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A solid sphere of mass ‘M’ and radius ‘a’ is surrounded by a uniform concentric spherical shell of thickness 2a and mass 2M. The gravitational field at distance ‘3a’ from the center will be:
- a)2GM/9α2
- b)GM/9α2
- c)GM/3α2
- d)2GM/3α2
Correct answer is option 'C'. Can you explain this answer?
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Gravitational Field due to a Solid Sphere:
The gravitational field at a point outside a solid sphere is equivalent to the gravitational field due to a point mass located at the center of the sphere. The gravitational field at a distance 'r' from the center of the sphere is given by the equation:
E = GM/r^2
where G is the universal gravitational constant, M is the mass of the sphere, and r is the distance from the center of the sphere.
Gravitational Field due to a Concentric Spherical Shell:
The gravitational field inside a uniform spherical shell is zero. This means that the gravitational field due to the shell at any point inside the shell is canceled out by the gravitational field due to the opposite side of the shell.
Gravitational Field at Distance '3a':
In this problem, we have a solid sphere of mass 'M' and radius 'a' surrounded by a uniform concentric spherical shell of thickness 2a and mass 2M. We need to find the gravitational field at a distance of '3a' from the center.
- Using the equation for the gravitational field due to a solid sphere, the field due to the solid sphere is given by:
E1 = GM/(3a)^2 = GM/9a^2
- Using the equation for the gravitational field due to a concentric spherical shell, the field due to the shell is zero.
Total Gravitational Field:
The total gravitational field at distance '3a' is the sum of the gravitational fields due to the solid sphere and the shell. Since the field due to the shell is zero, the total field is equal to the field due to the solid sphere.
Therefore, the gravitational field at distance '3a' from the center is:
E = GM/9a^2
Simplifying the equation, we get:
E = GM/(3a)^2
E = GM/3a^2
Thus, the correct answer is option 'C': GM/3a^2.
The gravitational field at a point outside a solid sphere is equivalent to the gravitational field due to a point mass located at the center of the sphere. The gravitational field at a distance 'r' from the center of the sphere is given by the equation:
E = GM/r^2
where G is the universal gravitational constant, M is the mass of the sphere, and r is the distance from the center of the sphere.
Gravitational Field due to a Concentric Spherical Shell:
The gravitational field inside a uniform spherical shell is zero. This means that the gravitational field due to the shell at any point inside the shell is canceled out by the gravitational field due to the opposite side of the shell.
Gravitational Field at Distance '3a':
In this problem, we have a solid sphere of mass 'M' and radius 'a' surrounded by a uniform concentric spherical shell of thickness 2a and mass 2M. We need to find the gravitational field at a distance of '3a' from the center.
- Using the equation for the gravitational field due to a solid sphere, the field due to the solid sphere is given by:
E1 = GM/(3a)^2 = GM/9a^2
- Using the equation for the gravitational field due to a concentric spherical shell, the field due to the shell is zero.
Total Gravitational Field:
The total gravitational field at distance '3a' is the sum of the gravitational fields due to the solid sphere and the shell. Since the field due to the shell is zero, the total field is equal to the field due to the solid sphere.
Therefore, the gravitational field at distance '3a' from the center is:
E = GM/9a^2
Simplifying the equation, we get:
E = GM/(3a)^2
E = GM/3a^2
Thus, the correct answer is option 'C': GM/3a^2.
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A solid sphere of mass ‘M’ and radius ‘a’ is surrounded by a uniform concentric spherical shell of thickness 2a and mass 2M. The gravitational field at distance ‘3a’ from the center will be:a)2GM/9α2b)GM/9α2c)GM/3α2d)2GM/3α2Correct answer is option 'C'. Can you explain this answer?
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