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Question
Write the three laws given by Kepler. How did they help Newton to arrive at the inverse square law of gravity?
Solution
The orbit of a planet moving around the sun
Kepler’s first law: The orbit of a planet is an ellipse with the Sun at one of the foci.
In the above figure, the elliptical orbit of a planet revolving around the sun. The position of the Sun is indicated by S.
Kepler’s second law: The line joining the planet and the Sun sweeps equal areas in equal intervals of time.
A → B, C → D and E → F are the displacements of the planet in equal intervals of time.
The straight lines AS and CS sweep equal areas in equal intervals of time, i.e. area ASB and CSD are equal.
Kepler’s third law: The square of its period of revolution around the Sun is directly proportional to the cube of the mean distance of a planet from the Sun.
Thus, if r is the average distance of the planet from the Sun and T is its period of revolution then,
`"T"^2 α "r"^3 "i.e.," ("T"^2)/("r"^3) = "constant" = "K"`
For simplicity, we shall assume the orbit to be a circle.
Circular motion of planet around the sun
S denotes the position of the Sun, P denotes the position of a planet at a given instant and r denotes the radius of the orbit (= the distance of the planet from the Sun). Here, the speed of the planet is uniform.
`"v" = "distance travelled"/"time taken"`
`= (2pir)/T`
If m is the mass of the planet, the centripetal force exerted on the planet by the Sun (= gravitational force),
`"F" = "mv"^2/r`
`"F" = (m((2pir)/"T")^2)/r`
`"F" = (4m pi^2r)/T^2`
Multiplying and dividing by r2 we get,
`"F" = (4 m pi^2)/r^2`
`"F" = (r^3/T^2)` ...(i)
According to Kepler’s third law,
`T^2/r^3 = K` ...(ii) [∵ K = constant]
Substituting (ii) in (i)
`"F" = (4m pi^2)/(K) xx 1/r^2`
F = `"constant" xx 1/r^2` [∵ where, `(4 m pi^2)/K = "constant"`]
∴ `F α 1/r^2`
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