physics : class11 Gravitational Force of a Spherical Shell : Gravitational Force : Gravitation MCQ Questions and Answers
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Gravitational Force of a Spherical Shell
MCQ questions and answers.
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MCQ Questions
1. A uniform thin spherical shell has mass M and radius R. What is the gravitational force on a point mass m placed at a distance R/2 from its centre?
A.
GMm/(4R^2), directed toward the centre
B.
2GMm/R^2, directed toward the centre
C.
Zero
D.
GMm/R^2, directed away from the centre
Correct Answer:
Zero
Explanation:
By the shell theorem, the net gravitational field at every point inside a uniform spherical shell is zero because the attractions from all parts of the shell cancel vectorially. Hence the force on m is zero.
2. A point mass m is placed at a distance 2R from the centre of a uniform spherical shell of mass M and radius R. What is the gravitational force on the point mass?
A.
GMm/(2R^2)
B.
GMm/(4R^2), directed toward the centre
C.
Zero
D.
4GMm/R^2, directed toward the centre
Correct Answer:
GMm/(4R^2), directed toward the centre
Explanation:
For a point outside a uniform spherical shell, the shell attracts the point mass as though its entire mass were concentrated at its centre. At distance 2R, the force is GMm/(2R)^2 = GMm/(4R^2).
3. Two uniform concentric spherical shells have masses M and 2M and radii R and 2R, respectively. A point mass m is located at distance 3R/2 from their common centre. What is the magnitude of the net gravitational force on m?
A.
4GMm/(9R^2)
B.
8GMm/(9R^2)
C.
GMm/(9R^2)
D.
Zero
Correct Answer:
4GMm/(9R^2)
Explanation:
The point lies outside the inner shell, so that shell produces a force GMm/(3R/2)^2 = 4GMm/(9R^2). It lies inside the outer shell, whose net force is zero by the shell theorem. Therefore, the net force is 4GMm/(9R^2).
4. For a uniform spherical shell of mass M and radius R, the gravitational forces on an identical point mass placed at distances 2R and 3R from the centre are F2 and F3, respectively. What is the ratio F2:F3?
A.
2:3
B.
3:2
C.
9:4
D.
4:9
Correct Answer:
9:4
Explanation:
Outside the shell, the force varies inversely as the square of the distance from the centre. Thus F2/F3 = (3R)^2/(2R)^2 = 9/4, giving F2:F3 = 9:4.
5. A uniform spherical shell has mass M and radius R. What is the work required to move a point mass m quasistatically from the centre to any point inside the shell?
A.
GMm/R
B.
−GMm/R
C.
Zero
D.
GMm/(2R)
Correct Answer:
Zero
Explanation:
The gravitational field inside a uniform spherical shell is zero everywhere, so no work is done in moving a mass between any two interior points. Although the potential inside is −GM/R, it is constant, making the potential difference and work zero.
6. A uniform spherical shell of radius R and mass M produces a gravitational potential V at an interior point and at an exterior point located at distance 2R from its centre. What are these potentials?
A.
V_inside = −GM/R and V_2R = −GM/(2R)
B.
V_inside = 0 and V_2R = −GM/(2R)
C.
V_inside = −GM/(2R) and V_2R = −GM/R
D.
V_inside = −GM/R and V_2R = 0
Correct Answer:
V_inside = −GM/R and V_2R = −GM/(2R)
Explanation:
The potential inside a uniform spherical shell is constant and equal to the surface value, −GM/R. Outside, the shell behaves like a point mass at its centre, so at 2R the potential is −GM/(2R).
7. A particle is released from rest at a distance 3R from the centre of a uniform spherical shell of mass M and radius R. Neglecting other forces, what speed will it have when it reaches the shell's surface?
A.
sqrt(GM/(3R))
B.
sqrt(2GM/(3R))
C.
sqrt(4GM/(3R))
D.
sqrt(2GM/R)
Correct Answer:
sqrt(4GM/(3R))
Explanation:
Using conservation of energy, the initial potential energy is −GMm/(3R), while the final potential energy at the surface is −GMm/R. Therefore, (1/2)mv^2 = GMm(1/R − 1/(3R)) = 2GMm/(3R), giving v = sqrt(4GM/(3R)).
8. A point mass m is placed at the centre of a uniform spherical shell of mass M and radius R. If the shell is expanded quasistatically to radius 2R without changing its mass, what is the change in the shell's gravitational potential energy with the point mass?
A.
−GMm/(2R)
B.
GMm/(2R)
C.
−GMm/R
D.
Zero
Correct Answer:
GMm/(2R)
Explanation:
The potential due to the shell at any interior point is −GM/R initially and −GM/(2R) after expansion. Thus the interaction energy changes from −GMm/R to −GMm/(2R), so the change is +GMm/(2R).
9. A uniform spherical shell of mass M and radius R attracts an external point mass m with force F at distance r, where r > R. At what distance from the centre will the force become F/4?
A.
2r
B.
r/2
C.
r/4
D.
4r
Correct Answer:
2r
Explanation:
For an external point, the shell behaves like a point mass at its centre, so F is proportional to 1/r^2. To reduce the force to F/4, the distance must be doubled: F' = GMm/(2r)^2 = F/4.
10. A uniform spherical shell of mass M and radius R surrounds a point mass m located anywhere inside it. Which statement correctly describes the gravitational interaction between the shell and the point mass?
A.
The force is GMm/r^2, where r is the distance from the centre
B.
The force is zero, but the interaction potential energy is −GMm/R
C.
Both the force and the interaction potential energy are zero
D.
The force is GMm/R^2 and is directed toward the nearest point of the shell
Correct Answer:
The force is zero, but the interaction potential energy is −GMm/R
Explanation:
Inside a uniform spherical shell, the gravitational field and hence the force on the point mass are zero. However, the potential is constant throughout the interior and equals the surface potential −GM/R, so the interaction energy is −GMm/R.
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