Gravitation Class 11 Notes: Key Concepts & Formulas

Key Definitions in Gravitation

Gravitation

The universal attractive force acting between any two objects having mass, which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

Universal Gravitational Constant (G)

The constant of proportionality in Newton's Law of Gravitation, numerically equal to 6.67 × 10^-11 N m²/kg², representing the strength of gravitational interaction.

Acceleration due to Gravity (g)

The acceleration experienced by a body due to the Earth's gravitational pull near its surface. Its value is approximately 9.8 m/s².

Gravitational Field Intensity (E)

The gravitational force experienced per unit mass at a point in a gravitational field. It is a vector quantity, E = F/m = GM/r².

Gravitational Potential (V)

The work done per unit mass in bringing a test mass from infinity to a point in the gravitational field without acceleration. It is a scalar quantity, V = -GM/r.

Gravitational Potential Energy (U)

The work done in bringing a mass 'm' from infinity to a point in the gravitational field of another mass 'M'. It is a scalar quantity, U = -GMm/r.

Escape Velocity (ve)

The minimum velocity with which a body must be projected vertically upwards from the surface of a planet so that it escapes the planet's gravitational field and never returns. ve = sqrt(2GM/R).

Orbital Velocity (vo)

The velocity required by a satellite to remain in a stable orbit around a planet. vo = sqrt(GM/r).

Newton's Universal Law of Gravitation

Newton's Law of Gravitation is a cornerstone of classical mechanics, describing the attractive force between any two objects with mass. It states that every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

The mathematical expression for this force (F) between two point masses, m₁ and m₂, separated by a distance 'r', is:

F = G (m₁ m₂) / r²

Here, G is the Universal Gravitational Constant, a fundamental constant with an approximate value of 6.67 × 10⁻¹¹ N m²/kg². It's important to remember that G is universal, meaning its value is the same everywhere in the universe. The gravitational force is always attractive and acts along the line joining the centers of the two masses.

Key properties of Gravitational Force:

• It is a universal force, acting between all objects with mass.
• It obeys the inverse square law (F ∝ 1/r²).
• It is a central force, acting along the line connecting the centers of the two bodies.
• It is a conservative force, meaning the work done by gravitation depends only on the initial and final positions, not on the path taken.
• It is the weakest of the four fundamental forces of nature. Despite this, its long range and cumulative effect over massive bodies make it dominant on astronomical scales.
• It forms an action-reaction pair; if mass m₁ exerts a force on m₂, then m₂ exerts an equal and opposite force on m₁.

Understanding these properties is crucial for solving problems and conceptual questions related to gravitation. Remember to distinguish between the universal constant G and the acceleration due to gravity g.

Kepler's Laws of Planetary Motion

Variation of Acceleration due to Gravity (g)

1. —
2. —
3. —

Worked Examples & Application of Formulas

• {"heading":"Example 1: Gravitational Force Calculation","bodyMarkdown":"Q: Calculate the gravitational force between two spheres each of mass 100 kg, whose centers are 1 m apart. (G = 6.67 × 10⁻¹¹ N m²/kg²)\n\nA: Given: m₁ = 100 kg, m₂ = 100 kg, r = 1 m, G = 6.67 × 10⁻¹¹ N m²/kg².\nUsing formula F = G (m₁ m₂) / r²\nF = (6.67 × 10⁻¹¹) (100 100) / 1²\nF = 6.67 × 10⁻¹¹ * 10⁴\nF = 6.67 × 10⁻⁷ N"}
• {"heading":"Example 2: Variation of 'g' with Height","bodyMarkdown":"Q: At what height above Earth's surface would the acceleration due to gravity be 1/4th of its value on the surface? (Assume Earth's radius R = 6400 km)\n\nA: Given g' = g/4. We use g' = g / (1 + h/R)².\ng/4 = g / (1 + h/R)²\n4 = (1 + h/R)²\n2 = 1 + h/R\nh/R = 1\nh = R = 6400 km\nSo, at a height equal to the Earth's radius, 'g' becomes 1/4th."}

Must Remember: Key Points for Gravitation

• Newton's Law of Gravitation: F = G * (m₁m₂) / r².
• Universal Gravitational Constant G (scalar, 6.67 × 10⁻¹¹ N m²/kg²) is different from g (vector, acceleration due to gravity, ~9.8 m/s²).
• Acceleration due to gravity g = GM/R² for a planet of mass M and radius R.
• Kepler's Laws (Orbits, Areas, Periods) describe planetary motion around the Sun.
• Value of g decreases with increasing altitude and increasing depth from Earth's surface.
• g is maximum at poles and minimum at the equator due to Earth's rotation.
• Gravitational Potential V = -GM/r and Gravitational Potential Energy U = -GMm/r are always negative and scalar quantities.
• Escape Velocity ve = sqrt(2GM/R) and Orbital Velocity vo = sqrt(GM/r). Note ve = sqrt(2) * vo for an orbit very close to the surface.
• Geostationary satellites have a time period of 24 hours, orbit at a specific height (~36,000 km), and appear stationary relative to Earth's surface.

Exam Tip: Avoiding Common Traps

When solving problems in Gravitation, pay close attention to units and dimensions. Always convert values to SI units (meters, kilograms, seconds) before calculation. Differentiate clearly between scalar quantities (like Gravitational Potential, Potential Energy, G) and vector quantities (like Force, Field Intensity, g). Remember the signs for potential and potential energy; they are negative, indicating an attractive force and a bound system. Don't confuse the inverse square law of gravitation with other inverse square laws. For Kepler's 3rd Law, ensure T² ∝ a³ is applied correctly, especially for different planets or satellites. Carefully read if a question asks for value at a specific height or distance from the center.

Practice Questions with Solutions

• Q: State two characteristic properties of the gravitational force. A: It is always attractive, acts along the line joining the centers of the masses, and obeys the inverse square law. (Any two)
• Q: How does the acceleration due to gravity change if the Earth suddenly stops rotating? A: If Earth stops rotating, the centrifugal effect Rω²cos²(λ) would become zero. This would cause 'g' to increase everywhere except at the poles (where λ=90° and cos(90°)=0). The increase would be maximum at the equator.
• Q: What is the significance of the negative sign in the expression for gravitational potential energy? A: The negative sign indicates that the gravitational force is attractive. It also implies that the system is a bound system, meaning work must be done against the gravitational force to separate the masses to infinity.
• Q: Distinguish between geostationary and polar satellites based on their orbits. A: Geostationary satellites orbit in the equatorial plane with a 24-hour period, appearing stationary from Earth. Polar satellites orbit in a north-south direction, passing over poles, with a much shorter period and covering different parts of the Earth at different times.

Frequently Asked Questions