physics : class11 Gravitational Force of a Uniform Sphere : Gravitational Force : Gravitation MCQ Questions and Answers
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Gravitational Force of a Uniform Sphere
MCQ questions and answers.
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MCQ Questions
1. A uniform solid sphere of mass M and radius R attracts a small mass m placed at a distance 2R from its centre. If the gravitational force on m at the surface of the sphere is F, what is the force at 2R?
A.
F/2
B.
F/4
C.
F/8
D.
F/16
Correct Answer:
F/4
Explanation:
For a point outside a uniform sphere, the sphere acts gravitationally as a point mass M concentrated at its centre. Thus, the force varies as 1/r². Comparing distances R and 2R gives F(2R) = F(R)(R/2R)² = F/4.
2. A uniform solid sphere has mass M and radius R. What is the magnitude of the gravitational force exerted by the sphere on a particle of mass m placed at a distance R/3 from its centre?
A.
GMm/(9R²)
B.
GMm/(3R²)
C.
GMm/R²
D.
3GMm/R²
Correct Answer:
GMm/(9R²)
Explanation:
Inside a uniform sphere, only the mass enclosed within radius r contributes to the net force. Since enclosed mass is M(r³/R³), the force is Gm[M(r³/R³)]/r² = GMmr/R³. At r = R/3, this becomes GMm/(9R²).
3. A particle of mass m is located at a distance R/2 from the centre of a uniform solid sphere of mass M and radius R. What is the ratio of the actual gravitational force to the force that would be produced if the entire mass M were concentrated at the sphere's centre?
A.
1/2
B.
1/4
C.
1/8
D.
1/16
Correct Answer:
1/8
Explanation:
At r = R/2, the enclosed mass is M(r/R)³ = M/8. The actual force is therefore one-eighth of the force produced by a point mass M at the same distance.
4. A uniform solid sphere has density ρ and radius R. At an internal point whose position vector from the centre is r⃗, what is the gravitational force on a particle of mass m?
A.
−(4πGρm/3)r⃗
B.
−(4πGρm/3R³)r⃗
C.
−(Gρm/R²)r⃗
D.
−(4πGρm/3r²)r⃗
Correct Answer:
−(4πGρm/3)r⃗
Explanation:
The mass enclosed within radius r is (4πρr³/3). The field inside is −G[4πρr³/3]r⃗/r³ = −(4πGρ/3)r⃗. Multiplying by m gives the stated force.
5. For a uniform solid sphere, how does the magnitude of the gravitational force on a particle vary as the particle moves from the centre to the surface along a radius?
A.
It remains constant throughout
B.
It decreases linearly with distance
C.
It increases linearly with distance
D.
It increases as the square of the distance
Correct Answer:
It increases linearly with distance
Explanation:
Within a uniform sphere, the enclosed mass is proportional to r³, while the inverse-square factor contributes 1/r². Hence the net force is proportional to r, increasing linearly from zero at the centre to its maximum value at the surface.
6. Two uniform solid spheres have the same density. Their radii are R and 2R, respectively. What is the ratio of the gravitational force at the surface of the larger sphere to that at the surface of the smaller sphere for identical test masses?
A.
1/2
B.
1
C.
2
D.
4
Correct Answer:
2
Explanation:
For a uniform sphere, surface acceleration is GM/R². Since M = (4πρR³/3), surface acceleration becomes (4πGρ/3)R. At the same density, it is proportional to radius, so doubling the radius doubles the force.
7. A uniform solid sphere has mass M and radius R. A test mass m is placed at a distance R/2 from its centre. If the gravitational force at the surface is F, what is the force on m at that internal point?
A.
F/4
B.
F/2
C.
F
D.
2F
Correct Answer:
F/2
Explanation:
Inside a uniform sphere, force varies linearly with distance from the centre. Therefore, at r = R/2, the force is (R/2)/R times the surface force, giving F/2.
8. A spherical cavity of radius a is carved from a uniform solid sphere of density ρ. The centre of the cavity is displaced from the centre of the original sphere by vector d⃗. What is the gravitational force on a mass m placed anywhere inside the cavity?
A.
(4πGρm/3)d⃗
B.
−(4πGρm/3)d⃗
C.
(4πGρm/3)(a/R)d⃗
D.
Zero for every noncentral cavity
Correct Answer:
(4πGρm/3)d⃗
Explanation:
Use superposition: the field of the complete sphere is −(4πGρ/3)r⃗, while the removed sphere contributes −(4πGρ/3)(r⃗ − d⃗). Subtracting the latter from the former leaves the uniform field (4πGρ/3)d⃗, so the force is independent of position inside the cavity.
9. A uniform solid sphere of mass M and radius R exerts a force F on a test mass at a point outside the sphere, at distance 3R from its centre. If the sphere is replaced by another spherically symmetric body having the same total mass M but a different radius, while the test point remains at distance 3R from the centre, what is the new force?
A.
F/9
B.
F/3
C.
F
D.
It depends on the new radius
Correct Answer:
F
Explanation:
For any spherically symmetric mass distribution, the gravitational force at an external point depends only on the total mass and the distance from the centre. Since both remain unchanged, the force remains F.
10. A small particle is placed exactly at the centre of a uniform solid sphere. Which statement correctly describes the net gravitational force on the particle?
A.
It is GMm/R² toward any selected surface point
B.
It is GMm/(2R²) because only half the sphere contributes
C.
It is zero because of spherical symmetry
D.
It is infinite because the distance from some mass elements is zero
Correct Answer:
It is zero because of spherical symmetry
Explanation:
At the centre, every mass element has an equal and oppositely placed counterpart producing an equal opposite force. All contributions cancel, so the net gravitational force is zero.
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