Laws of Motion

• Introduction

  • Aristotle's Fallacy
    • Force needed to keep a body moving
    • Neglected friction
  • Galileo's Law of Inertia
    • State of rest or uniform motion is natural
    • No force needed for uniform motion

• Newton's First Law of Motion

  • Definition: Body continues in state of rest or uniform motion unless acted upon by external force
  • Inertia
    • Resistance to change in state
    • Depends on mass (greater mass = greater inertia)
  • Inertial & Non-Inertial Frames
    • Inertial: Newton's laws hold valid
    • Non-inertial: Accelerated frames, requires pseudo forces

• Linear Momentum

  • Definition: Product of mass and velocity ($\vec{p} = m\vec{v}$)
  • Vector quantity
  • SI Unit: $kg \cdot m/s$

• Newton's Second Law of Motion

  • Definition: Rate of change of momentum is directly proportional to applied force ($\vec{F} = \frac{d\vec{p}}{dt}$)
  • Equation: $\vec{F} = m\vec{a}$ (for constant mass)
  • Impulse
    • Large force acting for a short time ($\vec{J} = \vec{F} \cdot \Delta t = \Delta \vec{p}$)
    • Examples: Hitting a ball with a bat

• Newton's Third Law of Motion

  • Definition: To every action, there is an equal and opposite reaction
  • Formula: $\vec{F}{AB} = -\vec{F}{BA}$
  • Key points:
    • Act on different bodies
    • Simultaneous occurrence
    • No cause-effect delay

• Conservation of Momentum

  • Law: Total momentum of an isolated system remains constant
  • Equation: $\sum \vec{p} = \text{constant}$
  • Applications:
    • Recoil of a gun
    • Explosion of a bomb
    • Rocket propulsion

• Equilibrium of a Particle

  • Concurrent forces acting on a point
  • First Condition: Vector sum of all forces is zero ($\sum \vec{F} = 0$)
  • Lami's Theorem: $\frac{A}{\sin \alpha} = \frac{B}{\sin \beta} = \frac{C}{\sin \gamma}$

• Common Forces in Mechanics

  • Weight ($W = mg$): Force of gravity acting downwards
  • Normal Force ($N$): Perpendicular contact force
  • Tension ($T$): Pulling force in strings, ropes, or cables
  • Spring Force ($F = -kx$): Restoring force governed by Hooke's Law

• Friction

  • Definition: Opposing force to relative motion between surfaces in contact
  • Types of Friction
    • Static Friction ($f_s$): Prevents motion; self-adjusting
      • Limiting friction ($f_{s,max} = \mu_s N$)
    • Kinetic Friction ($f_k$): Opposes actual relative motion ($f_k = \mu_k N$)
    • Rolling Friction: Resistance when a body rolls over a surface ($f_r < f_k < f_s$)
  • Angle of Friction ($\theta$): Angle between normal reaction and resultant contact force ($\mu = \tan \theta$)
  • Angle of Repose ($\alpha$): Minimum angle of an inclined plane where a body starts sliding ($\mu = \tan \alpha$)
  • Methods to Reduce Friction:
    • Polishing
    • Lubrication
    • Using ball bearings
    • Streamlining

• Circular Motion Dynamics

  • Centripetal Force
    • Force required to keep a body in a circle
    • Formula: $F_c = \frac{mv^2}{r} = m\omega^2 r$
    • Directed towards the center
  • Applications of Circular Motion
    • Motion of a car on a level road ($v_{max} = \sqrt{\mu r g}$)
    • Bending of a cyclist ($\tan \theta = \frac{v^2}{rg}$)
    • Banking of roads ($\tan \theta = \frac{v^2}{rg}$)
    • Skidding condition ($v > v_{max}$)

• Solving Mechanics Problems (Strategy)

  • Isolate the system (Draw Free Body Diagram - FBD)
  • Identify all forces (Gravity, Normal, Tension, Friction, Applied)
  • Choose a convenient coordinate system
  • Apply Newton's equations ($F_x = m a_x$, $F_y = m a_y$)