Notes from Toppers
Gravitation : JEE Detailed Notes
1. Newton’s Law of Gravitation

Gravitational force: According to Newton’s inverse square law of gravitation, the force of attraction (gravitational force) between any two point masses M and m is directly proportional to the product of their masses and inversely proportional to the square of the distance (r) between their centers.

Mathematical Formulation: $$ F = \frac{Gm_1 m_2}{r^2}$$

G is the gravitational constant $$G = 6.674 × 10^{−11}\ Nm^2 kg^{−2}$$

Significance of G:
 Measures the strength of the gravitational force between two masses.
 Same value throughout the universe.
Important Points
 Gravitational force is a central force acting along the line joining the centers of masses.
 Gravitational force is always attractive.
2. Gravitational Potential and Field
 Gravitational Potential (V):
 Defined as the work done in bringing a unit positive test mass from infinity to the point concerned without acceleration.
 SI unit: Joules/Kilogram (J/Kg).
 Gravitational Field Intensity:
 Defined as the force experienced by a unit positive test mass placed at the point.
 Gravitational field is conservative.
 Mathematical expression: Gravitational field intensity at a point is the negative of the gradient of potential at that point. $$\overrightarrow{E_g} = \overrightarrow{\nabla}V$$
$$\overrightarrow{E_g}=\frac{GM}{r^2}$$
 Gravitational Field Lines:
 Imaginary lines representing the direction and magnitude of gravitational field.
 Pointing towards the center of the attracting mass.
 Gauss’s Law for Gravitation:
 The total gravitational flux across a closed surface equals 4π times the enclosed mass divided by the permittivity of free space.
$$ \oint \vec{E_g} \cdot d\overrightarrow{A} = \frac{4\pi GM}{ε_0}$$
3. Motion of Satellites
 Orbital motion:
 Motion of a satellite in a circular or elliptical path around a central massive body.
 Kepler’s Laws of Planetary Motion:
 First Law: Orbits are elliptical with the sun at one focus.
 Second Law: Equal areas swept out in equal time intervals.
 Third Law: Square of orbital period is proportional to cube of mean distance from central body. $$ T^2 = k. r^3 $$
 Important formulas:
 Orbital velocity $$v=\sqrt{\frac{GM}{r}}$$
 Time Period $$T= 2\pi\sqrt{\frac{r^3}{GM}}$$
 Geostationary and Polar Satellites:
 Geostationary Satellites: Satellites in circular orbit at a height of approximately 36,000 km above Earth’s surface, with a period of 24 hours.
 Polar Satellites: Satellites in circular orbits passing over or near the poles of the Earth.
4. Gravitational Potential Energy
 Gravitational Potential Energy (PE): Energy possessed by an object due to its position in a gravitational field. Formula: $$PE = \frac{GMm}{r}$$
 Variation with Distance:
 Directly proportional to the product of masses.
 Inversely proportional to the distance between masses.
 Escape Velocity (V):
 Velocity required by an object to overcome gravitational attraction and escape from a gravitational field of a celestial body. $$v_e=\sqrt{\frac{2GM}{R}}$$
5. Weightlessness and Artificial Satellites
 Weightlessness:
 State in which an object does not experience any net gravitational force.
 Causes:
 In free fall.
 In orbit.
 Artificial Satellites:
 Manmade objects orbiting the Earth or another celestial body.
 Applications:
 Communication
 Weather forecasting
 Earth observation
 Remote sensing
6. Gravitational Field due to Spherical Objects
 Gravitational field due to a thin spherical shell:
 Same as if all its mass were concentrated at its center.
 Gravitational field inside and outside a solid sphere:
 Inside: Directly proportional to distance from center.
 Outside: Same as a point mass of equal mass at the center.
7. Gravitational Effects on Earth
 Tides:
 Rise and fall of sea level due to the combined effects of gravitational force exerted by the Moon and the Sun along with the Earth’s rotation.
 Spring tides:
 High tides are exceptionally high, and low tides are exceptionally low.
 Occur when the Moon, Earth, and the Sun are in a straight line.
 Neap tides:
 High tides are lower and low tides are higher than average.
 Occur when the angle between the Moon, Earth, and Sun is a right angle.
 Shape of the Earth:
 Earth is slightly bulging at the equator and flattened at the poles due to rotation.
8. Black Holes
 Characteristics:
 Region in space with such strong gravitational pull that nothing, not even light, can escape from it.
 Formed when a massive star collapses at the end of its life.
 Event Horizon:
 Boundary of a black hole beyond which the gravitational pull is so strong that escape is impossible.
 Significance:
 Important astrophysical phenomena.
 Insights into general relativity.
9. Gravitational Waves
 Introduction:
 Ripples in spacetime caused by the acceleration of massive objects.
 Properties:
 Travel at the speed of light.
 Cause minuscule distortion of spacetime as they pass.
 Significance:
 Detection confirmed existence of black holes and gravitational waves.
 Potential tool for studying the universe.
10. Applications of Gravitation
 Role in everyday life:
 Walking, Jumping
 Tides
 Importance in astronomy and astrophysics:
 Celestial motion
 Stellar Dynamics
 Practical applications in engineering and technology:
 Satellite communication
 Global Positioning System (GPS)
 Oil and gas exploration