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:
  • Man-made 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 space-time 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