Motion and Force

1. Gravitation

Definition: Gravitation is the force of attraction between any two objects in the universe.

Observed by Newton when an apple fell from a tree; led to the Universal Law of Gravitation (1687).

Newton’s Universal Law of Gravitation

The gravitational force (FFF) between two bodies is:

• F=Gm1m2d2F = G \frac{m_1 m_2}{d^2}F=Gd2m1​m2​​

Where:

m1,m2m_1, m_2m1​,m2​ = masses of objects

ddd = distance between their centers

G=6.67×10−11 N\cdotpm²/kg²G = 6.67 \times 10^{-11} \, \text{N·m²/kg²}G=6.67×10−11N\cdotpm²/kg² (gravitational constant)

Observations:

Doubling one mass → force doubles.

Doubling both masses → force quadruples.

Doubling distance → force becomes 1/4.

Halving distance → force becomes 4×.

Consequences:

Keeps planets in orbit around the Sun.

Causes tides (Moon’s gravitational effect on oceans).

Keeps objects on Earth’s surface.

2. Gravity & Weight

Mass (m): Amount of matter in an object (kg) → does not change with location.

Weight (W): Force due to gravity on a mass → changes with location.

• W=mgW = mgW=mg

SI unit: Newton (N)

Vector quantity → points toward the center of the planet.

Key Points:

Weight is directly proportional to mass: W∝mW \propto mW∝m

Weight is directly proportional to acceleration due to gravity: W∝gW \propto gW∝g

Examples of variation:

Earth poles (g = 9.83 m/s²) → weight more

Earth equator (g = 9.78 m/s²) → weight less

Moon (g = 1.62 m/s²) → weight ~1/6 of Earth

3. Acceleration Due to Gravity (g)

Denoted as g. Causes free fall.

Formula from Newton’s law:

• g=GMR2g = \frac{GM}{R^2}g=R2GM​

Depends on planet mass (M) and radius (R)

Independent of falling object’s mass

Earth:

Average g = 9.81 m/s²

Polar region → g ≈ 9.83 m/s²

Equator → g ≈ 9.78 m/s²

Variation with Height:

Higher from surface → g decreases

Example: ISS at 400 km → g ≈ 8.66 m/s²

Comparison:

Moon: g = 1.62 m/s² → ~6 times less than Earth

4. Free Fall

Definition: Motion of an object under gravity alone, ignoring air resistance.

All objects in vacuum fall with same acceleration, independent of mass.

Experiments:

Galileo’s Leaning Tower Experiment: Heavy & light balls fall together.

Feather & Coin Experiment: In vacuum, both fall together.

Equations of Motion for Free Fall:

v=u+gtv = u + gtv=u+gt → final velocity

v2=u2+2ghv^2 = u^2 + 2ghv2=u2+2gh → velocity from height

h=ut+12gt2h = ut + \frac{1}{2} g t^2h=ut+21​gt2 → height fallen

Weightlessness:

In free fall (e.g., astronauts in orbit), objects experience zero apparent weight.

5. Mass, Weight & Other Planets

Mass = constant everywhere; weight = depends on g.

Formula for weight on another planet:

• W=mgplanetW = mg_{\text{planet}}W=mgplanet​

Example: Lifting 100 kg on Earth → can lift 601 kg on Moon (g = 1.63 m/s²)

Weight comparison table (examples):

6. Parachutes & Air Resistance

Free fall → acceleration = g

Parachute fall: air resistance balances weight → uniform speed

Moon → no atmosphere → parachute cannot slow descent → free fall → dangerous landing

Seeds, hail, and parachutes demonstrate effect of air resistance

7. Important Formulas (Exam Focus)

Gravitational Force: F=Gm1m2d2F = G \frac{m_1 m_2}{d^2}F=Gd2m1​m2​​

Weight: W=mgW = mgW=mg

Acceleration due to gravity: g=GMR2g = \frac{GM}{R^2}g=R2GM​

Free fall equations:

v=u+gtv = u + gtv=u+gt

v2=u2+2ghv^2 = u^2 + 2ghv2=u2+2gh

h=ut+12gt2h = ut + \frac{1}{2} g t^2h=ut+21​gt2

Shortcut Observations:

Doubling mass → force doubles

Doubling distance → force 1/4

g decreases with height, increases toward poles

Free fall acceleration independent of mass

8. Quick Conceptual Points

Gravitational force is universal → acts between all objects

Mass = matter, scalar; Weight = force, vector

Free fall possible only if air resistance negligible

Astronauts feel weightless in orbit because all fall together

g on planets depends on mass and radius