1basicforcesandkeplers Laws

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Basic Forces and Kepler’s Laws

1.

Two identical spheres of gold are in contact with each other. The gravitational

1) Directly proportional to the square of their radius 2) Directly proportional to the cube of their radius

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3) Directly proportional to the fourth power of their radius

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force of attraction between them is

4) Inversely proportional to the square of their radius. 2.

Find the false statement

1) Gravitational force acts along the line joining the two interacting particles. 2) Gravitational force is independent of medium.

3) Gravitational force forms an action- reaction pair.

4) Gravitational force does not obey the principle of superposition. Among the following find the wrong statement

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3.

1) Law of gravitation is framed using Newton’s third law of motion.

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2) Law of gravitation cannot explain why gravity exists.

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3) Law of gravitation does not explain the presence of force even when the particles are not in physical contact.

Law of gravitation is not applicable if A) Velocity of moving objects are comparable to velocity of light.

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4.

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4) When the range is long, gravitational force becomes repulsive.

B) Gravitational field between objects whose masses are greater than the mass of sun.

1) A is true, B is false

2) A is false, B is true

3) Both A & B are true

4) Both A & B are false

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Statement A: Modification of space by a mass particle is called gravitational field Statement B: Law of gravitation is a consequence of “Action at a distance concept”. 2) A is false, B is true

3) Both A & B are true

4) Both A & B are false

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6.

1) A is true, B is false

The earth revolves round the sun in an elliptical orbit, its speed is

2) Greatest when it is closest to the sun

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1) Going on decreasing continuously

3) Greatest when it is farthest from the sun 4) Constant at all the points on the orbit 7.

How do you divide total mass M into two parts so that the gravitational force between them at a given distance is maximum? 1)

8.

M 3 , 4 4

2)

M 2M , 3 3

3)

M 4M , 5 5

4)

M M , 2 2

An infinite number of particles each of mass 1kg are placed on the positive xaxis at 1m, 2m, 4m, 8m…. from the origin. The magnitude of the resultant

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gravitational force on 1kg mass kept at the origin is 3)

3G 4

4)

4G 3

Two metal spheres of same material and radius ‘r’ are in contact with each

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9.

2) G

ks

1) 0

other. The gravitational force of attraction between the spheres is given by (k in

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a constant).

2) F = k/r2

3) F = k/4r2

4) F = kr2

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1) F = kr4

10. Three uniform spheres each having mass ‘m’ and radius ‘R’ kept in such a way that each two touches the other. The magnitude of the gravitational force on any sphere, due to the other two is 1) 3

Gm2 4R2

2)

Gm 2 4R2

3)

3Gm 2R2

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4) 3

Gm R2

www.sakshieducation.com 11. Two particles each of mass ‘m’ move in a circle of radius ‘r’ under the action of their mutual gravitational attraction. Then speed of each particle is Gm r

1)

2)

Gm 4r

3)

Gm 2r

4)

2Gm r

12. There are two bodies of masses 100kg and 10,000 kg separated by a distance of

gravitational field be zero? (1) 1/9m

(2) 1/10m

(3) 1/11m

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1 metre. At what distance from the smaller body, will the intensity of the

(4) 10/11m

the gravitational force is 1) Decreases by 36%

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13. If the distance between two bodies is increased by 25%, then the % change in

2) Increases by 36 %

3) Increases by 64%

4) Decreases by 64 %

14. The time period of revolution of a planet A around the sun is 8 times that of another planet B. The distance of planet A from the sun is how many times greater than that of the planet B from the sun 2) 3

3) 4

4) 5

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1) 2

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15. A planet revolves round the sun. Its velocity at the nearest point, distant d1 from sun, is v1. The velocity of the planet at the farthest point distant d2 from

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sun will be.

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d12 v1 (1) 2 d2

dv (2) 2 1 d1

dv (3) 1 1 d2

d 22 v1 (4) 2 d1

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16. A tunnel is dug along a diameter of earth. The force on a particle of mass m distant x from the centre in this tunnel will be

(1)

GM e m R3 x

(2)

GM e mR 3 x

(3)

GM e mx R2

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(4)

GM e mx R3

www.sakshieducation.com 17. Imagine a light planet is revolving round a very massive star in a circular orbit of radius R with a time period of revolution T. If the gravitational force of attraction between the star and planet is proportional to R-n, then T2 is proportional to 2) Rn+2

3) Rn–1

4) Rn–2

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1) Rn+1

18. If two planets have their radii in the ratio x:y and densities in the ratio m:n, then the acceleration due to gravity on them are in the ratio (2) mx / ny

(3) ny / mx

(4) my / nx

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(1) nx / my

19. If a planet of mass ‘m’ is revolving around the sun in a circular orbit of radius ‘r’ with time period T, then the mass of the sun is 1)

4π 2 r 3 GT

2)

4π 2 r 3 GT 2

3)

4π 2 r GT

4)

4π 2 r 3 G 2T 2

21. A satellite is launched into circular orbit of radius R around the earth while a second satellite is launched into an orbit of Radius 1.02R. The percentage

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change in the time periods of the two satellites is 2) 1.0

3) 1.5

4) 3

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1) 0.7

22. In a double star system, two stars of masses m1 and m2 separated by a distance

2)

Gm2 d3

3)

G ( m1 + m2 ) d3

4)

d3 G (m1 + m2 )

The magnitudes of the gravitational field at distance and from the centre of a

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23.

Gm1 d3

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1)

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d rotate about their centre of mass. Then their common angular velocity would be

uniform sphere of radius R and mass M are and respectively. Then:

E r 1. 1 = 1 if r1 < R and r2 < R E2 r2

3.

E1 r13 if r1 < R and r2 < R = E2 r23

E1 r22 2. = if r1 > R and r2 > R E2 r12

4.

E1 r12 = if r1 < R and r2 < R E2 r22

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www.sakshieducation.com 24.

A solid sphere of uniform density and radius R exerts attractive gravitational force F1 on a particle placed at a distance 2R from the centre of the sphere. Now a spherical cavity of radius

R is made as shown. The remaining part of 2

the sphere exerts a force F2 on the same particle. Then F2 F1 2) 7: 9

3) 5: 9

25. Newton's law of gravitation is universal because 1) It is always attractive

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2) It acts on all heavenly bodies and particles

4) 1: 9

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1) 2: 9

3) It acts on all the masses at all the distances and is not affected by the medium 4) None of these

Key

2) 4

3) 4

11) 2

12) 3

13) 1

4) 3

5) 3

6) 2

7) 4

8) 4

16) 4

17) 1 18) 2

9) 1

10) 1

15) 3

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14) 3

22) 3

23) 2

25) 3

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21) 4

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1) 3

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19) 2 20) 2

www.sakshieducation.com Hints

Let the mass M is divided into x and (M -x ) As F ∝ x(M − x) or F = K (Mx - x2)

∴

dF =0 dx

dF M = k ( M − 2 x) = 0 ⇒ x = dx 2

The two parts are 8.

F=

M M , 2 2

G (1× 1) G (1× 1) G (1×1) + + + ........ 2 1 22 42

⎛ 1 1 ⎞ F = G ⎜ 1 + + + ......... ⎟ ⎝ 4 16 ⎠

⎛ 1 ⎞ 4G =G⎜ ⎟= − 1 1/ 4 ⎝ ⎠ 3

a ⎞ ⎛ ⎜∴ SG − P = ⎟ 1− r ⎠ ⎝

⎛4 ⎞ ⎛4 ⎞ G ⎜ π r3 ⎟ ρ ⎜ π r3 ⎟ ρ Gm1m2 3 ⎠ ⎝3 ⎠ F= = ⎝ 2 2 r r

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F ∝ r 4 or F = Kr 4

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9.

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For Fmax,

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7.

Gm 2 Gm 2 3 = (2 R) 2 4R2

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= 3

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10. FR = 3 F FR = F 2 + F 2 + 2 F 2 cos 60

11. The gravitational force between the two particles provides the necessary centripetal

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force for rotation Gm 2 mv 2 Gm = ⇒V= 2 (2r ) r 4r

12. Let the intensity be zero at a distance from 100kg mass. Then, G × 100 G × 10, 000 = x2 (1 − x) 2

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www.sakshieducation.com Taking root 13.

1 10 1 or x = m = 11 x 1− x

F1 d 22 = F2 d12

14. T 2 ∝ R 3

v1d1 d2

16. F = G = Or F =

Or F =

M xm x2

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Or v2 =

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15. According to law of conservation of angular momentum v1d1 = v2 d 2

Gm M ⎛4 ⎞ . × ⎜ π x3 ⎟ 2 x ⎛4 3⎞ ⎝3 ⎠ ⎜ πR ⎟ ⎝3 ⎠ GMm x R3

GMm ⎛ 2π ⎞ 2 3 17. If =m R ⎜ ⎟ ⇒T ∝ R 2 R T ⎝ ⎠ 2

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2

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GMm ⎛ 2π ⎞ But =m R ⎜ ⎟ n R ⎝ T ⎠

T 2 ∝ R h +1

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18. g ∝ Rd , g1 = k .xm, g 2 = k . yn

GM s m =m r2

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19.

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∴ g1 : g 2 : : xm : yn

⇒ Ms =

⎛ 2π ⎞ r⎜ ⎟ ⎝ T ⎠

2

4π 2 r 3 GT 2

20. FR = 3 F =

3 Gm a2

But FR = Fcp for rotation

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www.sakshieducation.com Gm ⎛ a ⎞ 2 = m⎜ ⎟ω 2 a ⎝ 3⎠

3

ω=

3Gm a3

21. T 2 ∝ r 3 ΔT Δr =3 T r

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2

22.

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ΔT 3 ⎛ Δr ⎞ 3 ×100 = ⎜ × 100 ⎟ = (0.02) ×100 = 3 T 2⎝ r ⎠ 2

⎛ m2 d ⎞ 2 Gm1m2 = m1d1ω 2 = m1 ⎜ ⎟ω 2 d ⎝ m1 + m2 ⎠ G (m1 + m2 ) d3

⇒ω =

23. : If r ≤ R, then E =

GM ( r ) ⇒ E ∞r R3

E1 r22 = if r1 > R and r2 > R E2 r12

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⇒

24. Gravitational GMm

=

force

on

mass

m

due

to

whole

GMm 4R2

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F1 =

GM 1 (r ) ⇒ E∞ 2 2 r r

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If r ≥ R, then E =

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E1 r1 = if r1 < R and r2 < R E2 r2

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( 2R )

2

Gravitational force due to the removed sphere, M ×m GMm GMm GMm 8 F2 = F1 − F21 = − = F21 = 2 2 4R2 18R2 18R R⎞ ⎛ ⎜R+ ⎟ 2⎠ ⎝ G

F=

7 GM m F 7 ⇒ 2 = 2 36 R F1 9

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sphere