MP Board Class 11 Physics: Gravitation (गुरुत्वाकर्षण)

Comprehensive Revision Notes & Formula Sheet

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1. Universal Law of Gravitation (न्यूटन का सार्वत्रिक गुरुत्वाकर्षण नियम)

• Statement: Every particle of matter 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.
• Mathematical Form: F = G * (m1 * m2) / r^2
  • m1, m2: Masses of the bodies
  • r: Distance between centers
  • G: Universal Gravitational Constant
• Gravitational Constant (G):
  • G = 6.67 x 10^-11 N m^2 kg^-2 (or m^3 kg^-1 s^-2)
  • Dimensional Formula: [M^-1 L^3 T^-2]

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2. Acceleration Due to Gravity (g) (गुरुत्वीय त्वरण)

• Acceleration gained by an object due to Earth's gravitational pull.
• Formula on Earth's Surface: g = (G * M) / R^2
  • In terms of density (ρ): g = (4/3) * π * R * ρ * G
• Standard Value: g = 9.8 m/s^2 or 980 cm/s^2
• Dimensional Formula: [M^0 L T^-2]

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3. Variations in Acceleration Due to Gravity (g में परिवर्तन)

A. Effect of Altitude (Height 'h')

• For any height h: g_h = g * [R / (R + h)]^2
• For small height (h << R): g_h ≈ g * (1 - (2h / R))
• Note: g decreases as height increases.

B. Effect of Depth (Depth 'd')

• At depth d below Earth's surface: g_d = g * (1 - (d / R))
• At Center of Earth (d = R): g_center = 0
• Note: g decreases linearly as depth increases.

C. Effect of Latitude / Rotation of Earth (अक्षांश का प्रभाव)

• At latitude λ (or angle φ): g_λ = g - R * ω^2 * cos^2(λ)
  • ω: Angular velocity of Earth's rotation
• At Equator (λ = 0°): g_eq = g - R * ω^2 (Minimum value)
• At Poles (λ = 90°): g_pole = g (Maximum value)

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4. Gravitational Field & Potential (गुरुत्वीय क्षेत्र एवं विभव)

A. Gravitational Field Intensity (E)

• Force experienced by a unit mass placed at a point in the gravitational field.
• Formula: E = F / m = (G * M) / r^2
• Unit: N/kg or m/s^2
• Vector quantity, directed towards the mass M.

B. Gravitational Potential (V)

• Work done in bringing a unit mass from infinity to a point in the gravitational field.
• Formula: V = - (G * M) / r
• Unit: J/kg
• Scalar quantity. Always negative (zero at infinity).

C. Relation Between Field and Potential

• E = - (dV / dr)

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5. Gravitational Potential Energy (U) (गुरुत्वीय स्थितिज ऊर्जा)

• Work done in bringing a mass m from infinity to a distance r from mass M.
• Formula: U = - (G * M * m) / r
• Change in Potential Energy (raising height h from surface): ΔU = (m * g * h) / (1 + (h / R))
  • If h << R, ΔU ≈ m * g * h

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6. Escape Velocity (v_e) (पलायन वेग)

• Minimum velocity with which a body must be projected vertically upwards so that it escapes Earth's gravitational field permanently.
• Formula: v_e = √(2 * G * M / R) = √(2 * g * R)
• Value for Earth: v_e ≈ 11.2 km/s
• Independent of the mass of the projected body and angle of projection.

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7. Satellite Motion (उपग्रह की गति)

A. Orbital Velocity (v_o) (कक्षीय वेग)

• Velocity required to keep a satellite in its orbit around a planet.
• Formula: v_o = √(G * M / r) = √(G * M / (R + h))
• For satellite near Earth's surface (h ≈ 0): v_o = √(g * R) ≈ 7.92 km/s

B. Relation between Escape Velocity and Orbital Velocity

• v_e = √2 * v_o ≈ 1.414 * v_o

C. Time Period of Satellite (T) (परिक्रमण काल)

• T = Circumference / Orbital Velocity = (2 * π * r) / v_o
• T = 2 * π * √((R + h)^3 / (G * M)) = 2 * π * √((R + h)^3 / (g * R^2))
• Near Earth's surface: T ≈ 84.6 minutes (or 1.4 hours)

D. Energies of a Revolving Satellite

• Kinetic Energy (KE): KE = (G * M * m) / (2 * r)
• Potential Energy (PE): PE = - (G * M * m) / r
• Total Energy (E): E = KE + PE = - (G * M * m) / (2 * r)
• Binding Energy (BE): BE = -E = + (G * M * m) / (2 * r)

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8. Kepler's Laws of Planetary Motion (केप्लर के नियम)

1. First Law (Law of Orbits): All planets move in elliptical orbits with the Sun at one of the two foci.
2. Second Law (Law of Areas): The line joining the planet to the Sun sweeps out equal areas in equal intervals of time.
   • Areal Velocity: dA / dt = L / (2 * m) = Constant
   • Direct consequence of the Conservation of Angular Momentum.
3. Third Law (Law of Periods): The square of the time period of revolution of a planet is directly proportional to the cube of the semi-major axis of its elliptical orbit.
   • T^2 ∝ a^3 or T^2 / r^3 = Constant

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9. Satellites: Types & Comparison

Parameter	Geostationary Satellite (भू-स्थिर उपग्रह)	Polar Satellite (ध्रुवीय उपग्रह)
Orbit Plane	Equatorial plane	Polar plane (North-South)
Height above Earth	~36,000 km	~500 km to 800 km
Time Period	24 hours	~100 minutes
Direction of Rotation	West to East	North to South
Primary Uses	Communication, Weather forecasting	Remote sensing, Meteorology, Military

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10. Quick Memory Check Tips for MP Board Exams

• Weightlessness (भारहीनता): Occurs when the effective reaction force on a body is zero (e.g., inside a freely falling lift or in an orbiting satellite).
• Center of Gravity vs Center of Mass: Center of gravity depends on g, center of mass depends purely on mass distribution.
• Inertial Mass vs Gravitational Mass: Inertial mass is measured via F = ma; Gravitational mass is measured via Newton's law of gravitation F = (G M m)/r^2. Both are numerically equal.