This MCQ module is based on: Keplers Laws Universal Gravitation
TOPIC 29 OF 33
Keplers Laws Universal Gravitation
🎓 Class 11
Physics
CBSE
Theory
Ch 7 – Gravitation
⏱ ~14 min
🌐 Language: [gtranslate]
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Keplers Laws Universal Gravitation
7.1 Introduction
Early humans gazed at the night sky and wondered why the Sun, Moon and stars do not fall down. Ancient Indian astronomers, the Greeks, and later Renaissance scientists each built models to explain celestial motion. Aristotle taught that heavenly bodies move in perfect circles because they are made of a special "fifth element". A heliocentric model placing the Sun at the centre was proposed by Aryabhata in India, and rediscovered in Europe by Nicolas Copernicus in 1543. Detailed observations by Tycho Brahe and analysis by Johannes Kepler finally produced the three empirical laws that paved the way for Newton's law of universal gravitation.
7.2 Kepler's Laws
Tycho Brahe (1546–1601) recorded planetary positions with the naked eye for decades. His assistant Johannes Kepler (1571–1630) spent years analysing the Mars data and extracted three beautifully simple laws.
7.2.1 Law of Orbits (First Law)
First Law: Every planet revolves around the Sun in an elliptical orbit with the Sun at one of the two foci of the ellipse.
7.2.2 Law of Areas (Second Law)
Second Law: The line joining a planet to the Sun sweeps out equal areas in equal intervals of time.
This means a planet moves faster when it is close to the Sun (perihelion) and slower when far away (aphelion). The areal velocity \(\frac{dA}{dt}\) is constant. We can derive this from angular momentum:
\[\frac{dA}{dt}=\frac{1}{2}|\vec{r}\times\vec{v}|=\frac{L}{2m}=\text{constant}\]
Because gravity is a central force, the torque on a planet about the Sun is zero, so its angular momentum \(L\) is conserved. This is exactly Kepler's Second Law in disguise.
7.2.3 Law of Periods (Third Law)
Third Law: The square of the orbital period T is proportional to the cube of the semi-major axis a:
\[T^2 \propto a^3 \quad\text{or}\quad T^2 = k\,a^3\]
For all planets in the same solar system, the constant k is the same.
| Planet | a (×10¹⁰ m) | T (years) | T²/a³ (×10⁻³⁴) |
|---|---|---|---|
| Mercury | 5.79 | 0.241 | 2.95 |
| Venus | 10.8 | 0.615 | 2.99 |
| Earth | 14.96 | 1.000 | 2.96 |
| Mars | 22.79 | 1.881 | 2.98 |
| Jupiter | 77.83 | 11.86 | 3.01 |
| Saturn | 142.7 | 29.46 | 2.98 |
The remarkable constancy of T²/a³ across the solar system confirms Kepler's Third Law to high precision.
Interactive Simulation: Kepler's Third Law Calculator
Adjust the semi-major axis \(a\) (in AU) and instantly see the orbital period \(T\) via \(T^2 = a^3\) (Earth units).
Period T = 1.00 years
(Try a = 5.2 for Jupiter or a = 19.2 for Uranus)
7.3 Universal Law of Gravitation
Sir Isaac Newton (1665–1687) realised that the same force which makes an apple fall also keeps the Moon in orbit. He showed that to produce Kepler's Third Law with circular orbits, the force must vary as the inverse square of distance.
Newton's Law of Universal Gravitation: Every particle in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
\[F = G\,\frac{m_1 m_2}{r^2}\]
where \(G = 6.67\times 10^{-11}\,\text{N m}^2\text{kg}^{-2}\) is the universal gravitational constant.
The force is always attractive and acts along the line joining the two masses. In vector form, the force on mass 1 due to mass 2 is:
\[\vec{F}_{12} = -G\,\frac{m_1 m_2}{r^2}\,\hat{r}_{12}\]
The negative sign shows it points from 1 toward 2 (attraction). For an extended body like Earth, we may treat its entire mass as concentrated at its centre when computing the force on an external particle — this is the powerful Shell Theorem.
7.3.1 Cavendish's Experiment — Measuring G
In 1798 Henry Cavendish suspended a light rod with two small lead spheres at its ends from a fine quartz fibre. Two large lead spheres were brought near the small ones. The gravitational attraction caused the rod to rotate; the twist of the fibre measured the tiny force, yielding the first value of G.
Worked Example 1: Force between two students
Two friends of masses 60 kg and 55 kg stand 0.5 m apart. Find the gravitational force between them.
\[F = G\frac{m_1 m_2}{r^2} = \frac{6.67\times10^{-11}\times 60\times 55}{(0.5)^2}\]
\[F = \frac{6.67\times10^{-11}\times 3300}{0.25} = \boxed{8.8\times 10^{-7}\,\text{N}}\]
This is roughly one-millionth of the weight of a mosquito — far too small to feel, which is why we never notice gravity between everyday objects.
Worked Example 2: Earth–Moon force
Earth mass M = 6.0×10²⁴ kg, Moon mass m = 7.4×10²² kg, distance r = 3.84×10⁸ m. Find F.
\[F = \frac{(6.67\times10^{-11})(6.0\times10^{24})(7.4\times10^{22})}{(3.84\times10^{8})^2}\]
\[= \frac{2.96\times10^{37}}{1.47\times10^{17}} \approx \boxed{2.0\times10^{20}\,\text{N}}\]
This colossal force keeps the Moon in orbit and is the main driver of ocean tides.
Activity 7.1 — Drawing an Ellipse with StringL3 Apply
Materials: Cardboard, two pins, string loop, pencil.
- Stick two pins about 10 cm apart on the cardboard.
- Loop a string of length 20 cm loosely around both pins.
- Pull the string taut with a pencil tip and move the pencil all around — keeping the string tight.
Predict: What happens to the shape if the two pins are moved closer together? Farther apart?
Observation: The pencil traces an ellipse. The two pins are the foci. As the pins move closer, the ellipse becomes more like a circle; as they move farther apart (with fixed string length), the ellipse flattens.
Conclusion: An ellipse is the locus of points whose distances from two fixed foci sum to a constant. For a circle, the two foci coincide. Planetary orbits are ellipses with small eccentricity — close to circles but not exactly.
Competency-Based Questions
A NASA mission team studies a hypothetical exoplanet "Kalpana-X" orbiting a Sun-like star. Observations show the planet has a semi-major axis of 4 AU and an orbital eccentricity of 0.5. Use Kepler's laws to analyse its motion.
Q1. According to Kepler's Third Law, the orbital period of Kalpana-X is approximately:L3 Apply
Answer: (c) 8 years. T² = a³ ⇒ T = √(4³) = √64 = 8 years.
Q2. At which point does Kalpana-X move with the highest orbital speed?L2 Understand
At perihelion (closest approach to the star). By Kepler's 2nd law and conservation of angular momentum (\(mvr = \text{const}\)), the smaller r is, the larger v becomes.
Q3. True or False: Kepler's Third Law applies only to planets in our solar system. Justify.L5 Evaluate
FALSE. Kepler's law T²∝a³ applies to any two-body system bound by gravity. The constant k differs based on the central mass: for moons around Jupiter, k is different from that for planets around the Sun, but inside each system T²/a³ is constant. This universality let astronomers weigh extrasolar stars.
Q4. Fill in the blank: Kepler's Second Law is a direct consequence of conservation of __________.L1 Remember
Angular momentum. Because gravity is a central force, torque about the Sun is zero, so L is conserved, which implies dA/dt = L/2m is constant.
Q5. HOT: Design an experiment a high-school student could perform on Earth to estimate G using everyday materials.L6 Create
Sample design: Build a mini Cavendish balance — hang a light wooden dumb-bell (two table-tennis balls filled with sand) from a long thin nylon thread inside a glass box (to block air currents). Place two heavy iron weights on a turntable beside the small balls. Use a laser pointer reflected off a small mirror on the dumb-bell onto a distant wall. The slight deflection over hours gives F; with known masses and lever arm, calculate G. Modern student-built versions get G to within ~20% accuracy.
Assertion–Reason Questions
(A) Both true, R explains A. (B) Both true, R does NOT explain A. (C) A true, R false. (D) A false, R true.
A: A planet moves faster at perihelion than at aphelion.
R: Angular momentum of the planet about the Sun is conserved.
(A). Both true; R explains A — \(mvr=\) const, so smaller r implies larger v.
A: The gravitational constant G depends on the medium between the two masses.
R: G is a universal constant having the same value everywhere in the universe.
(D). A is FALSE — G is medium-independent, unlike electrostatic permittivity. R is TRUE.
A: Kepler's laws apply to artificial satellites orbiting the Earth.
R: Newton's law of gravitation governs all gravitational orbits, of which Kepler's laws are a special case.
(A). Both true; R is the correct explanation. Any orbit under an inverse-square force obeys Kepler's laws — including geostationary, GPS, and Moon orbits.
Frequently Asked Questions - Keplers Laws Universal Gravitation
What is the main concept covered in Keplers Laws Universal Gravitation?
In NCERT Class 11 Physics Chapter 7 (Gravitation), "Keplers Laws Universal Gravitation" covers core principles and equations needed for board exam success. The MyAiSchool lesson explains the topic with definitions, derivations, worked examples, and interactive simulations. Key formulas and dimensional analysis are included to build conceptual depth and problem-solving skills aligned with the CBSE 2025-26 syllabus.
How is Keplers Laws Universal Gravitation useful in real-life applications?
Real-life applications of Keplers Laws Universal Gravitation from NCERT Class 11 Physics Chapter 7 include engineering design, satellite mechanics, sports biomechanics, transportation safety, and electrical/electronic devices. The MyAiSchool lesson links every concept to a tangible example so students see physics as a problem-solving framework for the physical world, not as abstract formulas.
What are the key formulas in Keplers Laws Universal Gravitation?
Key formulas in Keplers Laws Universal Gravitation (NCERT Class 11 Physics Chapter 7 Gravitation) are derived step-by-step in the MyAiSchool lesson. Students should memorize the final formula AND understand its derivation for full board marks. Each formula is listed with its dimensional formula, SI unit, applicability range, and common pitfalls. The Summary section at the end of each part includes a quick-reference formula card.
How does this part connect to other parts of Chapter 7?
NCERT Class 11 Physics Chapter 7 (Gravitation) is structured so each part builds on the previous one. Keplers Laws Universal Gravitation connects directly to neighbouring parts via shared definitions, units, and methodology. The MyAiSchool lesson cross-references related concepts with internal links so students can navigate the whole chapter as one connected story rather than disconnected fragments.
What types of CBSE board questions come from Keplers Laws Universal Gravitation?
CBSE board questions from Keplers Laws Universal Gravitation typically include: (1) 1-mark MCQs on definitions and formulas, (2) 2-mark short-answer derivations or applications, (3) 3-mark numerical problems with units, (4) 5-mark long-answer derivations followed by application. The MyAiSchool lesson tags each Competency-Based Question (CBQ) with Bloom level (L1-L6) so students know how to study for each weight.
How can students use the interactive simulation effectively?
The interactive simulation in the Keplers Laws Universal Gravitation lesson allows students to adjust input parameters (sliders or selectors) and see physical quantities update in real time. To use it effectively: (1) try extreme values to understand limiting cases, (2) compare with the analytical formula, (3) check unit consistency, (4) test special configurations from worked examples. The simulation reinforces conceptual intuition that pure formula manipulation cannot.
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