Gravitation Test

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Gravitation Test - 3

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Question 1
1 / -0

If the Earth is replaced by a small point sized particle of the same mass, its geodesic centre being the same, how will the force of gravity change at the present surface of Earth? (Assuming Earth to be a perfect sphere)

A
Force of gravity increases.

B
Force of gravity decreases.

C
Force of gravity varies from place to place.

D
There is no change.

Solution

As the distance and mass remain the same, force of gravity also remains the same.
Question 2
1 / -0

At what height above the Earth's surface does the acceleration due to gravity fall to 1% of its value at the Earth's surface?

A
9R

B
10R

C
99R

D
100R

Solution

Acceleration due to gravity at the surface of the Earth is g =
Let 'h' is the height at which acceleration due to gravity

R + h = 10R
h = 9R
Question 3
1 / -0

Two planets have radii r₁ and r₂ and densities d₁ and d₂, respectively. The ratio of the values of acceleration due to gravity on them will be

A
r₁d₂ : r₂d₁

B
r₁d₁ : r₂d₂

C

D

Solution

If'd' is the density and 'R' is the radius of the of the planet, then acceleration due to gravity is
Or, we say
Question 4
1 / -0

Which of the following statements are true?

(i) Orbital velocity is more than escape velocity.
(ii) Escape velocity is times the orbital velocity.
(iii) Orbital velocity increases with radius.
(iv) A body with a velocity equal to the orbital velocity will orbit the planet in a circular path.

A
(i) and (ii)

B
(ii) and (iv)

C
(iii) and (ii)

D
(iii) and (iv)

Solution

The orbital velocity of a satellite is given by
. Hence, greater the value of r, lesser the orbital velocity.
Escape velocity is given by
If the satellite is orbiting near the surface of the planet, then the orbital velocity is given by
(where r = R )
Hence
If a body is projected with a speed equal to the orbital velocity, then it will move in a circular path.
Question 5
1 / -0

A satellite going around the Earth in a circular orbit loses some energy due to a collision. If its speed is v and distance from the Earth is d, which of the following is correct?

A
If d increases, v will increase.

B
If d increases, v will decrease.

C
If d decreases, v will decrease.

D
If d decreases, v will increase.

Solution

The orbital velocity of a satellite is given by . Hence, as the value of d decreases, orbital velocity increases.
Question 6
1 / -0

A satellite having circular orbit about the Earth has a kinetic energy E_x. What is the minimum amount of energy to be added so that it escapes from the Earth?

A

B

C
E_x

D
2 E_x

Solution

Orbital velocity of the satellite near the Earth's surface is
Kinetic energy of the satellite is
Escape velocity at the surface of the Earth is
Kinetic energy of the satellite is
K' = 2E_x
Hence, increase in the kinetic energy is
Question 7
1 / -0

Two planets 'A' and 'B' of same mass and same radius are shown in the figure. ρ₁and ρ₂ are the densities of the materials in the planets and ρ₁ > ρ₂. If the values of acceleration due to gravity on the surface of the planets A and B are g_Aand g_B respectively, then

A
g_A = g_B

B
g_A < g_B

C
g_A > g_B

D
Given information is not sufficient

Solution

Acceleration due to gravity g is given by g ∝ ρR, where R is the radius of the planet g ∝ ρR, where R is the radius of the planet
Hence, we cannot deduce the result.
Question 8
1 / -0

The gravitational force between two bodies is 'F'. If the mass of each body is doubled and distance between them is halved, then the force between them will be

A

B

C
16F

D

Solution

As
As
Force in the second case is

F' = 16F
Question 9
1 / -0

A satellite is revolving around the planet in an orbit of radius 'r' with orbital velocity 'v'. It is shifted to the new orbital radius 2r, the new orbital velocity is

A
v

B
2v

C

D

Solution

For the satellite revolving around the Earth, the gravitational force between the satellite and the Earth provides necessary centripetal force.

Hence,, let v' is the new orbital velocity.
Question 10
1 / -0

Two bodies - P and Q - of masses 1 kg and 16 kg, respectively, are placed at a distance of 10 m from each other. Where should a body of mass 1 kg be placed between them so that the gravitational forces due to P and Q on it are equal and opposite?

A
2 m from body P

B
4 m from body P

C
2 m from body Q

D
4 m from body Q

Solution

Let a body R of mass 1 kg be placed at distance x from P.
Then, R will be at a distance of (10 - x) from Q. Let force F₁ act between bodies P and Q.

Then, F₁ = Let force F₂ act between bodies R and Q.

Then, F₂ = G Since the forces acting between bodies P and R, and between bodies Q and R are equal and opposite, therefore
F₁ = F₂
i.e. GG

Or,

Taking square root on both sides, we get

or 10 - x = 4x

Or, 5x = 10 or x = 2 m
∴ Distance between P and R is 2 m. So, body of mass 1 kg should be placed 2 m from body P.
Hence, option (1) is correct.
Question 11
1 / -0

Consider a heavenly body whose mass is thrice that of Earth and radius is twice that of Earth. What is the weight of a block on this heavenly body if the weight of the block on Earth is 400 N?

A
600 N

B
300 N

C
200 N

D
1200 N

Solution

Weight of the block on Earth, W_e = = 400 N
Weight of the block on the heavenly body:
M = 3M_e
R = 2R_eW = W = W = 400 × W = 100 3
W = 300 N
Question 12
1 / -0

Earth is assumed to be a sphere of radius R. A platform is arranged at a height R from the surface of Earth. The escape velocity of a body from this platform is fv_e, where v_e is its escape velocity from the surface of Earth. The value of f is

A

B

C

D

Solution

The value of g at a height R_efrom Earth's surface is .

So,
Question 13
1 / -0

Which of the following graphs represents the motion of a planet moving around the sun?

A

B

C

D

Solution

T² is directly proportional to R³ (Kepler's third law).
Hence, option (1) is correct.
Question 14
1 / -0

The pressure of water at the bottom in a lake is times the pressure at half the depth, where the water barometer reads 10 m. The depth of the lake is

A
15 m

B
infinite

C
20 m

D
10 m

Solution

Let the height of the water column be 'h'.
Pressure at half depth, P₁ = 10
Here, 10 m is the reading of the barometer.
Pressure at the bottom, P₂ = h
According to the given condition,
Question 15
1 / -0

An iron ball is weighed in air and then in water by a spring balance. Where does it weigh more?

A
Its weight in air is more than in water.

B
Its weight in water is more that in air.

C
Its weight is the same both in air and water.

D
Its weight is zero in water.

Solution

Apparent weight of the body = True weight of the body - Up thrust
The greater is the up thrust, the lesser is the apparent weight.
Question 16
1 / -0

A boy carries a fish in one hand and a bucket of water in the other hand. If he places the fish in the bucket, the weight now carried by him

A
is less than before

B
is more than before

C
is the same as before

D
depends upon his speed

Solution

Total downward force in the both cases will be the same.

When the fish is placed in the bucket, then total downward force acting on the system of fish + bucket = (m + M)g + F_th - F_th
= (m + M)g
Question 17
1 / -0

A fisherman hooks an old log of wood of weight 12 N and volume 1000 cm³. He pulls the log half way out of water. The tension in the string at this instant is

A
7 N

B
8 N

C
10 N

D
12 N

Solution

Let T be the tension in the string.
Forces acting on the wooden log are: Weight (Mg) acting downwards, up thrust acting upwards and the tension in the string T acting in upward direction.

; where ρ is the density of water and V is the volume of the log

T = 7 N
Question 18
1 / -0

A solid cylinder floats on water with 4/5 of its volume inside water. The density of the material of the cylinder is

A
800 kg/m³

B
750 kg/m³

C
600 kg/m³

D
500 kg/m³

Solution

If the density of the solid cylinder is and is the density of water, then , where V_in is the volume of the cylinder inside the water.
Question 19
1 / -0

Two solids A and B float on a liquid. It is observed that A floats with (1/2) of its volume immersed and B floats with (2/3) of its volume immersed. Compare the densities of A and B.

A
4 : 3

B
2 : 3

C
3 : 4

D
1 : 3

Solution

For the first case:
If the density of the solid A is and is the density of the liquid, then
----- 1
Here, V_A is the volume of the solid A inside the liquid.
For the second case:
If the density of the solid B is and is the density of the liquid, then
----- 2
Here, V_A is the volume of the solid B inside the liquid.
From 1 and 2, we have
Question 20
1 / -0

The spring balance A reads 2 kg with a block m suspended from it. A balance B reads 5 kg when a beaker with liquid is put on the pan of the balance. The two balances are now so arranged that the hanging mass is inside the liquid in the beaker, as shown in the figure. In this situation,

A
the balance A will read more than 2 kg

B
the balance B will read more than 3 kg

C
the balance A will read less than 2 kg and balance B will read more than 5 kg

D
the balances A and B will read 2 kg and 5 kg, respectively

Solution

The liquid will apply an upthrust on block m. An equal force will be exerted (from Newton's third law) on the liquid. Hence, A will read less than 2 kg and B will read more than 5 kg. Therefore, the correct option is (3).