Gravitation

• Universal Law of Gravitation
  • Statement: Every object attracts every other object with a force proportional to their masses and inversely proportional to the square of the distance between them.
  • Mathematical Formula: $F = G \frac{M \times m}{d^2}$
  • Universal Gravitational Constant ($G$)
    • Value: $6.673 \times 10^{-11} \text{ N m}^2/\text{kg}^2$
    • Discovered by: Henry Cavendish
  • Significance
    • Binds us to the earth
    • Motion of moon around earth
    • Motion of planets around sun
    • Tides due to moon and sun
• Free Fall
  • Definition: Motion of an object under the influence of gravity alone (no other forces).
  • Acceleration due to Gravity ($g$)
    • Definition: Uniform acceleration produced in a freely falling body.
    • Value on Earth: $9.8 \text{ m/s}^2$
    • Formula: $g = \frac{G \times M}{R^2}$
  • Equations of Motion for Free Fall
    • $v = u + gt$
    • $s = ut + \frac{1}{2}gt^2$
    • $v^2 = u^2 + 2gs$
• Mass vs Weight
  • Mass
    • Definition: Measure of inertia / amount of matter.
    • Nature: Constant everywhere (scalar quantity).
    • SI Unit: Kilogram ($\text{kg}$)
  • Weight ($W$)
    • Definition: Force with which earth attracts an object ($W = mg$).
    • Nature: Varies with location (vector quantity).
    • SI Unit: Newton ($\text{N}$)
    • Moon's Weight: $\frac{1}{6}\text{th}$ of Earth's weight.
• Thrust and Pressure
  • Thrust: Force acting on an object perpendicular to the surface.
  • Pressure ($P$)
    • Definition: Thrust per unit area ($P = \frac{F}{A}$).
    • SI Unit: $\text{N/m}^2$ or Pascal ($\text{Pa}$).
    • Applications: Sharp knives, broad straps on school bags, wide tractor tires.
  • Pressure in Fluids
    • Fluids (liquids and gases) exert pressure in all directions.
    • Buoyancy (Upthrust): Upward force exerted by a fluid on an object immersed in it.
    • Archimedes' Principle
      • Statement: When a body is immersed fully or partially in a fluid, it experiences an upward force equal to the weight of the fluid displaced by it.
      • Applications: Designing ships/submarines, lactometers, hydrometers.
    • Relative Density
      • Definition: Ratio of density of a substance to the density of water.
      • Formula: $\text{Relative Density} = \frac{\text{Density of substance}}{\text{Density of water}}$
      • Unit: Dimensionless (no units).