Laws of Motion
Why This Chapter Matters
Newton's Laws is one of the highest-scoring JEE topics — 8-12 marks. FBD (Free Body Diagrams), friction, circular motion dynamics, and pseudo-force problems are tested every year.
Core Concepts
1. Newton's Three Laws
First Law (Inertia): A body remains at rest or uniform motion unless acted on by external net force.
Inertia = resistance to change. More mass = more inertia.
Second Law (F = ma):
Net force = mass × acceleration
F_net = ma (vector equation)
If multiple forces: vector sum = F_net. ΣF = ma.
Third Law (Action-Reaction):
For every action, there is an equal and opposite reaction.
F_AB = -F_BA (act on DIFFERENT bodies — never cancel each other)
2. Free Body Diagram (FBD)
Isolate a body. Draw ALL forces acting ON it:
Resolve forces. Apply ΣFx = max, ΣFy = may.
3. Friction
Static friction (f_s): prevents motion. f_s ≤ μ_s × N
Kinetic friction (f_k): during motion. f_k = μ_k × N (constant)
Always: μ_k < μ_s (kinetic friction < static friction)
μ = coefficient of friction (depends on surfaces, NOT area of contact, NOT normal force)
4. Atwood Machine
Two masses m₁ and m₂ over frictionless pulley (m₁ > m₂):
Acceleration: a = (m₁ - m₂)g/(m₁ + m₂)
Tension: T = 2m₁m₂g/(m₁ + m₂)
5. Circular Motion — Dynamics
For circular motion, centripetal force is needed:
F_c = mv²/r = mω²r (directed toward center)
Provided by: tension (for ball on string), normal force (for car on curved road), gravity (for satellite), friction (for car on flat road turning).
Banked road (no friction): tan(θ) = v²/rg → optimal banking angle
With friction: v_max = √[rg(tan θ + μ)/(1 - μ tan θ)]
6. Pseudo Force (Non-Inertial Frames)
In accelerating frame (acceleration a₀), a pseudo force F = -ma₀ acts on every mass m.
Direction: opposite to frame's acceleration.
Useful for: problems in lifts, accelerating cars, rotating frames.
Lift problems:
Apparent weight in lift accelerating up: N = m(g + a)
Apparent weight in lift accelerating down: N = m(g - a)
Free fall (a = g): N = 0 (weightlessness)
Solved Examples
Q1: Block of mass 10 kg on rough surface (μ=0.3). Force of 50 N applied. Find acceleration.
N = mg = 100 N. Friction = μN = 30 N. Net force = 50-30 = 20 N.
a = 20/10 = 2 m/s²
Q2: Two blocks m₁=3 kg and m₂=5 kg connected by string over pulley. Acceleration?
a = (5-3)×10/(5+3) = 20/8 = 2.5 m/s²
T = 2×3×5×10/(3+5) = 300/8 = 37.5 N
Q3: Car of mass 1000 kg takes circular turn of radius 100 m at 20 m/s on flat road. Friction needed?
F_c = mv²/r = 1000×400/100 = 4000 N
Friction provides centripetal force: f = 4000 N
μ = f/N = 4000/10000 = 0.4
PYQs
2024: Block on incline at angle θ. When does it just start sliding?
At threshold: mg sinθ = μ mg cosθ → tan θ = μ → θ = arctan(μ)
2023: Monkey of mass m hanging on rope. Rope can withstand Tmax. Min acceleration at which monkey must climb so rope doesn't break?
T - mg = ma → a = (Tmax - mg)/m (climbing up)
Actually: if monkey climbs up, tension increases. Max T: monkey accelerates down.
T = m(g - a). For rope not to break: T < Tmax → a > g - Tmax/m
2022: A ball of mass m tied to string of length l, rotated in vertical circle. Minimum speed at top?
At top: mg + T = mv²/r. Minimum when T=0: v_min = √(gr)