg.

Solution:

The distance between sun & Earth R= 150 million km = 150 × 1011m.

The mass of the sun Ms = 1.9 × 1030 kg

The mass of the Earth Me = 5.9 × 1024 kg

The gravitational potential energy U = ? U = -4.985 × 1032 J

P.E U = -49.84 × 1031 J

Ans: V = -49.84 × 1032 Joule

12. Earth revolves around the Sun at 30 km s−1. Calculate the kinetic energy of the Earth. In the previous example you calculated the potential energy of the Earth. What is the total energy of the Earth in that case? Is the total energy positive? Give reasons.

Solution:

The velocity of earth revolves around the sun = v = 30kms (or) 30 × 103 ms-1

K.E of the earth = 1/2 mv2 = ?

Mass of the earth = 5.9 × 1024 kg

K.E = 1/2 × 5.9 × 1024 × 30 × 30 × 1O+3 × 10+3

= 1/2 × 5.9 × 1024 × 900 × 106

= 2655 × 1030 (or) 26.55 × 1032J

T.E = P.E + K.E

= - 49.84 × 1032 + 26.55 × 1032

E = - 23.29 × 1032 J

Ans: K.E = 26.5 × 1032 J

E = −23.29 × 1032 J

( - ) ve implies that Earth is bounded with Sun

13. An object is thrown from Earth in such a way that it reaches a point at infinity with non-zero kinetic energy with what velocity should the object be thrown from Earth?  14. Suppose we go 200 km above and below the surface of the Earth, what are the g values at these two points? In which case, is the value of g small? Ans: gdown = 0.96 g

gup = 0.94 g

15.  Calculate the change in g value in your district of Tamil nadu. (Hint: Get the latitude of your district of Tamil nadu from the Google). What is the difference in g values at Chennai and Kanyakumari?

Solution:

Chennai

g'c = g - ω2 R cos2λ

= 9.8 - (3.4 × 10-2)

= 9.7677 ms-2

g'K =9.8 - (3.4x 10-2) × (cos0.1396)2 = 9.748 ms-2

Difference in g value Δg = 0.0331 ms-2

ω2R = (2 × (3.14/86400) )2 × 6400 × 103

Ans: gchennai = 9.767 m s-2

gKanyakumari = 9.798 m s-2

Δg = 0.031 m s-2

## Conceptual Questions

1. In the following, what are the quantities which that are conserved?

a) Linear momentum of planet

b) Angular momentum of planet

c) Total energy of planet

d) Potential energy of a planet

Answer: (b & d) Angular momentum of planet, Potential energy of a planet]

2. The work done by Sun on Earth in one year will be

a) Zero

b) Non zero

c) positive

d) negative

3. The work done by Sun on Earth at any finite interval of time is

a) positive, negative or zero

b) Strictly positive

c) Strictly negative

d) It is always zero

[Answer: (d) It is always zero]

4. If a comet suddenly hits the Moon and imparts energy which is more than the total energy of the Moon, what will happen?

If a comet hits the moon with large mass and with large velocity may destroy the moon completely or its impact makes the moon, go out of the orbit.

5.  If the Earth’s pull on the Moon suddenly disappears, what will happen to the Moon?

If the gravitational force suddenly disappears, moon will stop revolving around the earth and it will move in a direction tangential to its original orbit with a speed with which it was revolving around the earth.

6.  If the Earth has no tilt, what happens to the seasons of the Earth?

If the Earth has no tilt, there will be no seasons like now and the duration of day and night \vi! be equal throughout the year.

7.  A student was asked a question ‘why are there summer and winter for us? He replied as ‘since Earth is orbiting in an elliptical orbit, when the Earth is very far away from the Sun(aphelion) there will be winter, when the Earth is nearer to the Sun(perihelion) there will be winter’. Is this answer correct? If not, what is the correct explanation for the occurrence of summer and winter?

No, The seasons in the Earth arise due to tit; rotation of Earth around the Sun with 23.5° tilt Due to this 23.5° tilt, when the northern parto Earth is farther to the Sun, the southern part: nearer to the Sun. So when it is summer in tb northern hemisphere, the southern hemisphet experience winter.

8.  The following photographs are taken from the recent lunar eclipse which occurred on January 31, 2018. Is it possible to prove that Earth is a sphere from these photographs? 