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Newton's Laws
Foundations and Limits of Classical Mechanics
Newton's empirical laws accurately describe macroscopic motion but are limited to speeds much lower than the speed of light.
$$v \ll c$$
The validity range for Newtonian mechanics
$v$=Velocity of the object(m/s)
$c$=Speed of light(≈ 3 × 10⁸ m/s)
$v \approx c$
→Requires relativistic mechanics (Einstein)
Relativistic Limit: As objects approach $c$, mass effectively increases and time dilates, breaking classical assumptions.
Newton's First Law: The Law of Inertia
A body maintains its state of rest or uniform motion unless compelled to change by an unbalanced external force.
$$\sum F = 0 \implies a = 0$$
Equilibrium condition where motion state remains constant
$\sum F$=Vector sum of all external forces(N)
$a$=Acceleration(m/s²)
Static Equilibrium
→Object remains at rest ($v = 0$)
Dynamic Equilibrium
→Object maintains constant velocity ($v = $ constant)
Inertia: The inherent resistance of matter to changes in its motion; quantified by mass.
Inertial Frame of Reference: A coordinate system where this law holds true — must not be accelerating or rotating.
Constant Velocity ≠ Zero Force: A moving object with zero net force continues at the same speed and direction indefinitely.
Newton's Second Law: Quantifying Dynamics
Acceleration is directly proportional to the net force and inversely proportional to mass.
$$F = ma$$
The fundamental equation of motion connecting force and change in motion
$F$=Net external force(N (kg·m/s²))
$m$=Inertial mass(kg)
$a$=Acceleration(m/s²)
$F = 0$
→No acceleration — reduces to Newton's First Law
$a = g$ (free fall)
→$F = mg$ — the object's weight
Vector Direction: The acceleration vector always points in the exact direction of the resultant force vector.
Unit Definition: One Newton (N) is the force required to accelerate 1 kg by 1 m/s².
Proportionality Shortcut: If force and mass both change, find $a' = \frac{F'}{m'}$ directly — compare ratios instead of recalculating from scratch.
Weight vs Mass: Weight $W = mg$ is the gravitational force; mass is the intrinsic property. On the Moon ($g_{moon} \approx 1.6$ m/s²), weight decreases but mass stays the same.
Dimensional Check: $[F] = [m][a] = \text{kg} \cdot \text{m/s}^2 = [MLT^{-2}]$. If your answer doesn't have force dimensions, something went wrong.
Newton's Third Law: Interaction Pairs
Every interaction involves a pair of forces that are equal in magnitude and opposite in direction, acting on different bodies.
$$F_{AB} = -F_{BA}$$
Action and reaction forces act simultaneously on different bodies
$F_{AB}$=Force exerted by object A on object B(N)
$F_{BA}$=Force exerted by object B on object A(N)
Common misconception
→These forces NEVER cancel because they act on different objects.
Book on table
→Book pushes table down (weight); table pushes book up (normal force). The action-reaction pair is book→Earth and Earth→book, not book↔table.
Different Bodies: Action and reaction always act on two separate objects — never on the same body.
Simultaneity: The reaction occurs at the exact same instant as the action; one cannot exist without the other.
Friction and Normal Force
The normal force is the perpendicular contact force exerted by a surface on an object resting on it, adjusting to prevent penetration.
$$N = mg\cos\theta$$
Normal force on an inclined plane at angle θ to the horizontal
$N$=Normal (perpendicular) force from the surface(N)
$m$=Mass of the object(kg)
$g$=Gravitational acceleration(≈ 9.8 m/s²)
$\theta$=Angle of incline from horizontal(degrees or radians)
$\theta = 0°$ (flat surface)
→$N = mg$ — normal force equals weight
$\theta = 90°$ (vertical)
→$N = 0$ — no normal force, object is in free fall along surface
On flat surfaces: Normal force equals the object's weight ($N = mg$) only when no other vertical forces act.
Not always equal to weight: If someone pushes down on the object, $N > mg$. If a rope pulls upward, $N < mg$.
Perpendicular only: Normal force is always ⊥ to the surface, regardless of other forces.
Friction opposes relative motion (or attempted motion) between surfaces in contact. It depends on the normal force and the nature of the surfaces.
$$f = \mu N$$
Friction force is proportional to the normal force
$f$=Frictional force(N)
$\mu$=Coefficient of friction (static $\mu_s$ or kinetic $\mu_k$)(dimensionless)
$N$=Normal force(N)
Object at rest, small applied force
→Static friction matches applied force exactly ($f_s \leq \mu_s N$)
Object sliding
→Kinetic friction is constant at $f_k = \mu_k N$
Static vs Kinetic: Static friction adjusts up to a maximum $\mu_s N$; kinetic friction is a fixed value $\mu_k N$ once sliding begins.
Direction: Friction always acts opposite to the direction of motion (or attempted motion).
Limiting Case: $f_{s,max} = \mu_s N$ is the threshold — any applied force exceeding this initiates motion.
Friction on Inclines: On an incline, the component of gravity along the surface is $mg\sin\theta$. The block slides when $mg\sin\theta > \mu_s mg\cos\theta$, giving the critical angle $\theta_c = \tan^{-1}(\mu_s)$.
What Moves a Car?: The engine spins the tyres, which push backward on the road. By Newton's Third Law, the road pushes the tyres forward — this static friction is the force that accelerates the car.
Tension in Connected Systems
Tension is the pulling force transmitted through a string, rope, or cable when it is pulled tight by forces acting at each end.
Massless String: In idealized problems, the string has negligible mass, so tension is the same throughout its length.
Inextensible String: The string doesn't stretch, so connected objects share the same acceleration magnitude.
Direction: Tension always pulls toward the center of the string — it can only pull, never push.
When two or more objects are connected by a string, they form a system that can be analyzed as a whole or as individual free bodies.
$$a = \frac{F_{net}}{m_{total}}$$
System acceleration for connected bodies
$a$=Common acceleration of the system(m/s²)
$F_{net}$=Total unbalanced force on the entire system(N)
$m_{total}$=Combined mass of all connected objects(kg)
System Approach: First find the acceleration of the whole system using total net force and total mass.
Individual FBD: Then isolate one object and apply $F = ma$ to find tension in the connecting string.
Atwood Machine: Two masses on a pulley — $a = \frac{(m_1 - m_2)g}{m_1 + m_2}$ and $T = \frac{2m_1 m_2 g}{m_1 + m_2}$.
Limiting Case ($m_1 = m_2$): If both masses are equal, $a = 0$ and $T = mg$ — the system is in equilibrium and the tension simply supports the weight.
Limiting Case ($m_2 \to 0$): The heavier mass is essentially in free fall: $a \to g$ and $T \to 0$.
Kinematic Analysis via Velocity-Time Graphs
A velocity-time graph provides a complete history of motion, where slope identifies acceleration and area determines displacement.
$$s = \int v \, dt$$
The area under a v-t curve represents total distance
$s$=Total distance/displacement(m)
$v$=Velocity (function of time)(m/s)
$dt$=Infinitesimal time interval(s)
$a = 0$ (constant velocity)
→$s = v \times t$ — area forms a rectangle
Uniform acceleration from rest
→$s = \frac{1}{2}vt$ — area forms a triangle
Uniform acceleration from $v_i$
→$s = \frac{1}{2}(v_i + v_f)t$ — area forms a trapezoid
Gradient = Acceleration: The slope at any point equals the instantaneous acceleration.
Area = Distance: The geometric area between the curve and time axis gives the distance covered.
Negative Velocity: Area below the time axis represents displacement in the negative direction.