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Uniform Spheres Mass M Radius R Touch And Magnitude Gravitational
Best Answer Of Two Uniform Spheres Mass M Radius R Touch One Another Magnitude Gravitational Force Attraction
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Question:
Two Uniform Spheres Mass M Radius R Touch One Another Magnitude Gravitational Force Attraction
Answer:
Introduction:
Gravitation is one of the four fundamental forces of nature, and it is responsible for the attraction between any two masses in the universe. When two objects of mass m and m' are separated by a distance r, they exert a force of attraction on each other, known as the gravitational force. This force is directly proportional to the product of the masses and inversely proportional to the square of the distance between them.
In this article, we will discuss the gravitational force of attraction between two uniform spheres of mass m and radius r, which are touching each other. We will derive the expression for this force and discuss its properties.
Derivation Of The Gravitational Force Of Attraction:
Consider two uniform spheres of mass m and radius r, which are touching each other. We can assume that the spheres are symmetrical and have a uniform distribution of mass. Let's assume that the centers of the spheres are located at a distance of 2r from each other.
To calculate the gravitational force of attraction between these spheres, we need to integrate the force due to gravity acting on every infinitesimal mass element of one sphere, due to the other sphere. Let's consider an infinitesimal mass element dm on the surface of the first sphere. The force due to gravity acting on this element, due to the second sphere, can be calculated using the expression:
dF = G (m dm)/(4r^2)
where G is the gravitational constant, and 4r^2 is the square of the distance between the centers of the spheres. The direction of this force is towards the center of the second sphere.
Now, we can integrate this expression over the entire surface of the first sphere to get the total force due to gravity:
F = ∫dF = Gm^2/(2r^2)
Here, we have used the fact that the two spheres have the same mass and radius, and hence the gravitational force of attraction is the same on both the spheres.
Properties of the Gravitational Force of Attraction:
The expression for the gravitational force of attraction between two uniform spheres of mass m and radius r, which are touching each other, is given by:
F = Gm^2/(2r^2)
From this expression, we can see that the force of attraction is directly proportional to the square of the mass of each sphere. Hence, the force of attraction will be stronger for spheres with larger masses. Also, the force of attraction is inversely proportional to the square of the distance between the centers of the spheres. Hence, the force of attraction will be stronger for spheres that are closer to each other.
Another important property of the gravitational force of attraction is that it is a conservative force. This means that the work done by the force on an object moving between two points in space depends only on the positions of the points and not on the path taken between them. Hence, the gravitational force of attraction is a central force.
Conclusion:
The gravitational force of attraction is one of the fundamental forces of nature, and it is responsible for the attraction between any two masses in the universe. In this article, we have derived the expression for the gravitational force of attraction between two uniform spheres of mass m and radius r, which are touching each other. We have discussed the properties of this force and have seen that it is directly proportional to the square of the mass of each sphere and inversely proportional to the square of the distance between the centers of the spheres. This force is also a conservative force, and hence it is a central force. The gravitational force of attraction is an important concept in physics
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