Kepler’s Laws of Planetary Motion
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.
Test Your Knowledge
Newton’s Law of Gravitation Quiz
Conclusion
Understanding Newton’s Law of Gravitation through Kepler’s Laws helps in solving JEE, NEET, and CBSE Class 11 Board Exam problems efficiently.
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Written by: Neeraj Anand
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