Consider a thin uniform spherical shell in space with radius R and mass M. A three-dimensional object divides space into three pieces.
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Inside the spherical shell.
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On the surface of the spherical shell.
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Outside the spherical shell.
On the surface of the spherical shell, consider a unit test mass at a point P on the spherical shell’s surface at a distance r from the centre, then r = R.
E = -GM/R2
⇒ E = Constant
Gravitational field
The force field in space surrounding any mass or combination of masses is known as a gravitational field. The gravitational field extends in all directions and is measured in newtons per kilogramme (N/kg), a unit of force per mass.
The gravitational force decreases with the increase in the distance between the objects. A gravitational field is similar to electric and magnetic fields.
According to the Newton’s Law of Gravitation:
The gravitational force F between two point masses, M and m, separated by a distance r, acts along the line connecting their centres and is proportional to the masses and inversely proportional to the square of their separations.
F ∝ Mm/r²
The proportionality constant in the SI unit system is G, the gravitational constant, which has a value of 6.67 x 10-11 Nm2kg2.
Newton’s Law of Gravitation is re-written as follows:
F = GMm/r²
The gravitational field is the gravitational force per unit mass that a small mass would experience at that location. It is a vector field that points in the direction of the force experienced by the mass. The magnitude of the resultant gravitational field strength g, at a distance r from M, for a point particle of mass M is
g = GM/r²
The gravitational force exerted on a mass m, usually known as its weight, is as follows:
F = mg
Gravitational field intensity:
The force on a unit mass at any point in the gravitational field is the object’s gravitational field intensity or strength. So, suppose we transfer a unit test mass from infinity to a gravitational field. In that case, the gravitational force applied on that unit test mass due to a larger mass for which the gravitational field is produced is known as gravitational field intensity.
A gravitational field interacts between the source mass and the test mass in a non-contact force. If the force acting on a body of mass m at a point in the gravitational field is F, then the intensity of the gravitational field at that point is
g or E = F/m
Where g = gravitational field strength
F = gravitational force
M = mass of an object
Gravitational field intensity due to ring:
When a small mass element dm of a ring with radius R is selected, the gravitational field strength owing to dm at any point x is given by
dI = G (dmr)/(R² + r²)
where r is the unit vector along the line that connects x and dm. dI has two components in this case: along the x-axis and in the YZ plane, which are given by:
dIx = -G xdm/(R2+x2)3/2
dIx = -G Rdm/(R2+x2)3/2
Because of the ring’s symmetry, you can discover another YZ component with the same but opposite sign for each YZ component, resulting in a net impact of 0. Assume that the angle formed by the YZ component with the y axis is θ. The y and z components would therefore be as follows:
dIy = dIyz cos(θ)
dIz = dIyz sin(θ)
dIyz is a constant due to its symmetry.
Suppose dIyz = A
(So that you don’t get mixed up with differential d.) In the y and z directions, the net gravitational field intensity would now be:
2π
IY=∫ Acos(θ) = 0
0
2π
Iz= ∫ Asin(θ) = 0
0
To calculate the overall gravitational field intensity I, all you have to do now is integrate the dIx component from 0 to m.
m
I = ∫ -G xdm/(R2 + r2)3/2x = -Gm x/(R2 + r2)3/2x
0
Conclusion
A spherical shell divides space into three pieces: inside the spherical shell, on the surface of the spherical shell and outside the spherical shell. The gravitational field intensity on the surface of the spherical shell: r = R,
E = -GM/R2
⇒ E = Constant.
The force experienced by a unit mass put at any location in the gravitational field defines the object’s gravitational field intensity or strength. The gravitational force applied on the unit test mass due to a comparable larger mass from which the gravitational field is produced is the gravitational field intensity. Gravitational field intensity due to ring
I =m0∫ -G xdm/(R2 + r2)3/2x = -Gm x/(R2 + r2)3/2x.