Class 11 Physics Notes Chapter 8 (Chapter 8) – Examplar Problems (English) Book

Alright class, let's begin our detailed study of Chapter 8: Gravitation, focusing on the concepts crucial for your competitive government exams, drawing insights from the NCERT Exemplar problems.

Chapter 8: Gravitation - Detailed Notes

1. Introduction & Newton's Law of Universal Gravitation

• Concept: Every particle of matter in the universe attracts every other particle with a force.
• Newton's Law: The force of attraction between two point masses (m₁ and m₂) separated by a distance (r) is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
  • Formula: F = G (m₁m₂ / r²)
  • G: Universal Gravitational Constant. It's a scalar quantity.
    • Value: G ≈ 6.67 × 10⁻¹¹ Nm² kg⁻²
    • Dimensional Formula: [M⁻¹ L³ T⁻²]
  • Vector Form: The force exerted by m₁ on m₂ (F₂₁) is given by:
    F₂₁ = - G (m₁m₂ / r²₁₂) r̂₂₁ = - G (m₁m₂ / |r₂₁|³) r₂₁
    where r₂₁ is the position vector from m₁ to m₂, and r̂₂₁ is the unit vector along that direction. The negative sign indicates the force is attractive.
  • Key Properties:
    • Independent of the medium between the masses.
    • Obeys Newton's third law (F₁₂ = -F₂₁).
    • It's a central force (acts along the line joining the centers).
    • It's a conservative force (work done is path independent).
    • Obeys the principle of superposition (net force on a mass due to multiple masses is the vector sum of individual forces).

2. Acceleration Due to Gravity (g)

• Definition: The acceleration experienced by a body falling freely under the influence of Earth's gravity alone.
• Relation between g and G: Consider Earth as a uniform sphere of mass M and radius R. The force on a small mass m on its surface is F = GMm / R². By Newton's second law, F = mg.
  • Therefore, mg = GMm / R² => g = GM / R²
• Factors Affecting g:
  • Altitude (h): At a height h above the Earth's surface:
    g' = GM / (R+h)² = g [R / (R+h)]² = g (1 + h/R)⁻²
    If h << R (small altitude), using binomial expansion:
    g' ≈ g (1 - 2h/R) (g decreases with altitude)
  • Depth (d): At a depth d below the Earth's surface:
    Mass of the inner sphere of radius (R-d) is M' = M [(R-d)³/R³] (assuming uniform density ρ = M / (4/3 πR³)).
    g' = GM' / (R-d)² = G [M(R-d)³/R³] / (R-d)² = (GM/R³) (R-d)
    Since g = GM/R², we have g' = g (R-d)/R
    g' = g (1 - d/R) (g decreases with depth)
    At the center of the Earth (d=R), g' = 0.
  • Shape of Earth: Earth is not a perfect sphere but an oblate spheroid (bulges at the equator, flattened at the poles). R<0xE2><0x82><0x91> < R<0xE1><0xB5><0xA4>. Since g ∝ 1/R², g is maximum at the poles and minimum at the equator.
  • Rotation of Earth (Latitude λ): Due to Earth's rotation, a body experiences a centrifugal force. The effective acceleration due to gravity at latitude λ is:
    g' = g - Rω²cos²λ
    where ω is the angular velocity of Earth.
    At poles (λ = 90°), cosλ = 0, so g' = g (maximum).
    At equator (λ = 0°), cosλ = 1, so g' = g - Rω² (minimum).

3. Gravitational Field (E)

• Definition: The space around a massive body where its gravitational influence (force) can be experienced by another mass.
• Gravitational Field Intensity (E): The gravitational force experienced per unit test mass placed at that point.
  • Formula: E = F / m₀ = GM / r² r̂ (directed towards the source mass M)
  • Units: N kg⁻¹ or m s⁻²
  • Dimensional Formula: [M⁰ L T⁻²]
• It's a vector quantity.

4. Gravitational Potential (V)

• Definition: The work done per unit test mass in bringing it from infinity to a point in the gravitational field, without acceleration.
  • Formula: V = Work / m₀ = - GM / r
  • It's a scalar quantity.
  • Units: J kg⁻¹
  • Dimensional Formula: [M⁰ L² T⁻²]
• The negative sign indicates that the potential at infinity is taken as zero (maximum value), and potential decreases as we approach the mass.
• Relation between Field and Potential: E = - dV/dr (Gravitational field is the negative gradient of gravitational potential).

5. Gravitational Potential Energy (U)

• Definition: The work done in bringing a given mass (m) from infinity to a point in the gravitational field of another mass (M). Alternatively, it's the energy possessed by a system of masses due to their mutual gravitational attraction.
  • Formula: U = m × V = - GMm / r
  • It's a scalar quantity.
  • Units: Joule (J)
  • Dimensional Formula: [M L² T⁻²]
• Change in Potential Energy: ΔU = U₂ - U₁ = GMm (1/r₁ - 1/r₂)
• Potential Energy near Earth's Surface: If an object of mass m is raised by a height h (h<<R) near the Earth's surface, the change in potential energy is approximately ΔU ≈ mgh. This is derived from the general formula considering r₁ = R and r₂ = R+h.

6. Escape Velocity (vₑ)

• Definition: The minimum initial velocity required for an object to overcome the gravitational pull of a celestial body (like Earth) and escape to infinity, without any further propulsion.
• Derivation (Conservation of Energy):
  Total Energy on Surface = Total Energy at Infinity
  (K.E. + P.E.)<0xE2><0x82><0x92><0xE1><0xB5><0xA9><0xE1><0xB5><0xA3><0xE1><0xB5><0x93><0xE1><0xB5><0x84><0xE1><0xB5><0x8A> = (K.E. + P.E.)<0xE2><0x82><0x9E>
  (½ mvₑ² ) + (- GMm / R) = 0 + 0 (At infinity, potential energy is zero, and minimum velocity means K.E. is also zero)
  ½ mvₑ² = GMm / R
  vₑ = √(2GM / R)
• Alternative Form: Since g = GM / R², we have GM = gR².
  vₑ = √(2gR² / R) = √(2gR)
• Value for Earth: ≈ 11.2 km/s
• Key Points:
  • Independent of the mass (m) of the projected body.
  • Independent of the angle of projection (as long as it's projected away from the surface).
  • Depends on the mass (M) and radius (R) of the celestial body.

7. Satellites

• Definition: A body revolving in an orbit around a larger body (planet).
• Orbital Velocity (v<0xE1><0xB5><0xA2>): The velocity required to put a satellite into a stable circular orbit around a planet.
  • Derivation: The necessary centripetal force is provided by the gravitational force.
    mv<0xE1><0xB5><0xA2>²/r = GMm/r² (where r = R+h is the orbital radius)
    v<0xE1><0xB5><0xA2> = √(GM / r) = √(GM / (R+h))
  • Alternative Form: v<0xE1><0xB5><0xA2> = √(gR² / r)
  • For a satellite orbiting close to Earth (h << R, so r ≈ R):
    v<0xE1><0xB5><0xA2> ≈ √(GM / R) = √(gR)
  • Value for Earth (close orbit): ≈ 7.92 km/s (often approximated as 8 km/s)
  • Relation between vₑ and v<0xE1><0xB5><0xA2> (near surface): vₑ = √2 * v<0xE1><0xB5><0xA2>
• Time Period (T): Time taken for one complete revolution.
  • T = Distance / Speed = 2πr / v<0xE1><0xB5><0xA2>
  • T = 2πr / √(GM/r) = 2π √(r³/GM)
  • T² = (4π²/GM) r³ => T² ∝ r³ (This is Kepler's Third Law for circular orbits)
• Energy of an Orbiting Satellite:
  • Kinetic Energy (K.E.): K = ½ mv<0xE1><0xB5><0xA2>² = ½ m (GM/r) = GMm / 2r
  • Potential Energy (P.E.): U = - GMm / r
  • Total Mechanical Energy (E): E = K.E. + P.E. = (GMm / 2r) + (- GMm / r) = - GMm / 2r
  • Binding Energy: The energy required to remove the satellite from its orbit to infinity. Binding Energy = -E = GMm / 2r.
  • Note: Total energy is negative, indicating a bound system. K.E. = |E| = -U/2.

8. Kepler's Laws of Planetary Motion

• Based on observations by Tycho Brahe, formulated by Johannes Kepler.
  • First Law (Law of Orbits): Every planet revolves around the Sun in an elliptical orbit with the Sun situated at one of the foci of the ellipse.
  • Second Law (Law of Areas): The line joining a planet and the Sun sweeps out equal areas in equal intervals of time. (dA/dt = constant). This implies that the areal velocity is constant. This law is a consequence of the conservation of angular momentum (L = mvr sinθ = constant, as gravitational force is central, producing no torque). Planets move faster when closer to the Sun (perihelion) and slower when farther away (aphelion).
  • Third Law (Law of Periods): The square of the time period of revolution (T) of a planet around the Sun is directly proportional to the cube of the semi-major axis (a) of its elliptical orbit.
    T² ∝ a³
    For a circular orbit, a = r (radius), so T² ∝ r³.

9. Weightlessness

• Weight: The force with which a body is attracted towards the center of the Earth (W = mg). It's also the reading shown by a weighing scale, which measures the normal reaction force.
• Weightlessness: A state where the apparent weight of a body becomes zero. This occurs when the normal reaction force is zero.
  • Examples:
    • A body in a freely falling lift (a = g, Apparent weight = m(g-a) = 0).
    • A satellite orbiting the Earth (Both the satellite and the astronaut inside are constantly falling towards Earth with acceleration g', but their tangential orbital velocity keeps them in orbit. The astronaut doesn't press against the floor, hence experiences weightlessness).

10. Gravitational Shielding

• Gravitational force cannot be shielded. Unlike electric forces which can be shielded by conductors (Faraday cage), there is no known way to block gravitational influence.

Multiple Choice Questions (MCQs)

1. The value of acceleration due to gravity (g) is maximum at:
   (a) The Equator
   (b) The Poles
   (c) The Center of the Earth
   (d) A height R above the surface (R = Earth's radius)

2. If the distance between two masses is doubled, the gravitational attraction between them:
   (a) Is doubled
   (b) Becomes four times
   (c) Is reduced to half
   (d) Is reduced to a quarter

3. A satellite is orbiting the Earth close to its surface. What minimum additional speed is required for it to escape Earth's gravitational pull? (Let v<0xE1><0xB5><0xA2> be the orbital speed)
   (a) v<0xE1><0xB5><0xA2>
   (b) (√2 - 1) v<0xE1><0xB5><0xA2>
   (c) (√2 + 1) v<0xE1><0xB5><0xA2>
   (d) 2v<0xE1><0xB5><0xA2>

4. The time period T of a satellite revolving in a circular orbit of radius r around a planet of mass M is proportional to:
   (a) r
   (b) r²
   (c) r³/²
   (d) r⁻¹

5. Kepler's second law regarding the constancy of areal velocity of a planet is a consequence of the law of conservation of:
   (a) Energy
   (b) Linear momentum
   (c) Angular momentum
   (d) Mass

6. If the radius of the Earth were to shrink by 1% (its mass remaining the same), the acceleration due to gravity on the Earth's surface would:
   (a) Decrease by 1%
   (b) Increase by 1%
   (c) Decrease by 2%
   (d) Increase by 2%

7. The gravitational potential energy of a system of two particles of masses m₁ and m₂ separated by a distance r is given by:
   (a) G m₁m₂ / r
   (b) - G m₁m₂ / r
   (c) G m₁m₂ / r²
   (d) - G m₁m₂ / r²

8. An astronaut orbiting the earth in a circular orbit experiences weightlessness because:
   (a) There is no gravity in space.
   (b) The gravitational force is balanced by the centrifugal force.
   (c) The astronaut and the satellite are in a state of continuous free fall.
   (d) The mass of the astronaut becomes zero.

9. At what depth below the Earth's surface does the acceleration due to gravity become 75% of its value at the surface? (R = Radius of Earth)
   (a) R/4
   (b) R/2
   (c) 3R/4
   (d) R/3

10. The ratio of the kinetic energy to the total energy of a satellite revolving in a circular orbit is:
    (a) 1 : 1
    (b) 1 : -1
    (c) 1 : 2
    (d) 1 : -2

Answer Key for MCQs:

1. (b) The Poles
2. (d) Is reduced to a quarter (F ∝ 1/r²)
3. (b) (√2 - 1) v<0xE1><0xB5><0xA2> (Escape velocity vₑ = √2 v<0xE1><0xB5><0xA2>. Additional speed = vₑ - v<0xE1><0xB5><0xA2> = (√2 - 1)v<0xE1><0xB5><0xA2>)
4. (c) r³/² (Since T² ∝ r³, T ∝ r³/²)
5. (c) Angular momentum
6. (d) Increase by 2% (g = GM/R². Δg/g = -2 ΔR/R. If R decreases by 1%, ΔR/R = -0.01. So, Δg/g = -2(-0.01) = +0.02 or +2%)
7. (b) - G m₁m₂ / r
8. (c) The astronaut and the satellite are in a state of continuous free fall.
9. (a) R/4 (g' = g(1 - d/R). 0.75g = g(1 - d/R) => 0.75 = 1 - d/R => d/R = 0.25 = 1/4 => d = R/4)
10. (b) 1 : -1 (K.E. = GMm/2r, Total Energy E = -GMm/2r. Ratio K.E./E = (GMm/2r) / (-GMm/2r) = -1 or 1:-1)

Study these notes thoroughly. Pay close attention to the derivations of escape velocity, orbital velocity, and the variations in 'g'. Understanding the energy relations for satellites and Kepler's laws is also vital. Good luck with your preparation!