Two identical solid sphere of radius R placed in contact with each other, the gravitational attraction between them is proportionalto-(ans:R^4)?

Question

Answer

Force =Gm^2 /r^2 now mass = density x vol depending on r^3  therefore force will be proportional to m^6 /m^2 = m^4?

Answer

R2

Answer

F=Gm^2/(2R)^2
=Gm^2/4R^2
=G (volume charge density×volume)^2/4R^2
=G (volume charge density×4/3piR^3)/4R^2
=(4G×volume charge density×pi^2 R^6)/12R^2
=(4G×volume charge density ×pi^2 R^4)/12
So,F is proportional to R^4

Answer

Gravitational Attraction Between Two Spheres
The gravitational attraction between two solid spheres can be derived from Newton's law of gravitation. When considering two identical solid spheres of radius R in contact, we can analyze the forces at play.
Key Points of Gravitational Attraction
- Mass of the Spheres:
Each sphere has a mass M, which can be expressed in terms of its volume and density. The mass M for a sphere is given by:
M = (4/3) * π * R³ * ρ, where ρ is the density of the material.
- Gravitational Force Formula:
The gravitational force F between two point masses is described by:
F = G * (m1 * m2) / r², where G is the gravitational constant, m1 and m2 are the masses of the two spheres, and r is the distance between their centers.
- Distance Between Centers:
For two spheres in contact, the distance r between their centers is equal to 2R.
- Substituting Values:
By substituting the values of masses and distance into the gravitational force formula:
F = G * [(M) * (M)] / (2R)²
= G * [(4/3) * π * R³ * ρ]² / (4R²)
= (G * (16/9) * π² * R^6 * ρ²) / (4R²)
= (4/9) * G * π² * ρ² * R^4.
- Proportionality:
The gravitational attraction F between the two spheres is therefore proportional to R^4, confirming that the force increases significantly with the radius of the spheres.
Conclusion
In summary, the gravitational attraction between two identical solid spheres in contact is indeed proportional to R^4, illustrating how size significantly influences gravitational interactions.

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