Core Concepts of Gravitation and Fluid Mechanics

Overview

This lecture covers the core concepts and formulas from the "Gravitation" chapter, including Newton's law of gravitation, acceleration due to gravity, its variation, and the basics of pressure, thrust, buoyancy, Archimedes’ principle, and relative density.

Universal Law of Gravitation

• Every particle in the universe attracts every other particle with a force called gravity.
• Gravitational force (( F )) is directly proportional to the product of their masses (( m_1, m_2 )) and inversely proportional to the square of the distance (( d )) between their centers: ( F \propto \frac{m_1 m_2}{d^2} ).
• By adding the gravitational constant (( G )): ( F = G \frac{m_1 m_2}{d^2} ).
• ( G ) (universal gravitational constant) = ( 6.67 \times 10^{-11} ) N·m(^2)/kg(^2).

Properties & Importance of Gravitational Force

• Gravitational force is always attractive.
• It is the weakest of all fundamental forces.
• It is independent of the medium between bodies.
• Action and reaction forces are equal and opposite (Newton's third law).
• Explains phenomena like falling objects, planetary orbits, tides, etc.

Acceleration Due to Gravity (( g ))

• Acceleration due to gravity (( g )) at Earth's surface: ( g = G \frac{M_{\text{Earth}}}{R_{\text{Earth}}^2} ).
• Average ( g ) on earth ≈ ( 9.8 ) m/s(^2).
• ( g ) is greater at poles (≈ ( 9.83 ) m/s(^2)), less at equator (≈ ( 9.78 ) m/s(^2)) due to Earth's shape.
• At the center of Earth, ( g = 0 ).
• On the Moon, ( g ) ≈ ( 1/6 )th of Earth's ( g ).

Weight and Mass

• Mass is the amount of matter in a body; it is constant everywhere (unit: kg).
• Weight = ( m \times g ); it varies with location (unit: Newton).
• Weight on Moon is ( 1/6 )th of weight on Earth due to lower ( g ).

Motion Under Gravity

• Use equations of motion with ( a ) replaced by ( g ) for vertically moving bodies:
  • ( v = u + gt )
  • ( s = ut + \frac{1}{2}gt^2 )
  • ( v^2 = u^2 + 2gs )
• If object is dropped, initial velocity (( u )) = 0.

Pressure and Thrust

• Thrust is the force acting perpendicular to a surface.
• Pressure = Thrust/Area (( P = F/A )), unit: Pascal (Pa).
• In liquids, pressure increases with depth: ( P = \rho g h ) (( \rho ): density, ( h ): depth).

Buoyancy and Archimedes’ Principle

• Buoyant force (upthrust) is the upward force exerted by a fluid on an immersed object.
• Archimedes’ Principle: Upthrust = weight of fluid displaced by object.
• If object's density < liquid's density, it floats; if >, it sinks.

Relative Density

• Relative density = density of substance / density of water (no units).
• Objects with relative density < 1 float in water; > 1 sink.

Key Terms & Definitions

• Gravity — Force attracting objects towards the center of Earth or other massive bodies.
• Gravitational Constant (G) — Universal constant in Newton’s law of gravitation.
• Acceleration due to gravity (g) — Rate of increase of velocity of objects falling towards Earth.
• Mass — Quantity of matter in a body; does not change with location.
• Weight — Force with which gravity pulls an object; depends on location.
• Thrust — Force applied perpendicular to a surface.
• Pressure — Force per unit area.
• Buoyancy (upthrust) — Upward force by a fluid on an immersed object.
• Archimedes’ Principle — The upthrust on a body equals the weight of fluid displaced.
• Relative Density — Ratio of the density of a substance to the density of water.

Action Items / Next Steps

• Complete the homework: Explain why iron ships float even though iron is denser than water.
• Review equations and definitions.
• Prepare for next session on numericals and problem-solving in gravitation.