Chapter 3: Motion and Force

• Introduction

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  • Motion/rest: relative to observer.
  • Rest: position unchanged w.r.t. surroundings.
  • Motion: position changes w.r.t. surroundings.

  A moving body can be both at rest and in motion, depending on the observer.

• Displacement

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  • Vector: change in position (initial to final).

  • Distance

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    • Length of path between two points.

    For straight-line motion, displacement equals distance.

• Speed & Velocity

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  • Velocity: Time rate of change of displacement (Vector).
  • Speed: Magnitude of velocity (Scalar).

  • Speed

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    • Ratio: distance / time.
    • Scalar.
    • Dimensions: $[LT^{-1}]$.
    • Unit: $m~s^{-1}$.

  • Velocity

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    • Time rate of change of displacement.
    • Vector.

  For uniform speed, instantaneous and average speeds are equal. Relative velocity: e.g., boat in current.

• Acceleration

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  • Rate of change of velocity: $\vec{a} = \frac{\Delta \vec{v}}{\Delta t}$.
  • Produced by change in:
    • Magnitude of velocity (speed).
    • Direction of velocity.
    • Both magnitude and direction.
  • Vector.
  • Unit: $m/s^2$.
  • Dimensions: $[LT^{-2}]$.
  • Deceleration: velocity reduces (not zero).
  • Retardation: velocity reduces to zero.

  • Types of Acceleration

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    • Uniform
    • Variable
    • Average
    • Instantaneous
    • Positive
    • Due to gravity
    • Negative (retardation)

• Graphs (Motion)

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  • Displacement-Time Graph

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    • Slope = Velocity.
    • Parallel to x-axis: rest, zero velocity.
    • Inclined straight line: constant velocity.
    • Parallel to y-axis: infinite velocity (not possible).
    • Increasing slope: increasing velocity, accelerated motion.
    • Decreasing slope: decreasing velocity, decelerated motion.

  • Velocity-Time Graph

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    • Slope = Acceleration: $a = \frac{\Delta v}{\Delta t} = \tan\theta$.
    • Horizontal line: constant velocity, zero acceleration.
    • Inclined line: constant acceleration.
    • Curved line: variable acceleration.

  • Important Points (Velocity-Time Graph)

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    • Constant velocity $\implies$ zero acceleration/force.
    • Body falls freely/starts from rest: $v_i = 0$.
    • Velocity and acceleration not necessarily in same direction.
    • Velocity and acceleration not necessarily zero simultaneously.
    • Average speed (equal distance, $v_1, v_2$): $v_{av} = \frac{2v_1v_2}{v_1+v_2}$.
    • Average speed (equal time, $v_1, v_2$): $speed = \frac{v_1+v_2}{2}$.
    • Body thrown upward, $v_f=0$ at height $h$: $h = \frac{v^2}{2g}$.
    • Average acceleration ($a_1$ for $t_1$, $a_2$ for $t_2$): $a_{av} = \frac{a_1t_1+a_2t_2}{t_1+t_2}$.

• Equations of Motion (by Galileo)

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  • $v_f = v_i + at$ (General) or $v_f = v_i + gt$ (Vertical)
  • $s = v_i t + \frac{1}{2}at^2$ (General) or $s = v_i t + \frac{1}{2}gt^2$ (Vertical)
  • $v_f^2 = v_i^2 + 2as$ (General) or $v_f^2 = v_i^2 + 2gs$ (Vertical)
  • Distance in $n^{th}$ second: $s_n = v_i + \frac{1}{2}a(2n-1)$.

• Newton's Laws of Motion

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  • From Newton's PRINCIPIA.
  • Applicable for speeds not comparable to light speed.

  • 1^st Law (Law of Inertia)

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    • Body maintains rest/uniform motion unless external force acts.
    • Qualitatively defines force.
    • Inertia: inability to change state of motion.

  • 2^nd Law (Force and Acceleration)

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    • Force causes acceleration in its direction.
    • $F \propto ma \implies F = ma$ (if $k=1$).
    • $F = \frac{\Delta p}{\Delta t}$ (change in momentum per second).
    • Weight: Earth's gravitational force on a body.

  • 3^rd Law (Action and Reaction)

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    • Every action has equal and opposite reaction.
    • $\vec{F}_{AB} = -\vec{F}_{BA}$ or $\vec{F}_{action} = -\vec{F}_{reaction}$.
    • Action/reaction forces act on *different* bodies.

• Inertial and Non-Inertial Frame

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  • Inertial Frame

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    • Body at rest or uniform velocity ($\vec{a}=\vec{0}$).
    • Law of inertia holds.

  • Non-Inertial Frame

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    • Not uniform velocity ($\vec{a}\ne\vec{0}$).
    • Law of inertia does not hold.

• Momentum

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  • Linear Momentum

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    • Product of mass and velocity: $\vec{p} = m\vec{v}$.
    • Vector, points along velocity.
    • $\vec{p} = \vec{0}$ if $\vec{v}=\vec{0}$.
    • SI unit: $kg~m~s^{-1}$ or $Ns$.
    • Dimension: $[MLT^{-1}]$.

• Impulse

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  • Product of force and duration: $\vec{I} = \Delta \vec{p} = \vec{F}\Delta t$.
  • Same dimensions as linear momentum.
  • Change in momentum = Impulse.

  Thermore pieces protect dinnerware by increasing impact time (reducing force).

• Law of Conservation of Linear Momentum

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  • In isolated system, total linear momentum is constant: $m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'$.

  • Collisions

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    • Interaction due to closeness.

    Elastic Collision in One Dimension

    • Bodies remain on line of motion.
    • Speed of approach = Speed of recession: $v_1 - v_2 = -(v_1' - v_2')$.
    • After collision velocities:
      • $v_1' = \frac{(m_1 - m_2)v_1 + 2m_2v_2}{(m_1 + m_2)}$
      • $v_2' = \frac{(m_2 - m_1)v_2 + 2m_1v_1}{(m_1 + m_2)}$

    Cases of Elastic Collision:

    Case	Condition	Result
    I	$m_1=m_2$	$v_1'=v_2$ & $v_2'=v_1$
    II	$m_1=m_2$ & $v_2=0$	$v_1'=0$ & $v_2'=v_1$
    Velocities interchange in cases I & II.
    III	$m_1 \gg m_2$ & $v_2=0$	$v_1'=v_1$ & $v_2'=2v_1$
    IV	$m_1 \ll m_2$ & $v_2=0$	$v_1'=-v$ & $v_2'=0$

• Force Due to Water Flow

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  • Force exerted by water on wall: $F = \frac{m}{t}v$.
  • Basis for turbines (hydral energy).

• Momentum & Explosive Forces

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  • Conservation of momentum governs explosion fragments.
  • Recoil velocity of rifle: $v' = -\frac{m}{M}v$.
  • Recoil velocity is small fraction of bullet velocity ($M \gg m$).

• Propulsion of Rocket

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  • Based on conservation of linear momentum.
  • Fuel: ~80% of launch mass.
  • Fuel consumption: ~10,000 Kg/s.
  • Exhaust gas speed: >4000 m/s.
  • Instantaneous acceleration: $a = mv/M$.

  Multi-compartment rockets drop empty sections to reduce mass and increase acceleration.

• Friction

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  • Resisting force between rough bodies, opposes sliding.
  • Acts opposite to tendency of motion.
  • Max friction depends on: nature of surfaces, normal force.
  • Independent of contact area (within limits).
  • Decreases once motion starts.
  • Never zero in practice.
  • Walking: friction acts forward (opposes backward slip).
  • Caused by molecular interlocking/irregularities.

  • Coefficient of friction ($\mu$)

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    • $\mu = \frac{\text{Limiting friction (f)}}{\text{Normal reaction (R)}}$ or $f = \mu R$.
    • Depends on material, surface finish, film, temperature.
    • Horizontal surface: $R = mg \implies f = \mu mg$.

  • Static Friction

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    • Force before actual motion starts.
    • $F_s = \mu_s R_s$.

  • Kinetic Friction

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    • Force during motion.
    • $F_k = \mu_k R_k$.
    • Limiting friction: max force before sliding.

  • Angle of Friction ($\theta$)

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    • Angle of resultant of normal reaction (R) and limiting friction (f) with R.
    • $\tan\theta = \frac{f}{R}$ or $\mu = \tan\theta$.
    • Static: $\tan\theta_s = \mu_s$.
    • Kinetic: $\tan\theta_k = \mu_k$.
    • Note: $\mu_s > \mu_k$.

  • Graph for frictions

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• Normal Contact Force

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  • Perpendicular/parallel force between contacting bodies.
  • Number of forces = number of contact surfaces/points.

  • Few contact forces are here:

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    • Normal contact force
    • Frictional force
    • Tension in strings
    • Spring force

    Scenario	Image
    Single surface contact.
    Two surfaces contact on A.
    Three surfaces contact on A.
    Two points contact on rod.
    Three points contact on ring.
    No contact force (block in air).

• Masses in Contact/Connected by Strings

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  • 1) Masses in Contact

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    • $F=$ Horizontal force on $m_1$. $F_c=$ Contact force. $a=$ Acceleration.
    • $F - F_c = m_1a$
    • $F_c = m_2a$
    • Result: $F_c = \frac{m_2F}{m_1+m_2}$, $a = \frac{F}{m_1+m_2}$.

  • 2) Masses Connected by Strings

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    • $F=$ Force on $m_1$. $T=$ Tension. $a=$ Acceleration.
    • $F - T = m_1a$
    • $T = m_2a$
    • Result: $T = \frac{m_2F}{m_1+m_2}$, $a = \frac{F}{m_1+m_2}$.

    • For three masses ($m_1, m_2, m_3$):
    • $F - T_1 = m_1a$
    • $T_1 - T_2 = m_2a$
    • $T_2 = m_3a$
    • Result: $a = \frac{F}{m_1+m_2+m_3}$, $T_2 = \frac{m_3F}{m_1+m_2+m_3}$, $T_1 = \frac{(m_2+m_3)F}{m_1+m_2+m_3}$.

• Tension and Acceleration in a String (Pulley Systems)

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  • Case 1: Vertical motion (two bodies, frictionless pulley)

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    • If $m_1 > m_2$, $m_1$ moves down, $m_2$ moves up with same acceleration.
    • Equations: $m_1g - T = m_1a$, $T - m_2g = m_2a$.
    • Result: $a = \frac{m_1 - m_2}{m_1 + m_2}g$, $T = \frac{2m_1m_2}{m_1 + m_2}g$.

  • Case 2: One vertical, other horizontal

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    • Acceleration: $a = \frac{m_1g}{m_1+m_2}$.
    • Tension: $T = \frac{m_1m_2g}{m_1+m_2}$.

• Mechanical Advantage

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  • Ratio of load to effort.
  • Mechanical Advantage = $\frac{\text{LOAD}}{\text{EFFORT}}$
  • LOAD = MECHANICAL ADVANTAGE $\times$ EFFORT
  • EFFORT = $\frac{\text{LOAD}}{\text{MECHANICAL ADVANTAGE}}$

• Projectile Motion

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  • Instantaneous velocity: max at projection/landing, min at max height ($v_y=0$).
  • Horizontal velocity component: constant.
  • Vertical velocity component: decreases to zero (max height), then increases.
  • Trajectory: Parabola (no air friction).

  • Assumptions

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    • Earth is flat.
    • Gravity (g) constant.
    • Negligible aerodynamic effects.

  • Coordinates at time 't'

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    • $x = v_o \cos\theta_o~t$
    • $y = v_o \sin\theta_o~t - \frac{1}{2}gt^2$

  • Time to reach Maximum Height

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    • $t = \frac{v_o \sin\theta_o}{g}$

  • Total Time of Flight

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    • $T = \frac{2v_o \sin\theta_o}{g}$

  • Vertical Range (Maximum Height)

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    • $H = \frac{v_o^2 \sin^2\theta_o}{2g}$

  • Horizontal Range

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    • $R = \frac{v_o^2 \sin2\theta_o}{g}$ or $R = \frac{v_o^2 (2\sin\theta_o \cos\theta_o)}{g}$
    • Maximum range at $\theta = 45^\circ$: $R_{max} = \frac{v_o^2}{g}$.
    • Relation between range and height: $R \tan\theta = 4H$.
    • Equal ranges for $\theta_1 + \theta_2 = 90^\circ$.

    Range = Max Height when angle of projection is $76^\circ$.

  • Ballistic Missile Application

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    • Un-powered, un-guided.
    • Air friction affects motion.
    • Short range only.
    • Long range/precision: powered, remote-controlled guided missiles.

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