Question

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State and prove Newton's law of gravitation.

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Answer 1

Newton's law of gravitation states that every object in the universe attracts each other with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between the centres.

$\huge \bold{prove - }$

Let us consider a planet of mass m moving with constant speed v in a circular orbit.

T=2πrv

and v>0

Where r is the radius of circular path

And from Kepler;s 3rd law, where k is a constant of proportionality.

T²=kr³

From above two equations we can rewrite

4π^2r^2/v^2=kr^3

We know that an object in a circular path is accelerated and its acceleration towards the centre is

v^2/r

,and r>0

Now, F=mv²/r

F=m.4π^2/kr^2

F∝1/r²

F∝m

Now force on the planet due to the sun = force on the sun due to the planet.

If the force is proportional to the mas of the planet, it should be proportional to the mass of the sun.

F∝Mm/r^2

F=GMm/r^2

Where G is any constant.

Answer 2

the law states that every point mass attracts every other point mass by a force acting along the line intersecting the two points. The force is proportional to the product of the two masses, and inversely proportional to the square of the distance between them.[2]

The equation for universal gravitation thus takes the form:

{\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}\ } F=G{\frac {m_{1}m_{2}}{r^{2}}}\

where F is the gravitational force acting between two objects, m1 and m2 are the masses of the objects, r is the distance between the centers of their masses, and G is the gravitational constant.