Question
Write the three laws given by Kepler. How did they help Newton to arrive at the inverse square law of gravity?
✓
Answer
Kepler’s first law :
The orbit of a planet is an ellipse with the Sun at one of the foci.
Figure $1.5$ shows the elliptical orbit of a planet revolving around the Sun $(S)$.Kepler’s second law :
The line joining the planet and the Sun sweeps equal areas in equal intervals of time.
$A \rightarrow B, C \rightarrow D$ and $E \rightarrow F$ are the displacements of the planet in equal intervals of time.
The straight lines $AS, CS$ and ES sweep equal areas in equal intervals of time.
Area $ASB =$ area $CSD =$ area $ESF.$
Kepler’s third law :
The square of the period of revolution of a planet around the Sun is directly proportional to the cube of the mean distance of the 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,
$T^2 ∝ r^2 , i.e., \frac{T^{2}}{r^{3}} =$ constant $= K$
For simplicity, we shall assume the orbit to be a circle.
In Fig. $1.6,$
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.
If m is the mass of the planet, the centripetal force exerted on the planet by the Sun (= gravitational force),
According to Kepler’s third law,
Thus, $F ∝ \frac{1}{r^{2}}$ as $\frac{4 \pi^{2} m}{K}$ is constant in a particular case.
The orbit of a planet is an ellipse with the Sun at one of the foci.
Figure $1.5$ shows the elliptical orbit of a planet revolving around the Sun $(S)$.Kepler’s second law :
The line joining the planet and the Sun sweeps equal areas in equal intervals of time.
$A \rightarrow B, C \rightarrow D$ and $E \rightarrow F$ are the displacements of the planet in equal intervals of time.
The straight lines $AS, CS$ and ES sweep equal areas in equal intervals of time.
Area $ASB =$ area $CSD =$ area $ESF.$
Kepler’s third law :
The square of the period of revolution of a planet around the Sun is directly proportional to the cube of the mean distance of the 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,
$T^2 ∝ r^2 , i.e., \frac{T^{2}}{r^{3}} =$ constant $= K$
For simplicity, we shall assume the orbit to be a circle.
In Fig. $1.6,$
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.
If m is the mass of the planet, the centripetal force exerted on the planet by the Sun (= gravitational force),
According to Kepler’s third law,
Thus, $F ∝ \frac{1}{r^{2}}$ as $\frac{4 \pi^{2} m}{K}$ is constant in a particular case.
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