state the universal law of gravitation and derive its equation
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Answer:
This was done by measuring the acceleration due to gravity as accurately as possible and then calculating the mass of Earth M from the relationship Newton's universal law of gravitation gives mg=GmMr2 m g = G m M r 2 , where m is the mass of the object, M is the mass of Earth, and r is the distance to the center of ...
According to Newton’s law of gravitation, “Every particle in the universe attracts every other particle with a force whose magnitude is,
Directly proportional to the product of their masses i.e. F ∝ (M1M2) . . . . (1)
Inversely proportional to the square of the distance between their centre i.e. (F ∝ 1/r2) . . . (2)
On combining equations (1) and (2) we get,
F ∝ M1M2/r2
F = G × [M1M2]/r2 . . . . (7)
Or, f(r) = GM1M2/r2 [f(r)is a variable, Non-contact, and conservative force]
As f(r) varies inversely as a square of ‘r’ it is also known as inverse square law force. The proportionality constant (G) in the above equation is known as gravitational constant.
The dimension formula of G is [M-1L3T-2]. Also, the value of the gravitational constant,
In SI units: 6.67 × 10-11 Nm2 kg-2,
In CGS units: 6.67×10-8 dyne cm2 g-2
Derivation of Newton's Law of Gravitation
From the above figure, it can be seen that the two particles of masses and are placed at a distance, therefore according to Newton’s law of gravitation, the force on m1 due to m2 i.e. F12 is given by,
F12 = [-G m1m2]/|r12|2 r^12
F12 = [-Gm1m2/|r1 – r2|3] (r1 – r2) . . . . . . . . . (9)
Where r^12 is a unit vector pointing from m2 to m1.
The negative sign in Equation (9) indicates that the direction of force F12 is opposite to that of r^12.
Similarly, the force on m2 due to m1 i.e. F21 is given by,
F21 = -Gm1m2/|r21|2 r^12 . . . . . . . . . . (10)
From equations (9) and (10) we get,
F12 = – F21
As F12 and F21 are directed towards the centres of the two particles, so the gravitational force is conservative in nature.
Gravitational Force Formula
Gravitational force is explained using Newton’s law of gravitation. Gravitational force decides how much we weigh and how far a ball travels when thrown before it lands on the ground.
According to Newton’s law of gravitation, every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Mathematically it can be represented as,
F = Gm1m2/r2
Where,
F is the Gravitational force between two objects measured in Newton (N).
G is the Universal Gravitational Constant with a value of 6.674 × 10-11 Nm2kg-2.
m1 is the mass of one massive body measured in kg.
m2 is the mass of another massive body measured in kg.
r is the separation between them measured in kilometre (Km).