physics : class11 Gravitational Force in Spherical Cavities : Gravitational Force : Gravitation MCQ Questions and Answers
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Gravitational Force in Spherical Cavities
MCQ questions and answers.
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MCQ Questions
1. A uniform-density solid sphere of density ρ has a spherical cavity of radius a whose center C is displaced from the original sphere's center O by vector d. The cavity is completely inside the sphere. What is the gravitational field at any point inside the cavity?
A.
−(4πGρ/3)r, where r is measured from O
B.
(4πGρ/3)d, independent of the point inside the cavity
C.
−(4πGρ/3)d, independent of the point inside the cavity
D.
Zero only at the geometrical center of the cavity
Correct Answer:
(4πGρ/3)d, independent of the point inside the cavity
Explanation:
Use superposition: the original full sphere produces field −(4πGρ/3)r, while the removed spherical region produces the opposite of its own field, +(4πGρ/3)(r−d). Their sum is (4πGρ/3)d everywhere inside the cavity.
2. A spherical cavity is carved out of a uniform solid sphere. The cavity center is displaced from the original sphere center by distance d. If the material density is ρ, what is the magnitude of the gravitational force on a particle of mass m placed anywhere inside the cavity?
A.
(4πGρmd)/3
B.
(4πGρma)/3
C.
(4πGρm(d+a))/3
D.
(4πGρm(d−a))/3
Correct Answer:
(4πGρmd)/3
Explanation:
The gravitational field inside an off-center spherical cavity is uniform and has magnitude (4πGρd)/3. Multiplying by the test mass m gives the force magnitude, which is independent of the cavity radius a and the particle's position.
3. A spherical cavity of radius a is made concentrically in a uniform solid sphere of radius R and density ρ, with a<R. What is the gravitational field at every point inside the cavity?
A.
−(4πGρ/3)r
B.
(4πGρ/3)r
C.
Zero
D.
−(4πGρ/3)(R−a)r
Correct Answer:
Zero
Explanation:
The full sphere and the removed concentric sphere produce equal and opposite fields at every point inside the cavity. Therefore, the net gravitational field throughout a concentric spherical cavity is zero.
4. A spherical cavity in a uniform solid sphere produces a uniform gravitational field of magnitude g0 inside it. If the displacement between the cavity center and the original sphere center is doubled while the density remains unchanged, what happens to the field inside the cavity?
A.
Its magnitude becomes g0/2 and its direction reverses
B.
Its magnitude remains g0
C.
Its magnitude becomes 2g0 with the same direction
D.
Its magnitude becomes 4g0 with the same direction
Correct Answer:
Its magnitude becomes 2g0 with the same direction
Explanation:
Inside an off-center cavity, the field magnitude is g=(4πGρ/3)d. Hence it is directly proportional to the displacement d; doubling d doubles the field without changing its direction.
5. The center of a spherical cavity lies at position vector d relative to the center of a uniform solid sphere. A particle is placed at position vector r, measured from the original sphere center, inside the cavity. Which expression gives the net gravitational force on the particle?
A.
−(4πGρm/3)r
B.
(4πGρm/3)(r−d)
C.
(4πGρm/3)d
D.
−(4πGρm/3)(r+d)
Correct Answer:
(4πGρm/3)d
Explanation:
The full sphere contributes −(4πGρm/3)r. The missing sphere, represented by negative density, contributes +(4πGρm/3)(r−d). Adding these forces cancels r and leaves (4πGρm/3)d.
6. A uniform spherical shell has outer radius R and density ρ, and a smaller spherical cavity of radius a is carved from it. The cavity is completely inside the shell but its center is displaced from the shell center by vector d. What determines the gravitational field inside the cavity?
A.
Only the cavity radius a
B.
Only the outer radius R
C.
The displacement vector d and density ρ, giving (4πGρ/3)d
D.
The distance of the point from the cavity center
Correct Answer:
The displacement vector d and density ρ, giving (4πGρ/3)d
Explanation:
The shell can be represented as a full uniform sphere of radius R plus a negative-density spherical region for the cavity. The field from the full sphere and the negative sphere combine to give the uniform field (4πGρ/3)d inside the cavity.
7. A cavity is formed by removing a spherical region of density ρ from a uniform sphere. At a point inside the cavity, the gravitational field due to the original complete sphere is g1 and that due to the removed spherical region, if it were present, is g2. Which relation correctly gives the actual field?
A.
gactual = g1 + g2
B.
gactual = g1 − g2
C.
gactual = −g1 − g2
D.
gactual = g2 − g1 only when the cavity is concentric
Correct Answer:
gactual = g1 − g2
Explanation:
The cavity is created by subtracting the mass of the removed spherical region. Therefore, by superposition, the field of the body with the cavity equals the field of the complete sphere minus the field that the removed material would have produced.
8. An off-center spherical cavity lies inside a uniform solid sphere. Two equal test masses are placed at different points inside the cavity. How do the gravitational forces on them compare?
A.
They are equal in magnitude and direction
B.
They are equal in magnitude but opposite in direction
C.
The force is larger for the mass closer to the cavity surface
D.
The force is zero for the mass at the cavity center only
Correct Answer:
They are equal in magnitude and direction
Explanation:
The gravitational field inside a spherical cavity displaced from the parent sphere's center is uniform. Thus equal masses at any two positions inside the cavity experience equal forces with the same direction.
9. A uniform sphere of density ρ contains an off-center spherical cavity. The cavity center is displaced from the original sphere center toward the positive x-axis. In which direction is the gravitational field inside the cavity?
A.
Toward the original sphere center
B.
Away from the original sphere center, along the positive x-axis
C.
Toward the cavity center from every point
D.
Radially outward from the outer surface
Correct Answer:
Away from the original sphere center, along the positive x-axis
Explanation:
The field inside the cavity is (4πGρ/3)d, where d points from the original sphere center to the cavity center. Therefore, it points along the displacement vector, away from the original center and toward the displaced cavity region.
10. A uniform solid sphere of density ρ has a spherical cavity whose center is displaced from the sphere center by distance d. If the density is changed to 3ρ while the geometry remains unchanged, what is the new gravitational field inside the cavity?
A.
The original field divided by 3
B.
The original field unchanged
C.
Three times the original field, in the same direction
D.
Nine times the original field, in the same direction
Correct Answer:
Three times the original field, in the same direction
Explanation:
The cavity field is g=(4πGρ/3)d, so it is directly proportional to the material density. Replacing ρ by 3ρ therefore triples the field while preserving its direction.
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