Gravitation
Introduction to Gravitation
Key Points:
- Every object attracts every other object
- Force is always attractive (never repulsive)
- Acts between all masses, no matter how small
- Weakest of all fundamental forces
- Infinite range (works across universe)
Universal Law of Gravitation
"Every object in the universe attracts every other object with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers."
F = G × (m₁ × m₂) / r²
Where:
F = Gravitational force (Newton, N)
G = Universal gravitational constant = 6.67 × 10⁻¹¹ N m²/kg²
m₁ = Mass of first object (kg)
m₂ = Mass of second object (kg)
r = Distance between centers of objects (m)
Characteristics of Gravitational Force:
- Universal: Acts between all masses everywhere
- Always Attractive: Always pulls objects together
- Central Force: Acts along the line joining centers
- Inverse Square Law: F ∝ 1/r²
- Independent of Medium: Works in vacuum, air, water
- Long Range: Decreases with distance but never zero
Universal Gravitational Constant (G)
Unit: N m²/kg² or m³ kg⁻¹ s⁻²
Nature: Universal constant (same everywhere in universe)
Discovered by: Henry Cavendish in 1798
Physical Meaning:
G is the force of attraction between two objects of 1 kg mass each placed 1 meter apart.
Gravity
Gravity is a special case of gravitation where one object is Earth.
Acceleration Due to Gravity (g):
Value at Earth's surface: g = 9.8 m/s² (approximately 10 m/s²)
Direction: Always towards the center of Earth (downward)
Formula: g = GM/R²
Where M = Mass of Earth, R = Radius of Earth
Important Points About 'g':
- Independent of mass of falling object
- All objects fall with same acceleration (ignoring air resistance)
- Decreases with height above Earth's surface
- Decreases with depth below Earth's surface
- Varies slightly at different locations on Earth
- Maximum at poles, minimum at equator
Free Fall
Conditions for Free Fall:
- Only gravitational force acts
- Air resistance is negligible
- Initial velocity may or may not be zero
Equations of Motion for Free Fall:
1. v = u + gt
2. h = ut + ½gt²
3. v² = u² + 2gh
Where:
u = initial velocity
v = final velocity
g = 9.8 m/s²
t = time
h = height/distance
Mass and Weight
The amount of matter contained in an object is called mass.
Characteristics:
- Scalar quantity
- SI unit: kilogram (kg)
- Constant everywhere (doesn't change with location)
- Measured by beam balance
- Never zero
- Measure of inertia
The force with which Earth attracts an object is called weight.
Where:
W = Weight (Newton, N)
m = Mass (kg)
g = Acceleration due to gravity (9.8 m/s²)
Characteristics:
- Vector quantity (acts downward)
- SI unit: Newton (N)
- Changes with location (depends on 'g')
- Measured by spring balance
- Can be zero (in space)
- Type of force
| Property | Mass | Weight |
|---|---|---|
| Definition | Amount of matter | Gravitational force |
| Type | Scalar | Vector |
| SI Unit | kilogram (kg) | Newton (N) |
| Depends on location | No (constant) | Yes (varies) |
| Measured by | Beam balance | Spring balance |
| Can be zero | No | Yes (in space) |
Weight on Moon
The mass of Moon is 1/100 times the mass of Earth, and its radius is 1/4 times the radius of Earth.
Therefore, acceleration due to gravity on Moon:
g_moon = (1/6) × g_earth = 9.8/6 ≈ 1.63 m/s²
Implication:
Weight on Moon = (1/6) × Weight on Earth
If your weight on Earth is 600 N, on Moon it would be 100 N!
But your mass remains the same (60 kg)
Thrust and Pressure
SI Unit: Newton (N)
P = F / A
SI Unit: Pascal (Pa) or N/m²
1 Pa = 1 N/m²
Applications of Pressure:
- Sharp knife: Small area → High pressure → Cuts easily
- Broad handles: Large area → Low pressure → Comfortable
- Railway tracks: Large area → Low pressure on ground
- Camel feet: Broad feet → Low pressure on sand
- Needles: Sharp point → High pressure → Penetrates easily
Pressure in Fluids
Key Points:
- Fluids exert pressure in all directions
- Pressure increases with depth
- Pressure is same at same depth
- Pressure acts perpendicular to surface
P = ρgh
Where:
ρ (rho) = Density of fluid (kg/m³)
g = Acceleration due to gravity (9.8 m/s²)
h = Depth (m)
Buoyancy and Archimedes' Principle
Characteristics:
- Acts vertically upward
- Due to pressure difference in fluid
- Depends on volume of object submerged
- Depends on density of fluid
"When an object is immersed in a fluid, it experiences an upward force (buoyant force) equal to the weight of the fluid displaced by it."
F_b = ρ × V × g
Where:
ρ = Density of fluid
V = Volume of fluid displaced
g = Acceleration due to gravity
Applications of Archimedes' Principle:
- Design of ships and submarines
- Hot air balloons and airships
- Hydrometers (measure density)
- Swimming and floating
Conditions for Floating and Sinking:
- Object Floats: When buoyant force ≥ weight of object
- Object Sinks: When buoyant force < weight of object
- Density criterion:
- If density of object < density of fluid → Floats
- If density of object > density of fluid → Sinks
- If density of object = density of fluid → Neutral buoyancy
Relative Density (Specific Gravity)
RD = ρ_substance / ρ_water
Since density of water = 1000 kg/m³ or 1 g/cm³
RD = ρ_substance / 1000 (if ρ in kg/m³)
RD = ρ_substance (if ρ in g/cm³)
Note: Relative density has no unit (it's a ratio)
Important Formulas Summary
F = G(m₁m₂)/r²
2. Acceleration due to gravity:
g = GM/R²
3. Weight:
W = mg
4. Pressure:
P = F/A
5. Pressure in fluid:
P = ρgh
6. Buoyant force:
F_b = ρVg
7. Relative Density:
RD = ρ_substance / ρ_water
Important Constants and Values
- G (Universal gravitational constant) = 6.67 × 10⁻¹¹ N m²/kg²
- g (Acceleration due to gravity on Earth) = 9.8 m/s²
- g on Moon = 1.63 m/s² = (1/6) × g on Earth
- Mass of Earth = 6 × 10²⁴ kg
- Radius of Earth = 6.4 × 10⁶ m
- Density of water = 1000 kg/m³ = 1 g/cm³
Multiple Choice Questions (MCQ)
Subjective Questions
Practice these questions to strengthen your understanding. Write your answers in the space provided.