'Gravitation'

Newtons Universal Law of Gravitation

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 them.

F = G (m_1 m_2)/r^2

Where G = 6.67 x 10^(-11) N m^2/kg^2 is the universal gravitational constant.

Vector Form

vecF_12 = -G (m_1 m_2)/r^2 hatr_12

The negative sign indicates the force is attractive.

Distinction Between G and g

• G is universal constant (same everywhere).
• g varies with location.
• G = 6.67 x 10^(-11) N m^2/kg^2
• g = 9.8 m/s^2 at Earth's surface.

Acceleration Due to Gravity and Its Variation

At Earth's Surface

g = GM/R^2, where M is Earth's mass and R is Earth's radius.

Variation with Height (h above surface)

g_h = g (R/(R+h))^2 For h << R: g_h = g(1 - 2h/R)

Variation with Depth (d below surface)

g_d = g(1 - d/R) At the centre of Earth (d = R): g = 0.

Variation with Latitude

Due to Earth's rotation, effective g is less at the equator and maximum at the poles. g' = g - R omega^2 cos^2 phi Where phi is latitude and omega is Earth's angular speed.

Variation Due to Earth's Shape

Earth is oblate (flattened at poles). g_pole > g_equator.

Gravitational Potential and Potential Energy

Gravitational Potential: Work done per unit mass to bring a mass from infinity to that point. V = -GM/r

Gravitational Potential Energy: U = -GMm/r

• U = 0 at r = infinity.
• U is always negative (attractive force).

Orbital Velocity

Velocity required to put a satellite into orbit around Earth. v_0 = sqrt(GM/r) = sqrt(gR^2/(R+h))

For a satellite close to Earth's surface (h << R): v_0 = sqrt(gR) approx 7.92 km/s.

Escape Velocity

Minimum velocity required to escape Earth's gravitational pull. v_e = sqrt(2GM/R) = sqrt(2gR)

v_e approx 11.2 km/s for Earth.

Relation between orbital and escape velocity: v_e = sqrt(2) * v_0

Escape Velocities of Other Bodies

• Moon: 2.38 km/s
• Mars: 5.02 km/s
• Jupiter: 59.5 km/s
• Sun: 618 km/s

Keplers Laws of Planetary Motion

First Law (Law of Orbits)

Each planet moves around the Sun in an elliptical orbit with Sun at one focus.

Second Law (Law of Areas)

The line joining a planet to the Sun sweeps equal areas in equal intervals of time. This is a consequence of conservation of angular momentum.

Areal velocity: dA/dt = L/(2m) = constant

Third Law (Law of Periods)

The square of the time period of revolution is proportional to the cube of the semi-major axis. T^2 prop a^3

For circular orbits: T^2 = (4pi^2/GM) r^3

Geostationary Satellites

Satellites that appear stationary relative to Earth.

Conditions

1. Orbital period = Earth's rotation period (24 hours).
2. Orbit is in the equatorial plane.
3. Orbit is circular.
4. Orbital radius r = 4.23 x 10^4 km (height h = 3.59 x 10^4 km above surface).

Uses

Communications, television broadcasting, weather monitoring, GPS.

Worked Examples

Example 1: Two bodies of masses 5 kg and 10 kg are 1 m apart. Find gravitational force. Solution: F = G*5*10/1^2 = 6.67x10^(-11)*50 = 3.335 x 10^(-9) N.

Example 2: Find g at a height equal to Earth's radius. Solution: At h = R: g_h = g (R/(2R))^2 = g/4 = 2.45 m/s^2.

Example 3: Find escape velocity from Earth. (Given R = 6400 km, g = 10 m/s^2). Solution: v_e = sqrt(2*10*6.4x10^6) = sqrt(1.28x10^8) = 11314 m/s approx 11.3 km/s.

Common Mistakes

1. G vs g: G is constant; g varies. Do not confuse them.
2. Escape velocity formula: v_e = sqrt(2gR), not sqrt(gR).
3. Negative PE does not mean negative energy: PE is negative due to choice of reference (zero at infinity).
4. g decreases with height: It decreases, following inverse square law.

ISC Exam Focus

• Theory (70%): Newton's law, variation of g, derivation of escape/orbital velocity, Kepler's laws.
• Application (30%): Numerical problems on g variation, orbital velocity, satellite period.
• ISC frequently asks: "Find the height at which g becomes g/4" or "Find orbital velocity of a satellite."
• Kepler's laws and their derivations are important for theory questions.

Self-Test Questions

Q1: State Newton's law of gravitation. Write the value of G with units. Answer: F = G m_1 m_2/r^2. G = 6.67 x 10^(-11) N m^2/kg^2.

Q2: Find the percentage decrease in g at a height of 32 km from Earth's surface (R = 6400 km). Answer: Delta g/g = -2h/R = -64/6400 = -0.01 = -1%. So g decreases by 1%.

Q3: Find orbital velocity of a satellite at 400 km above Earth's surface. (R = 6400 km, g = 10 m/s^2). Answer: v_0 = sqrt(gR^2/(R+h)) = sqrt(10*(6.4x10^6)^2/(6.8x10^6)) = sqrt(6.02x10^7) = 7760 m/s approx 7.76 km/s.

Q4: State Kepler's three laws of planetary motion. Answer: (1) Elliptical orbits with Sun at focus. (2) Equal area in equal time. (3) T^2 prop a^3.

Q5: Find the escape velocity from the Moon (M = 7.4 x 10^22 kg, R = 1740 km). Answer: v_e = sqrt(2GM/R) = sqrt(2*6.67x10^(-11)*7.4x10^22/(1.74x10^6)) = sqrt(5.67x10^6) = 2380 m/s.

Q6: Derive the relation between orbital velocity and escape velocity. Answer: v_0 = sqrt(GM/R), v_e = sqrt(2GM/R) = sqrt(2) * v_0.