Gravitation Test

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Gravitation Test - 5

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Question 1
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A satellite S is moving in an elliptical orbit around the earth. Mass of satellite is very small compared to the mass of earth.

A
Acceleration of S is always directed towards the centre of the earth.

B
Angular momentum of S about the centre of the earth changes in direction, but its magnitude remains constant.

C
Total mechanical energy of S varies periodically with time.

D
Linear momentum of S remains constant in magnitude.

Solution

As the satellite is moving in a circle about the center of Earth with constant speed, so it has centripetal acceleration only, which is always directed towards the centre of earth.

Force on the satellite is gravitational attraction, which is always directed towards the centre of earth. So torque of this gravitational force about centre of earth is zero. As there is no net torque on the satellite, so angular momentum of satellite remains the same both in magnitude

and direction wrt centre of earth.

Since gravitational force is conservative in nature, mechanical energy of satellite remains constant.

Because angular momentum of satellite is conserved, so speed of satellite is maximum when it is nearest to earth and minimum when it is farthest.
Question 2
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Change in potential energy of a point mass if left free to itself, with time,

A
may be negative

B
may be positive

C
must be negative

D
must be zero

E
may be positive

Solution

Change in potential energy, with time, of a point mass if left free to itself, may be zero (point mass on horizontal surface) or negative (point mass falling freely vertically) .
Question 3
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Orbital angular momentum of Earth,

A
Depends on mass of Earth

B
Does not depend on the direction of projection

C
Changes with time

D
May change from point to point

Solution

Torque on earth = 0 ⇒ Angular momentum is constant

However Angular momentum of Earth does depend on its mass
Question 4
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Statement-1: If earth suddenly stops rotating about its axis, then the value of acceleration due to gravity will become same at all the places.

Statement-2: The value of acceleration due to gravity is independent of rotation of earth.

A
Statement-1 is true but Statement-2 is false.

B
Statement-1 is false but Statement-2 is true.

C
Both Statement-1 and Statement-2 are true and Statement-2 is not the correct explanation of Statement-1.

D
Both Statement-1 and Statement-2 are true and Statement-2 is the correct explanation of Statement-1.

Solution

The value of g at any place is given by the relation, g′ = g – R_eω² cos² λ

Where λ is angle of latitude and ω is the angular velocity of earth. If earth suddenly stops rotating, then

ω = 0 ∴ g′ = g

i.e., the value of g will be same at all places.
Question 5
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Considering earth to be a homogeneous sphere but keeping in mind its spin, Acceleration due to gravity,

A
May change from point to point

B
Does not depend on the direction of projection

C
Remains constant

D
Changes with time

Solution

Due to rotation of Earth,

g_eff = g₀ - ω²R cos2λ

∴ g changes with λ (altitude angle) and g₀ depends on earth mass.