Chapter 10 — Gravitation

What is Gravitation?
Gravitation (or gravity) is the attractive force that acts between all masses. Every object in the universe attracts every other object. On Earth, gravitation is the reason why objects fall to the ground, why the Moon orbits the Earth, and why the planets orbit the Sun.

Direction and effect of gravity on Earth
Near Earth’s surface, the gravitational force on an object is directed toward the centre of the Earth. This force gives an object weight. If there were no gravity, objects would float away and there would be no atmosphere.

Mass vs Weight
Mass is the amount of matter in an object and is constant everywhere (measured in kg). Weight is the gravitational force on that mass and depends on the local acceleration due to gravity g (measured in newton, N). For example, a person’s mass remains the same on Earth and Moon, but the weight is smaller on the Moon because the Moon’s gravity is less.

Free fall and apparent weightlessness
When an object falls under the influence of gravity alone (no support or air resistance), it is said to be in free fall and all parts accelerate equally at g. Astronauts in an orbiting spacecraft feel weightless because both they and the spacecraft are falling around the Earth together with the same acceleration (free fall), even though gravity still acts on them.

Universal nature — Newton’s insight
Isaac Newton proposed that gravitational force is universal: every pair of masses attracts each other. The magnitude of this force depends on the masses and the distance between them. This law explains terrestrial phenomena (falling objects) and celestial phenomena (orbits, tides).

• Key points to remember:
  • Gravitation is always attractive.
  • Gravitational force acts between any two masses, however small, regardless of the medium between them.
  • Weight depends on g; mass does not.
  • g on Earth ≈ 9.8 m/s² (varies slightly with location, altitude, and depth).

Definition: The Universal Law of Gravitation (proposed by Sir Isaac Newton) states that every two point masses in the universe attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.

Mathematical statement: If two masses m1 and m2 are separated by a distance r (centre to centre), the gravitational force between them is

F = G · (m1 · m2) / r²

Here G is the universal gravitational constant (G = 6.674 × 10^-11 N·m²/kg²). The force acts along the line joining the two masses and is always attractive.

Key points & consequences:

• Universality: The law applies to all masses anywhere in the universe (planets, stars, apples, humans).
• Inverse-square dependence: If the distance between the masses is doubled, the force becomes one-fourth. In general F ∈ 1/r².
• Proportional to mass product: Doubling either mass doubles the force.
• Action-at-a-distance: The force acts without physical contact, mediated by the gravitational field.
• Always attractive: Gravity never repels.
• Superposition principle: The net gravitational force on a mass from many other masses equals the vector sum of forces from each mass individually.

Gravitational acceleration (g) near a planet: Consider a mass m near a planet of mass M and radius R. Using the law with r = R (centre to surface), the gravitational force on m is F = G M m / R². This force per unit mass is the acceleration due to gravity, g:

g = G M / R²

On Earth, M ≈ 5.97 × 10²⁴ kg and R ≈ 6.37 × 10⁶ m, so g ≈ 9.8 m/s². Weight W of an object of mass m is W = m g.

Why all bodies fall with same acceleration (neglecting air resistance): From F = G M m / R² = m g, the mass m of the falling object cancels when using Newton's second law (F = m a), giving a = g. Thus different masses have the same acceleration under gravity.

Measurement of G: The constant G is very small, so gravitational forces between laboratory-sized masses are tiny. It was first measured by Henry Cavendish using a torsion balance in the late 18th century.

Range: Gravity has infinite range but becomes extremely weak at large distances.

Common misunderstandings: Gravity is not caused by air or being 'heavy' only; it depends on masses and distance. The gravitational force between two small everyday objects is present but negligible compared to Earth's pull.

Mass: Mass is the amount of matter in a body. It is a scalar quantity and does not change with location. SI unit: kilogram (kg). Mass is measured using a balance (e.g., beam balance) which compares an unknown mass with standard masses — a beam balance gives the same reading anywhere because it compares gravitational forces on two masses.

Weight: Weight is the gravitational force with which a planet (e.g., Earth) attracts a body. It is a vector (acts vertically downwards on Earth) and depends on both the mass of the body and the local acceleration due to gravity. SI unit: newton (N). Weight is measured using a spring balance (which measures the force exerted by gravity through extension of the spring) and can change with location (altitude, latitude) or with acceleration of the reference frame (e.g., elevator).

Key differences (summary): Mass is intrinsic and constant (kg). Weight is a force and variable (N). A person’s mass remains the same on Earth, Moon or in space; their weight changes.

Why weight changes with place: The acceleration due to gravity g depends on the planet’s mass and the distance r from the planet’s centre: g = GM/r². On the Moon g is about 1/6 that on Earth, so the same mass weighs less. In free-fall or orbit (microgravity) apparent weight can be zero even though mass is unchanged.

Measurement notes: Beam balances compare masses and are unaffected by local g; spring balances measure weight (force) and therefore read different values if g changes or if the object accelerates with the scale.

Definition: Acceleration due to gravity (symbol g) is the acceleration experienced by an object because of the gravitational pull of the Earth. It is directed towards the centre of the Earth. Near Earth's surface its magnitude is approximately 9.8 m/s².

Origin (Newtonian view): From Newton's law of gravitation, the gravitational force on a mass m at distance R from Earth's centre is F = G M m / R². The acceleration produced (independent of m) is

g = GM / R²

Here G is the universal gravitational constant, M is the mass of Earth and R is Earth's radius. Substituting M ≈ 5.97×10²⁴ kg and R ≈ 6.37×10⁶ m gives g ≈ 9.8 m/s².

Free fall (neglecting air resistance): Any body released from rest falls with acceleration g. The kinematic equations (taking downward as positive) are:

• v = u + g t
• s = u t + 1/2 g t²
• v² = u² + 2 g s

For objects dropped from rest u = 0, so v = g t and s = 1/2 g t².

Weight and apparent weight: Weight of a body of mass m is the gravitational force W = m g. If the system accelerates (e.g., in an elevator), apparent weight changes: W' = m (g ± a) depending on direction of acceleration.

Variations of g: g is not exactly same everywhere. It decreases with height: g(h) = g (R/(R + h))². It also varies slightly with latitude because Earth is not a perfect sphere and due to Earth's rotation (centrifugal effect reduces effective g slightly at equator).

Measurement: A common classroom method to determine g is the simple pendulum. For small oscillations of length L, the period T is T = 2π √(L/g), so g can be found from g = 4π² L / T².

Practical notes: In real situations air resistance acts and eventually an object may reach terminal velocity when drag equals weight, so acceleration becomes zero. For most Class 9 problems, air resistance is neglected and g = 9.8 m/s² (often approximated as 10 m/s² for calculations).

Free fall is the motion of an object when the only force acting on it is gravity (air resistance is negligible). Near Earth's surface this gravitational acceleration is nearly constant and is denoted by g. Its magnitude is about 9.8 m/s² (often approximated as 10 m/s² for calculations).

Key ideas

• Acceleration: In free fall the object has a constant acceleration of magnitude g directed toward the centre of the Earth (downwards). This acceleration is independent of the mass of the object.
• Sign convention: Choose a positive direction (commonly upward positive). Then acceleration due to gravity is a = −g (if up is positive) or a = +g (if down is positive).
• Two common situations:
  • Object dropped from rest (initial velocity u = 0).
  • Object thrown vertically upward (initial velocity u upward > 0) and slows down under gravity until it momentarily stops at maximum height, then falls back.
• Mass independence: In absence of air resistance, all objects accelerate equally under gravity (Galileo’s result / demonstrated by vacuum experiments).

Equations of motion (constant acceleration with a = ±g). Using standard kinematic relations, where u = initial velocity, v = velocity after time t, s = displacement in time t:

• v = u + g t (if downward is positive). If upward positive, use v = u − g t.
• s = u t + (1/2) g t² (downward positive). If upward positive, s = u t − (1/2) g t².
• v² = u² + 2 g s (downward positive). If upward positive, v² = u² − 2 g s.

Special results (use g = 9.8 m/s² or 10 m/s²):

• Time to fall from height h (dropped from rest): t = sqrt(2h/g).
• Velocity on hitting ground after falling height h: v = sqrt(2 g h) (directed downward).
• For object thrown vertically up with initial speed u: maximum height H = u² / (2 g); time to reach top = u / g; total time (up and down to same level) = 2 u / g.

Limitations: These relations hold when air resistance is negligible. In real-life situations (feathers, paper) air drag significantly alters motion; parachutes reduce acceleration so descent is not free fall.

What is g? g is the acceleration due to gravity at a point near Earth. At Earth's surface g = GM/R^2 (G = gravitational constant, M = mass of Earth, R = radius of Earth).

Variation of g with altitude (above Earth's surface)

If a point is at height h above the surface, its distance from Earth's centre is r = R + h. Using Newton's law of gravitation:

g_h = GM/(R + h)^2

But g (at surface) = GM/R^2, so

g_h = g · (R/(R + h))^2

For small heights (h << R) you can use binomial approximation:

g_h ≈ g (1 - 2h/R) (approximate linear decrease)

So g decreases as altitude increases. In the limit h → ∞, g → 0.

Variation of g with depth (below Earth's surface)

Use the shell theorem: a uniform spherical shell exerts no net gravitational force on an object inside it. Thus at a depth d below the surface, only the mass of the inner sphere of radius r = R - d contributes.

Mass of inner sphere M' = M · (R - d)^3 / R^3. The acceleration at that point is

g_d = G M' / (R - d)^2 = G M (R - d) / R^3

Since g = GM/R^2,

g_d = g · (1 - d/R)

So g decreases linearly with depth and becomes zero at the centre (d = R).

Key assumptions

• Earth is taken as a perfect sphere.
• For the linear depth formula we assume uniform density (or use shell theorem to justify dependence solely on inner mass).
• Rotational effects (centrifugal force) and local density variations are neglected here; these cause small additional differences (e.g., pole vs equator).

Quick physical picture

• Above the surface: distance from centre increases → gravitational pull falls off roughly as 1/r^2.
• Below the surface: outer spherical shells cancel out → only inner mass pulls, and effective g falls roughly linearly to zero at the centre.

When does this matter?

At everyday heights (buildings, mountains) the change in g is very small. At large heights (satellite orbits) the change is significant. Inside deep mines or towards Earth’s centre the change is linear and can become large (g → 0 at centre).

Definitions:

Inertial mass is a measure of an object's resistance to change in its motion when a force is applied. It appears in Newton's second law: F = m_i a, where m_i is the inertial mass.

Gravitational mass measures how strongly an object interacts with a gravitational field. It appears in Newton's law of gravitation: F_g = G (m_g M)/r^2, where m_g is the gravitational mass of the small body and M is the mass of the attracting body (for example, Earth).

Why the distinction? Conceptually, the two masses describe different roles: one resists acceleration (inertia), the other produces/experiences gravitational attraction. If m_i and m_g were different, objects with different compositions would accelerate differently under the same gravitational field.

Connection and equivalence: In practice experiments show that m_i and m_g are proportional to extremely high precision. If we set units so the proportionality constant is 1, m_i = m_g for all tested materials. This empirical equality is the basis of the equivalence principle, which was a key idea leading Einstein to general relativity.

Simple derivation showing independence of mass in free fall: An object of gravitational mass m_g in the gravitational field of Earth (mass M) feels a gravitational force F_g = G m_g M / r^2. Using Newton's second law F = m_i a gives m_i a = G m_g M / r^2. Thus a = (m_g/m_i) (G M / r^2). If m_g = m_i then a = G M / r^2 = g and is independent of the falling object's mass. This is why all bodies fall with the same acceleration in vacuum.

How they are measured:

• Inertial mass can be measured by applying a known force and measuring the resulting acceleration (accelerometer method).
• Gravitational mass is measured by comparing the gravitational attraction or by weighing the object in a gravitational field (balance or torsion balance methods, e.g. the Eötvös experiment).

Practical significance: Because m_i and m_g are equal to high precision, we use a single quantity called "mass" in everyday physics and engineering. However, the conceptual difference is important in advanced physics (tests of fundamental principles, general relativity, precision experiments).

Units: Both inertial and gravitational mass are measured in kilograms (kg) in the SI system.

Historical/experimental note: Experiments by Galileo (qualitative), and precise torsion-balance tests by Eötvös and later experiments, show no measurable difference between m_i and m_g within extremely small experimental limits.

Definition: Weightlessness (apparent zero weight) occurs when there is no contact force (normal force) supporting an object — the object does not press on any support. Microgravity refers to a condition of very small residual acceleration (typically 10^-6 to 10^-3 g) experienced by objects in near-weightless environments.

Why weightlessness happens: When an object and its surroundings are in free fall under gravity, every part of the system accelerates equally downward. Because there is no support force, occupants and loose objects float relative to the spacecraft or container — they are weightless. Important point: weightlessness does not mean gravity is absent. For example, astronauts in a space station feel weightless while gravity at that altitude is still about 0.88 g; they are continuously falling around the Earth (orbital free fall).

Orbit and continual free fall: In a circular orbit the gravitational force provides the exact centripetal acceleration required to make an object move in a circle. The spacecraft and everything inside fall toward Earth at the same rate; hence internal normal forces vanish and occupants float. This is why the International Space Station (ISS) appears to be a microgravity environment.

Microgravity — why it is not perfectly zero: Several small effects produce tiny accelerations inside spacecraft, so gravitation is not exactly cancelled everywhere. Sources of residual acceleration include:

• Gravity gradient or tidal forces (gravity slightly stronger on the near side than the far side).
• Atmospheric drag and occasional thruster firings or attitude control maneuvers.
• Rotations, mechanical vibrations, crew movement and fluid sloshing.

These small influences produce microgravity (very small but nonzero apparent accelerations). Microgravity is sufficient to change fluid behavior, convection, combustion and biological processes compared with Earth conditions.

Common classroom explanation (elevator model): If an elevator accelerates downward with acceleration a = g, then the normal force on a scale becomes zero and occupants will feel weightless. More generally, if the elevator accelerates with acceleration a (upward positive), the apparent weight (normal force) is N = m(g + a) for upward acceleration, and N = m(g - a) for downward acceleration. When a = g downward, N = 0.

Key conceptual takeaway: Weightlessness = absence of support force / apparent weight zero. Microgravity = very small residual accelerations in an environment that is nearly free fall.

Overview
A satellite is any body that moves around a larger body due to gravitational attraction. The Moon is a natural satellite of the Earth. Artificial satellites are launched by humans and move around Earth (or other planets) in controlled orbits. The motion of satellites is governed by gravity which provides the centripetal force required to keep them moving in curved paths.

Gravity as Centripetal Force
For a satellite of mass m orbiting a planet of mass M at a distance r from the planet's centre, the gravitational force provides the centripetal force that keeps the satellite in orbit. In the special case of a circular orbit:

Gravitational force: F_g = G M m / r^2
Centripetal force required: F_c = m v^2 / r

Equating F_g = F_c and cancelling m gives the orbital speed for a circular orbit:

v = sqrt(G M / r)

The time taken to complete one orbit (period T) is the circumference divided by speed. For a circular orbit:

T = 2π r / v = 2π sqrt(r^3 / (G M))

These relations show that orbital speed decreases with increasing orbital radius (v ∝ 1/√r) and the period increases with radius (T ∝ r^{3/2}). These are consequences of Newtonian gravity and are consistent with Kepler's laws.

Important special orbits

• Low Earth Orbit (LEO): Altitudes ~160–2000 km (example: ISS ~408 km). Orbital speed ~7.6–7.9 km/s.
• Medium Earth Orbit (MEO): For example GPS satellites at ~20,200 km altitude (period ~12 h).
• Geostationary orbit (GEO): Circular orbit above the equator with period equal to Earth’s rotation (24 h). Radius ≈ 42,164 km from Earth's centre, altitude ≈ 35,786 km. From the ground a GEO satellite appears fixed in the sky.

Natural satellite: The Moon
The Moon orbits Earth at an average distance ~384,400 km with orbital speed ≈ 1.02 km/s and a sidereal period ≈ 27.3 days. The same face of the Moon faces Earth because the Moon is tidally locked (its rotation period equals its orbital period).

Weightlessness and microgravity
Astronauts inside an orbiting spacecraft feel weightless because both they and the spacecraft are in free fall around Earth (falling toward Earth but moving forward fast enough to keep missing it). Small residual accelerations from atmospheric drag or manoeuvres make it microgravity rather than perfect zero gravity.

Other effects in the Earth–Moon–Satellite system
In reality, orbits are influenced by more than one massive body (Earth, Moon, Sun), atmospheric drag (for low orbits), and Earth's nonspherical mass distribution. These cause perturbations, precession, and long-term changes. Interactions among Earth, Moon and satellites also cause tides and tidal locking effects.

Key numerical constants
Universal gravitational constant G = 6.67×10^{-11} N·m^2/kg^2. Earth's mass M_earth ≈ 5.97×10^{24} kg. Earth's mean radius R_earth ≈ 6.371×10^6 m. Standard gravitational parameter μ = G M_earth ≈ 3.986×10^{14} m^3/s^2.

What are tides?
Tides are the periodic rise and fall of sea level produced mainly by the gravitational pull of the Moon and the Sun on the Earth, together with the inertia (centrifugal effects) of the rotating Earth–Moon system.

How tides are produced (simple picture)

• The Moon pulls on every part of the Earth. The gravitational pull is slightly stronger on the side of the Earth nearer to the Moon and slightly weaker on the far side. This difference in pull produces two opposite bulges of water on the Earth's oceans: one on the side facing the Moon and one on the opposite side.
• The bulge toward the Moon is caused by the stronger direct pull of the Moon. The bulge on the far side arises because the Earth's centre is pulled more strongly toward the Moon than the far-side water, or equivalently because of the centrifugal effect of the Earth–Moon rotation about their common centre of mass.
• As the Earth rotates, a given coastal location passes through these bulges, producing (normally) two high tides and two low tides every lunar day.

Timing and frequency
A lunar day (time for the Earth to rotate once relative to the Moon) is about 24 h 50 min. Thus high tides at a place are about 50 minutes later each solar day. Because there are two bulges, the interval between two successive high tides is roughly 12 h 25 min.

Spring and neap tides

• Spring tides: When the Sun, Moon and Earth are almost in a line (at new moon and full moon), the tidal effects of the Sun and Moon add. This gives higher high tides and lower low tides (large tidal range).
• Neap tides: When the Sun and Moon are at right angles with respect to the Earth (first and third quarter), the Sun’s tidal effect partially cancels the Moon’s. This gives smaller tidal range (lower high tides and higher low tides).

Other related effects

• Tidal bore: In some rivers with funnel-shaped estuaries a sudden surge of seawater (a tidal bore) can travel upstream at high tide (examples: Qiantang River in China, Severn in UK).
• Tidal power: Tides can be harnessed for electricity in suitable locations (tidal barrages, tidal turbines).
• Earth tides: The solid Earth also deforms slightly under lunar and solar tides (vertical shifts of the ground by up to ~30–40 cm in some places).
• Variation of weight: The gravitational pull of the Moon and Sun causes tiny changes in the apparent weight of objects on Earth; these are measurable only with sensitive instruments.
• Tidal locking: Over long times, tidal friction can synchronise a satellite’s rotation with its orbit (the Moon always shows nearly the same face to Earth).

Key points to remember

• Moon’s tidal effect on Earth is larger than the Sun’s tidal effect even though the Sun’s gravitational pull is stronger—this is because tidal effect depends on the gradient of gravity and varies roughly as 1/R^3 (distance cubed), so the much smaller distance to the Moon makes its tidal influence dominant.
• Tides are periodic and predictable; local coastline shape and seabed depth strongly affect actual tidal heights (amplitude) and times.

Overview
Typical numerical problems in Class 9 Gravitation center on applying Newton's law of gravitation and derived expressions for acceleration due to gravity (g) at or away from Earth's surface. Key derivations include g on the surface (g = GM/R^2), g at a height h above the surface, g at a depth d below the surface, and the variation of g with radial distance inside the Earth.

1. From universal law to g on Earth's surface
Newton's universal law: F = G m1 m2 / r^2. For a mass m near Earth (mass M, radius R) the gravitational force (weight) is W = G M m / R^2. Define acceleration due to gravity g by W = m g, so g = G M / R^2. This is the basic derivation used in most numerical problems.

2. g at a height h above the surface
At height h (distance from Earth's centre = R + h), g_h = G M / (R + h)^2. Using g = GM/R^2, we get g_h = g (R/(R + h))^2. For small h (h << R) use binomial approximation: g_h ≈ g (1 - 2h/R).

3. g at a depth d below the surface
Assuming uniform Earth density, mass enclosed within radius (R - d) is M_enclosed = M (R - d)^3 / R^3. Gravitational acceleration at that depth is g_d = G M_enclosed / (R - d)^2 = g (1 - d/R). (To first order, g decreases linearly with depth.)

4. g at distance r inside the Earth (general radial dependence)
For r < R (inside Earth): g(r) = G (M_enclosed) / r^2 = G M r / R^3 = g (r/R). So g increases linearly from centre (g = 0) to surface (g = g).

5. Typical numerical problem types

1. Compute g at a given height h or depth d using g_h = g (R/(R + h))^2 and g_d = g (1 - d/R).
2. Compute weight change: W_h = m g_h or W_d = m g_d.
3. Compute gravitational force between two masses separated by distance r: F = G m1 m2 / r^2.
4. Compare g on different planets or Moon using g = GM/R^2 or using known ratios (e.g., Moon's g ≈ 1/6 Earth's).
5. Estimate small fractional changes using approximations (e.g., for small h, Δg/g ≈ -2h/R).

Hints for solving numerical problems
• Use consistent SI units: mass in kg, distances in meters, G = 6.67 × 10^-11 N m^2 kg^-2.
• For heights much smaller than Earth's radius use binomial approximations to simplify calculations.
• For depth problems assume uniform density unless otherwise stated (g_d = g (1 - d/R)).
• Check whether the problem expects an approximate or exact answer (exact uses (R/(R+h))^2 formula).