Gravitation

Practice Questions:

Question 131: easy

Two masses each equal to \(M\) are moving on a circular path of radius \(R\) about another fixed mass \(M\) (at the centre of the circular path). The gravitational potential energy of the system is:

1. \(-\frac{GM^2}{2R}\)

2. \(-\frac{GM^2}{R}\)

3. \(-\frac{2GM^2}{R}\)

4. \(-\frac{5GM^2}{2R}\)

View Answer

The total GPE of the three-mass system is \(U = -\frac{GMM}{R} - \frac{GMM}{R} - \frac{GMM}{2R} = -\frac{5GM^2}{2R}\) since the outer masses are at a distance of \(2R\) from each other and \(R\) from the center.

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Question 132: easy

An object is weighed at the equator using a physical balance and a spring balance. When the same object is taken to the pole, then corresponding readings on the physical balance and spring balance (also taken there) will respectively:

1. Remain same, increase

2. Increase, remain same

3. Both remain same

4. Both increase

View Answer

A physical balance measures mass, which is constant everywhere. A spring balance measures weight, \(W = mg\). Since gravity \(g\) is greater at the poles, the spring balance reading increases.

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Question 133: easy

If potential energy is assumed to be zero at infinity, then

1. The total energy of an orbiting satellite is negative of its potential energy.

2. The potential energy of an orbiting satellite is twice of its total energy

3. The potential energy of an orbiting satellite is negative of its kinetic energy

4. The total energy of an orbiting satellite is twice of its kinetic energy

View Answer

For an orbiting satellite, Potential Energy \(U = -\frac{GMm}{r}\), Kinetic Energy \(K = \frac{GMm}{2r}\), and Total Energy \(E = -\frac{GMm}{2r}\). This shows that \(U = 2E\).

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Question 134: easy

Two masses each equal to \(M\) are moving on a circular path of radius \(R\) about a common centre. The gravitational force of attraction between the masses has magnitude

1. \(F = \frac{GM^2}{R^2}\)

2. \(F = \frac{GM^2}{4R^2}\)

3. \(F = \frac{4GM^2}{R^2}\)

4. \(F = \frac{GM^2}{2R^2}\)

View Answer

For two identical masses to move on a circular path of radius \(R\) about a common centre, they must always be diametrically opposite. The distance between them is \(2R\). Thus, \(F = \frac{GM^2}{(2R)^2} = \frac{GM^2}{4R^2}\).

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Question 135: easy

The ratio of escape velocity at earth (\(v_e\)) to the escape velocity at a planet (\(v_p\)) whose radius and mean density are twice as that of earth is

1. \(1 : 4\)

2. \(1 : \sqrt{2}\)

3. \(1 : 2\)

4. \(1 : 2\sqrt{2}\)

View Answer

Escape velocity is given by \(v_e = R\sqrt{\frac{8}{3}\pi G\rho}\). Thus, \(v_e \propto R\sqrt{\rho}\). The ratio is \(frac{v_e}{v_p} = \frac{R_e}{R_p}\sqrt{\frac{\rho_e}{\rho_p}} = \frac{1}{2}\sqrt{\frac{1}{2}} = \frac{1}{2\sqrt{2}}\).

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Question 136: easy

Statement A: If a particle is outside a uniform spherical shell or solid sphere with a spherically symmetric internal mass distribution, the sphere attracts the particle.

Statement B: If a particle is inside a uniform spherical shell, the gravitational force on the particle is zero.

Statement C: If a particle is inside a uniform solid sphere, the gravitational force on the particle is zero.

In light of above statements choose the correct option.

1. Only statement C is correct

2. Only statements A and B are correct

3. Only statements C and B are correct

4. Only statements A and C are correct

View Answer

Statements A and B are correct (shell theorem). Inside a uniform solid sphere, the gravitational force is non-zero (except at the center) and varies linearly with distance from the center.

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Question 137: easy

By what percentage will the acceleration due to gravity at a height of 1600 km from the surface of the Earth differ from that on the surface of the Earth? (Take radius of Earth to be 6400 km)

1. 20%

2. 15%

3. 24%

4. 36%

View Answer

Acceleration due to gravity at height \(h\) is \(g' = g\left(\frac{R}{R+h}\right)^2 = g\left(\frac{6400}{8000}\right)^2 = 0.64g\). The percentage difference is \(\frac{g - 0.64g}{g}\times 100% = 36%\).

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Question 138: easy

Assertion (A): The mechanical energy of earth-moon system remains same when a heavenly body passes nearby the earth-moon system.

Reason (R): Force exerted by heavenly body on the earth-moon system is non-conservative.

1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)

2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)

3. (A) is true but (R) is false

4. Both (A) and (R) are false

View Answer

Assertion (A) is false:

An external heavenly body exerts a gravitational force on the Earth-Moon system, causing its mechanical energy to change.

Reason (R) is false: Gravitational force is a conservative force by nature. Therefore, both Assertion (A) and Reason (R) are false.

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Question 139: easy

Assertion (A): Comets move around the sun in elliptical orbits. The gravitational force on the comet due to sun is not normal to the comet’s velocity but the work done by the gravitation force over every complete orbit of the comet is zero.

Reason (R): Gravitational force is a conservative force.

1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)

2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)

3. (3) (A) is true but (R) is false

4. (4) Both (A) and (R) are false

View Answer

Gravitational force is a conservative force. For a conservative force, the work done over a closed path (like a complete elliptical orbit) is zero.

Therefore, both Assertion and Reason are true, and Reason is the correct explanation of the Assertion.

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Question 140: easy

Assertion (A): Two satellites A and B are in the same orbit around the earth, B being behind A. Satellite B can overtake satellite A by increasing its speed.

Reason (R): Orbital speeds of two satellite in same orbit may different

1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)

2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)

3. (A) is true but (R) is false

4. Both (A) and (R) are false

View Answer

Assertion (A) is false. For a satellite to remain in a given orbit, its speed must be constant. Increasing speed will cause the satellite to move to a higher orbit or escape. Reason (R) is false. Satellites in the same orbit must have the same orbital speed to maintain that orbit. Therefore, both (A) and (R) are false.

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