Laws of Motion

4.1 Introduction

• Earlier focus: quantitative description of motion of a particle in space (velocity for uniform motion; acceleration for non-uniform motion).
• Core question of this chapter: What governs the motion of bodies? Identify the external agency that can cause motion.
• Everyday examples show that an external force is needed to start motion or slow/stop motion (e.g., kick a football, push a stone upward, wind causes branches to swing).
• Some forces act at a distance (e.g., gravity, magnetic attraction). So external agency may or may not be in contact with the body.
• Question raised: Is an external force required to keep a body in uniform motion on a frictionless surface? The intuitive answer from common experience is negative only if friction is absent; otherwise friction opposes motion.
• Chapter outline (major topics):
  • 4.2 Aristotle’s fallacy
  • 4.3 The law of inertia
  • 4.4 Newton’s first law of motion
  • 4.5 Newton’s second law of motion
  • 4.6 Newton’s third law of motion
  • 4.7 Conservation of momentum
  • 4.8 Equilibrium of a particle
  • 4.9 Common forces in mechanics
  • 4.10 Circular motion
  • 4.11 Solving problems in mechanics
  • 4.12 Summary and exercises
• Key takeaway: development from Aristotelian views to Galileo’s inertia and Newton’s laws, forming the foundation of classical mechanics.

4.2 Aristotel’s Fallacy

• Central question: Does a body in motion require a continuous external force to keep moving?
• Aristotle’s view (384–322 B.C.): An external force is required to keep a body in motion (e.g., air behind arrow pushes it; external force maintaining motion).
• Natural experience seemed to support this view: objects slow due to opposing forces like friction; to maintain motion one must counteract friction.
• Flaw in Aristotelian argument: Moving toy car on a floor comes to rest due to friction opposing motion; to keep moving, a force must be applied in the direction of motion to cancel friction.
• Corollary: If there were no friction, the external force would not be needed to keep uniform motion.
• Opposing forces (friction for solids; viscous drag for fluids) are always present in the real world.
• Galileo’s insight: imagine a world with uniform motion possible without friction; this sets the stage for a true law of motion.

4.3 The Law of Inertia

• Galileo studied motion on inclined planes:
  • Objects moving down incline accelerate; those moving up retard; motion on a horizontal plane is an intermediate case.
• Conclusion (frictionless horizontal plane): an object should move with constant velocity (no acceleration).
• Double inclined plane experiment: a ball released from rest on one plane, rolls down and climbs the other; with smooth planes, the final height equals initial height (ignoring small losses).
• If the second plane’s slope decreases, the ball travels a longer distance but reaches the same final height; in the limit of zero slope, the ball travels indefinitely (ideal frictionless case).
• In practice, friction stops the ball after finite distance, but in the frictionless limit, the motion would continue indefinitely.
• Inertia: state of rest and state of uniform linear motion are equivalent (no net external force).
• Definition: inertia = resistance to change; a body does not change its state of rest or uniform motion unless an external force compels it to change.

4.4 Newton’s First Law of Motion

• Galileo’s ideas lead to a formal statement: Every body remains in its state of rest or of uniform motion in a straight line unless compelled by some external force to act otherwise.
• Equivalently: If the net external force on a body is zero, its acceleration is zero.
• Practical examples:
  • A spaceship far from external forces experiences zero net external force; its acceleration is zero; if already in motion, it continues with uniform velocity.
• Important nuance: in practice, a zero net external force does not imply the absence of all forces; multiple forces can cancel to give zero net external force (e.g., weight and normal force on a book at rest).

4.5 Newton’s Second Law of Motion

• Extends the first law to non-zero net external force.
• Momentum p is defined as the product of mass and velocity: $p = m \boldsymbol{v}$ (vector quantity).
• Observational basis: changes in momentum relate to applied forces; mass distribution affects how forces change motion.
• Key statements:
  • The rate of change of momentum is directly proportional to the applied force and occurs in the direction of the force:
    $\frac{d \boldsymbol{p}}{dt} = \boldsymbol{F}$
  • For constant mass, this reduces to $\boldsymbol{F} = m \boldsymbol{a}$ where $\boldsymbol{a} = \frac{d \boldsymbol{v}}{dt}$.
• In differential form, with a constant mass, the second law becomes:
  F = rac{d p}{d t} = rac{d}{d t} (m v) = m a
  
• SI unit: the newton (N), defined by the force required to produce acceleration of $1 ext{ m s}^{-2}$ in a mass of $1 ext{ kg}$: $1 ext{ N} = 1 ext{ kg m s}^{-2}$
• Important points about the second law:
  1. If $F = 0$ then $a = 0$, consistent with the first law.
  2. The law is a vector law, with component form: $F<em>x = \frac{d p</em>x}{dt} = m a<em>x, \, F</em>y = \frac{d p<em>y}{dt} = m a</em>y, \, F<em>z = \frac{d p</em>z}{dt} = m a_z$
     A force not parallel to velocity changes only the velocity component along the force; the perpendicular component remains unchanged.
  3. The law applies to a single particle, but also to systems: F refers to the total external force on the system, and a is the acceleration of the system’s center of mass; internal forces cancel within the system.
  4. The law is a local relation: a at a point and time is determined by the force at that same point and time, independent of past motion.
• Examples:
  • A bullet entering a block: average resistive force relates to deceleration through the second law; example provides numerical computation of average force.
  • Gravity acting on a particle in a prescribed motion: the force is gravity, giving acceleration g.
• Impulse (see 4.7): force acting for a short duration can yield a finite change in momentum, linking force, time, and momentum change.

4.6 Newton’s Third Law of Motion

• Action-reaction pairs: forces arise from mutual interaction between two bodies.
• For two bodies A and B, the force on A by B is equal and opposite to the force on B by A:
  $\boldsymbol{F}<em>{A ext{ on } B} = - \boldsymbol{F}</em>{B ext{ on } A}$
• Key clarifications:
  • The terms action and reaction refer to forces, not causal precedence; both forces act simultaneously.
  • Action-reaction forces act on different bodies and should not be summed when analyzing a single body’s motion.
  • Internal forces within a system cancel when considering the system as a whole; only external forces drive the motion of the system.
• Example: Two billiard balls hitting a wall reflect with equal and opposite impulses; analysis uses second law on the ball and third law to infer wall forces.

4.7 Conservation of Momentum

• From the second and third laws: in an isolated system (no external impulse), total momentum is conserved.
• For a bullet-firegun scenario: the momentum gained by the bullet is balanced by an equal and opposite momentum imparted to the gun, yielding total momentum zero if initially at rest.
• General statement: The total momentum of an isolated system of interacting particles remains unchanged:
  $ext{initial } p<em>{ ext{total}} = ext{final } p</em>{ ext{total}}$
• Collision examples:
  • Elastic collisions: total initial kinetic energy is conserved in addition to momentum conservation.
  • Inelastic collisions: momentum is conserved, but kinetic energy is not necessarily conserved.
• Mathematical expression for two-body collision: if two bodies with momenta $p<em>A, p</em>B$ interact and end with $p'<em>A, p'</em>B$, then
  $p'<em>A + p'</em>B = p<em>A + p</em>B$
• This conservation applies to the system as a whole, regardless of whether collisions are elastic or inelastic.

4.8 Equilibrium of a Particle

• Equilibrium means zero net external force on the particle.

• Translational equilibrium: net force is zero; the particle is at rest or moving with constant velocity.

• For multiple concurrent forces, equilibrium requires the vector sum to be zero: oldsymbol{F}1 + oldsymbol{F}2 + oldsymbol{F}3 + \n oldsymbol{F}4 +
  abla
  abla = 0

• How to determine forces in equilibrium:

  • Draw free-body diagrams (FBDs) showing the system and forces acting on it from the rest of the environment.
  • Use the first law to deduce that the net force is zero; then apply the conditions for components along x, y, z to vanish.

• Example 4.6 (free-body analysis): a rope suspends a 6 kg mass with a horizontal force applied at the rope’s midpoint; fate of the rope angle requires balancing vertical and horizontal components to satisfy equilibrium.

4.9 Common Forces in Mechanics

• In mechanics, forces include gravitational attraction, contact forces, normal reactions, friction, buoyancy, viscous drag, tension, and spring forces.
• Gravitational force: acts at a distance; universal in terrestrial and celestial contexts.
• Contact forces arise when bodies touch each other:
  • Normal reaction (perpendicular to contact surface).
  • Friction (parallel to the contact surface) opposing relative motion.
• Fluid contact forces: buoyancy equals the weight of fluid displaced (Archimedes' principle); viscous drag and other fluid forces can act on surfaces.
• Tension in a string and restoring force in a spring: for a spring, F = -k x, where x is displacement from rest; negative sign indicates opposition to displacement.
• The four fundamental forces are summarized up to gravitational and electromagnetic interactions; non-contact forces in mechanics usually reduce to gravitational and electrical (via microscopic origin, but treated macroscopically as empirical forces).
• Free-body-diagram practice is emphasized for problem solving.

4.9.1 Friction

• Friction opposes relative motion between contacting surfaces; it is a component of the contact force parallel to surfaces in contact.
• Static friction (fs) opposes impending motion (the motion that would occur if friction were absent).
• For impending motion, the maximum static friction is:
  $f<em>s^{\max} = \mu</em>s N$
  where $\mu_s$ is the coefficient of static friction and N is the normal force.
• The static friction law: $f<em>s \le \mu</em>s N$
• If the applied horizontal force exceeds the maximum static friction, the body begins to slide; once sliding occurs, kinetic friction fk comes into play.
• Kinetic (sliding) friction is typically less than static friction; it is approximately described by:
  $f<em>k = \mu</em>k N$
  where $\mu_k$ is the coefficient of kinetic friction (surface-dependent).
• Kinetic friction is almost independent of velocity; frictional force opposes relative motion and is nearly independent of contact area.
• On a horizontal surface with a net external force F, the acceleration when sliding is given by:
  $a = \frac{F - f_k}{m}$
• If the applied force is removed, acceleration becomes $a = -f_k / m$ and the body eventually stops.
• Practical note: friction laws are empirical, not fundamental; they are highly useful for engineering and everyday calculations.
• Static friction example: a box on a train floor has the train’s acceleration provided by static friction until the threshold is exceeded; beyond that, the box slips.
• Practical implications: static friction is self-adjusting up to its limit; it can provide the exact needed force to accelerate the object with the same acceleration as the surface (e.g., floor accelerating with train).
• Examples: calculating maximum acceleration of a train before slipping begins (Example 4.7); angle at which an incline block begins to slide (Example 4.8).

4.9.2 Rolling Friction

• Rolling friction is the frictional resistance experienced when a wheel, ring, or sphere rolls without slipping.
• In principle, rolling without slipping experiences no kinetic friction, but in practice rolling friction exists due to deformation at the contact point.
• Rolling friction is typically much smaller than static or kinetic friction, which explains the efficiency of wheels and bearings.
• Practical measures to reduce friction: ball bearings, lubricants, and air cushions between surfaces.

4.10 Circular Motion

• Uniform circular motion of a body of mass $m$ and speed $v$ on a circle of radius $R$ has acceleration $a = v^2 / R$ directed toward the center.
• The centripetal force required to provide this acceleration is:
  $f_c = \frac{m v^2}{R}$
• This centripetal force is not a separate kind of force; it is the net force component toward the center provided by real forces (tension, gravity, friction, etc.).
• Example: a stone on a string: centripetal force provided by the string’s tension; a planet around the sun: centripetal force provided by gravity.
• On a car turning in a circle on a level road, the centripetal force is provided by the frictional force between tires and road.
• Banked curves: a car on a banked road can rely on the normal force component and friction to provide the centripetal force. The speed limit is set by friction and geometry; the conditions can be derived from vertical and horizontal force components:
  • Vertical balance: $N \cos \theta = mg + f \sin \theta$
  • Horizontal (centripetal) balance: $N \sin \theta + f \cos \theta = \frac{m v^2}{R}$
  • With the maximum static friction, set $f = \mu<em>s N$ to obtain the maximum speed on a banked curve: $v</em>{\max}^2 = \frac{R g (\tan \theta + \mu<em>s)}{1 - \mu</em>s \tan \theta}$
• A special case: if $\mu<em>s = 0$, the optimum banked speed reduces to $v</em>0 = \sqrt{R g \tan \theta}$
• Flat road (unbanked) maximum speed is governed by
  $v<em>{\max} = \sqrt{\mu</em>s g R}$
• Key interpretation: the centripetal force is not a separate force; it is the resultant of real forces toward the center.

4.11 Solving Problems in Mechanics

• Systematic approach to problems involving multiple bodies:
  1) Draw a schematic diagram of the assembly with links, supports, etc.
  2) Choose a convenient part of the assembly as the system.
  3) Draw a free-body diagram (FBD) for the system showing all forces acting on it due to the environment and other bodies; do not include forces exerted by the system on the environment.
  4) Record known forces and directions; treat unknowns as variables to be solved using Newton’s laws.
  5) If needed, repeat for another choice of the system and use Newton’s third law to relate the forces.
• Example 4.12 (free-body method): a wooden block on a yielding floor; an iron cylinder placed on top causes the floor to yield; the system accelerates downward; identify action–reaction pairs.
• Practical notes: free-body diagrams help clearly define the system and the forces; use them to organize problem-solving steps and to apply Newton’s laws consistently.

4.12 Examples and Exercises (overview)

• The chapter includes numerous worked examples and exercises to reinforce the concepts, including:
  • Free-body diagram construction for a simple block on an inclined plane, a rope and pulley system, and block–c cylinder interactions.
  • Impulse problems, momentum conservation in collisions (elastic and inelastic), and applications to everyday scenarios (e.g., a car braking, a rocket launch, a bat hitting a ball).
  • Circular motion problems on level and banked roads, with frictional limits and maximum speed calculations.
  • Problems involving energy-like quantities such as tension, normal force, and friction in various configurations.

4.13 Summary of Key Concepts

• Aristotle’s view (force needed to keep motion) is incorrect; real friction opposes motion, not an essential requirement to maintain it.
• Galileo’s law of inertia: rest and uniform straight-line motion are equivalent states (zero net external force).
• Newton’s First Law: If the net external force on a body is zero, its acceleration is zero (state of rest or uniform motion).
• Momentum: $\boldsymbol{p} = m \boldsymbol{v}$; momentum is a vector.
• Second Law (Newton): The rate of change of momentum equals the net external force; for constant mass, $\boldsymbol{F} = \frac{d \boldsymbol{p}}{dt} = m \boldsymbol{a}$; in general, $\boldsymbol{F} = \frac{d \boldsymbol{p}}{dt}$.
• Impulse: $\text{Impulse} = \boldsymbol{F} \Delta t = \Delta \boldsymbol{p}$.
• Third Law: Forces occur in equal-and-opposite pairs acting on two bodies in contact: $\boldsymbol{F}<em>{A on B} = -\boldsymbol{F}</em>{B on A}$.
• Conservation of Momentum: In an isolated system, total momentum is conserved: $\sum \boldsymbol{p}<em>{\text{initial}} = \sum \boldsymbol{p}</em>{\text{final}}$; applies to collisions regardless of elasticity; elastic collisions also conserve kinetic energy.
• Equilibrium of a particle: Net external force is zero; F1 + F2 + … = 0; represented graphically by closed polygons (for multiple forces).
• Friction: Static and kinetic; static friction up to a maximum value, kinetic friction generally smaller; key relations: $f<em>s^{\max} = \mu</em>s N, \quad f<em>s \le \mu</em>s N, \quad f<em>k = \mu</em>k N$; friction is independent of contact area.
• Rolling friction: smaller than sliding friction; rolling improves efficiency; mechanisms include deformation at the contact surface and presence of rolling elements (bearings, lubricants).
• Circular motion: centripetal acceleration toward the center; centripetal force is provided by real forces (tension, gravity, friction); limits on speed from friction and road geometry; banked road formulas as above.

4.14 Points to Ponder (highlights)

• Force can be not in the direction of motion; still parallel to acceleration in Newtonian mechanics.
• Zero velocity does not imply zero force or zero acceleration (e.g., when an object momentarily at the top of its trajectory).
• The second law refers to net external force, not any internal or historical forces.
• The centripetal force is not a new kind of force but the resultant of existing forces providing inward acceleration.
• Static friction is self-adjusting up to its limit; do not blindly set fs = µs N.
• The distinction between action and reaction is about simultaneous forces on different bodies, not a temporal order.
• The same framework applies to both inanimate and animate systems; walking requires frictional interaction with the ground.
• The microscopic origin of contact forces is electromagnetic in origin, though macroscopic equations treat them as empirical forces.

4.15 Exercises (sample themes and problem types)

• Net force magnitudes and directions for various scenarios (rain, cork on water, kite stationary, car on rough road, free-falling electron).
• Vertical motion with gravity and impulses (pebble thrown upward; forces during ascent, descent, and at the apex).
• Free-fall scenarios from a train window under different speeds; effect of motion on perceived force.
• Circular motion questions on string tension, centripetal forces, and choosing the correct net force expression for a particle in circular motion.
• Impulse problems: impact with walls, bouncing balls, and momentum changes for different masses.
• Banked turn problems: optimum speed, maximum permissible speed, and frictional limits.
• Problems involving multi-body systems connected by strings and pulleys; tension and acceleration calculations.
• Disintegration or recoil problems: momentum conservation in nuclear or explosive contexts.
• Free-body diagram practice: constructing and analyzing FBDs for complex assemblies.
• Conceptual questions reinforcing action–reaction, momentum conservation, and friction.

Key formulae to remember

• Momentum: $\boldsymbol{p} = m \boldsymbol{v}$
• Newton’s Second Law (vector form): $\boldsymbol{F} = \frac{d \boldsymbol{p}}{dt}$
• Special case (constant mass): $\boldsymbol{F} = m \boldsymbol{a}$
• Impulse: $\text{Impulse} = \boldsymbol{F} \Delta t = \Delta \boldsymbol{p}$
• Newton’s Third Law (action–reaction pair): $\boldsymbol{F}<em>{A on B} = - \boldsymbol{F}</em>{B on A}$
• Conservation of Momentum (two-body collision): $\boldsymbol{p}'<em>A + \boldsymbol{p}'</em>B = \boldsymbol{p}<em>A + \boldsymbol{p}</em>B$
• Static friction: $f<em>s \le \mu</em>s N, \quad f<em>s^{\max} = \mu</em>s N$
• Kinetic friction: $f<em>k = \mu</em>k N$
• Centripetal force: $f_c = \frac{m v^2}{R}$
• Banked curve with friction (maximum speed): $v<em>{\max}^2 = \frac{R g (\tan\theta + \mu</em>s)}{1 - \mu_s \tan\theta}$
• Banked curve with μs = 0 (no friction): $v_0 = \sqrt{R g \tan\theta}$
• Level-road friction-limited speed: $v<em>{\max} = \sqrt{\mu</em>s g R}$
• Free-body diagram (FBD) practice: represent all external forces on the chosen system; use Newton’s laws to solve for unknowns.