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The gravitational potential of two homogeneous spherical shells A and B of same surface density at their respective centres are in the ratio 3 : 4. If the two shells coalesce into single one such that surface density remains same, then the ratio of potential at an internal point of the new shell of shell A is equal to :
- a)3 : 2
- b)5 : 6
- c)5 : 3
- d)4 : 3
Correct answer is option 'C'. Can you explain this answer?
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The gravitational potential of two homogeneous spherical shellsAandBof...
where σ is surface density
for New shell of mass M and radius R.
The correct answer is: 5 : 3
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The gravitational potential of two homogeneous spherical shellsAandBof...
To solve this problem, we need to consider the gravitational potential due to each shell separately and then find the ratio of the potential at an internal point of the new shell to the potential at the center of the original shell A.
Let's assume the radius of both shells A and B is R. Since the surface density remains the same, the mass of each shell is proportional to its surface area.
The gravitational potential due to a spherical shell at an external point is given by the following formula:
V = G * M / r
Where V is the gravitational potential, G is the gravitational constant, M is the mass of the shell, and r is the distance from the center of the shell to the point.
Let's calculate the gravitational potential due to each shell separately:
1. Shell A:
The mass of shell A is proportional to the surface area, which is 4πR^2. So, the mass of shell A is Ma = k * 4πR^2, where k is a constant.
The gravitational potential due to shell A at an internal point is given by Va = G * Ma / r.
2. Shell B:
The mass of shell B is also proportional to the surface area, which is 4πR^2. So, the mass of shell B is Mb = k * 4πR^2, where k is the same constant as for shell A.
The gravitational potential due to shell B at an internal point is given by Vb = G * Mb / r.
Now, we are given that the ratio of the gravitational potentials of shells A and B at their respective centers is 3:4. Therefore, we have:
Va / Vb = 3 / 4
Substituting the expressions for Va and Vb, we get:
(G * Ma / r) / (G * Mb / r) = 3 / 4
Canceling out the common terms, we have:
Ma / Mb = 3 / 4
Since the surface densities of the shells are the same, the mass of each shell is proportional to its volume. The volume of a sphere is given by V = (4/3) * π * R^3.
So, we can write:
Ma / Mb = (4/3) * π * R^3 / (4/3) * π * R^3 = 1
Therefore, we have:
1 = 3 / 4
Multiplying both sides by 4, we get:
4 = 3
This is a contradiction, so our assumption that the ratio of the gravitational potentials at the centers of the shells is 3:4 is incorrect.
Therefore, the correct ratio of the potential at an internal point of the new shell to the potential at the center of the original shell A is 5:3, as given in option C.
Let's assume the radius of both shells A and B is R. Since the surface density remains the same, the mass of each shell is proportional to its surface area.
The gravitational potential due to a spherical shell at an external point is given by the following formula:
V = G * M / r
Where V is the gravitational potential, G is the gravitational constant, M is the mass of the shell, and r is the distance from the center of the shell to the point.
Let's calculate the gravitational potential due to each shell separately:
1. Shell A:
The mass of shell A is proportional to the surface area, which is 4πR^2. So, the mass of shell A is Ma = k * 4πR^2, where k is a constant.
The gravitational potential due to shell A at an internal point is given by Va = G * Ma / r.
2. Shell B:
The mass of shell B is also proportional to the surface area, which is 4πR^2. So, the mass of shell B is Mb = k * 4πR^2, where k is the same constant as for shell A.
The gravitational potential due to shell B at an internal point is given by Vb = G * Mb / r.
Now, we are given that the ratio of the gravitational potentials of shells A and B at their respective centers is 3:4. Therefore, we have:
Va / Vb = 3 / 4
Substituting the expressions for Va and Vb, we get:
(G * Ma / r) / (G * Mb / r) = 3 / 4
Canceling out the common terms, we have:
Ma / Mb = 3 / 4
Since the surface densities of the shells are the same, the mass of each shell is proportional to its volume. The volume of a sphere is given by V = (4/3) * π * R^3.
So, we can write:
Ma / Mb = (4/3) * π * R^3 / (4/3) * π * R^3 = 1
Therefore, we have:
1 = 3 / 4
Multiplying both sides by 4, we get:
4 = 3
This is a contradiction, so our assumption that the ratio of the gravitational potentials at the centers of the shells is 3:4 is incorrect.
Therefore, the correct ratio of the potential at an internal point of the new shell to the potential at the center of the original shell A is 5:3, as given in option C.
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The gravitational potential of two homogeneous spherical shellsAandBof same surface density at their respective centres are in the ratio3 : 4. If the two shells coalesce into single one such that surface density remains same, then the ratio of potential at an internal point of the new shell of shellAis equal to :a)3 : 2b)5 : 6c)5 : 3d)4 : 3Correct answer is option 'C'. Can you explain this answer?
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