Kepler formulated three laws that describe the motion of planets around the Sun:
Law of Orbits: Planets move in elliptical orbits with the Sun at one focus.
Law of Areas: The line joining a planet and the Sun sweeps out equal areas in equal intervals of time.
Law of Periods: The square of the period of a planet is proportional to the cube of the semi-major axis of its orbit.
Derivation of Newton’s Law of Gravitation
To derive Newton’s Law of Gravitation from Kepler’s Third Law, consider:
Step 1: Kepler’s Third Law
\begin{array}{l} T^2 \propto r^3 \end{array}
where T is the orbital period and r is the semi-major axis.
Step 2: Centripetal Force due to Gravity
For a planet orbiting the Sun, the centripetal force is provided by the gravitational attraction:
\begin{array}{l} F = \frac{G M m}{r^2} \end{array}
where:
G is the universal gravitational constant,
M is the mass of the Sun,
m is the mass of the planet,
r is the orbital radius.
Step 3: Equating Centripetal Force and Gravitational Force
From circular motion, centripetal force is:
\begin{array}{l} F = m \frac{v^2}{r} \end{array}
Using orbital velocity
v=2πr/T
we substitute in the equation:
\begin{array}{l} m \frac{(2 \pi r)^2}{T^2 r} = \frac{G M m}{r^2} \end{array}
Simplifying,
\begin{array}{l} \frac{4 \pi^2 r}{T^2} = \frac{G M}{r^2} \end{array} Using Kepler’s Third Law,
\begin{array}{l} T^2 \propto r^3 \end{array}
and substituting,
\begin{array}{l} G M = 4 \pi^2 k \end{array}
[Where, K = 4π2/GM]
Thus, we arrive at Newton’s Law of Universal Gravitation:
\begin{array}{l} F = \frac{G M m}{r^2} \end{array}
Multiple-Choice Questions (MCQs)
Q1: Kepler’s first law states that planetary orbits are:
A) Circular B) Parabolic C) Elliptical D) Hyperbolic
Answer: C) Elliptical Explanation: Kepler’s First Law states that planets move in elliptical orbits with the Sun at one focus.
Q2: Newton’s Law of Gravitation states that force is:
A) Inversely proportional to the square of the distance B) Directly proportional to the cube of the mass C) Inversely proportional to the mass D) Directly proportional to the distance
Answer: A) Inversely proportional to the square of the distance Explanation: The force of gravity follows an inverse square law, meaning that as distance increases, force decreases.
Frequently Asked Questions (FAQs)
Q1: Why does Kepler’s Law lead to Newton’s Law of Gravitation?
Kepler’s Laws describe planetary motion, while Newton’s Laws explain the cause of that motion using gravitational attraction.
Q2: What is the significance of Kepler’s Third Law?
It helps in determining the mass of celestial bodies and verifying Newton’s Law of Gravitation.
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