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1800-102-2727This is the complete JEE Main Physics Formula Sheet and Class 11 Formula Sheet for Laws of Motion — Chapter 03 from the Aakash Rapid Revision & Formula Bank. Laws of Motion is the cornerstone chapter of JEE Main Physics, covering: Newton's Three Laws — inertia, F=ma, action-reaction; Linear Momentum — p=mv, impulse F·Δt=Δp; Equilibrium of Concurrent Forces — Lami's theorem P/sinα=Q/sinβ=R/sinγ; Constraint Relations — fixed middle end, fixed side end, three free ends; Pulley Systems — Atwood's machine (a, T), moving pulley, inclined plane pulley; Contact Forces — tension in blocks, force distribution in multi-block systems; Machine Gun / Bullet / Liquid Jet Forces — F=nmv (bullets), F=ρAv² (liquid jet); Rope Tension — uniform rope under gravity and acceleration; Friction — static (fs≤μsN), kinetic (fk=μkN), angle of repose (tanλ=μs), minimum force to move; Wedge-Block Problems — block on accelerating wedge, minimum F for free fall; Two-Block Systems — F on lower/upper block, slipping conditions; Dynamics of Circular Motion — conical pendulum (T, time period), banked road (vmax, smooth incline v=√(rg tanθ), horizontal road v=√(μrg)); and Vertical Circle Motion — minimum speed at top √(gl), TL–TH=6mg, condition for completing vertical circle. Laws of Motion contributes 4–6 questions in every JEE Main session. Download the Free PDF for all Laws of Motion formulas in one JEE Main exam-ready reference.
Scroll to explore all Laws of Motion formulas — JEE Main Physics Formula Sheet
Laws of Motion is the chapter where kinematics meets forces — the transition from describing motion to explaining it. Newton's three laws form the foundation of classical mechanics: the first law defines inertia and equilibrium, the second law quantifies the relationship F=ma between force and acceleration, and the third law establishes action-reaction pairs. Every JEE Main problem in mechanics — pulleys, friction, inclined planes, circular motion dynamics — is ultimately a Laws of Motion problem.
For JEE Main physics, Laws of Motion contributes 4–6 questions per session, making it the single highest-weightage chapter alongside Electrostatics. Questions test: Newton's second law (F=ma) in multi-block systems, constraint relations for pulleys, tension in strings and ropes, friction (static and kinetic), wedge-block problems, conical pendulum, banked road maximum velocity, and vertical circle minimum speed. Each question has a fixed solving algorithm: draw FBD → write constraint equations → apply F=ma to each body → solve.
Download the Free PDF for Laws of Motion to access all Newton's law formulas, all pulley system results, all friction formulas, all banking and circular motion dynamics results, and all vertical circle conditions in one structured JEE Main physics revision reference.
Newton's First Law of Motion (from Aakash PDF — Laws of Motion):
A body continues in its state of rest or of uniform motion in a straight line unless acted upon by a net external force. This law defines inertia — the tendency of a body to resist change in its state of motion. A body at rest has inertia of rest; a body in motion has inertia of motion. First law also defines the condition of equilibrium: F_net = 0 ↔ a = 0 (body either at rest or moving with constant velocity).
Newton's Second Law of Motion (from Aakash PDF — Laws of Motion JEE Main):
The rate of change of momentum of a body is proportional to the applied net external force and takes place in the direction of the net force.
F_net = ma = dp/dt
For constant mass: F = ma (magnitude: F = ma, direction: same as a).
Linear Momentum: p = mv (vector, direction same as velocity).
Impulse: J = F·Δt = Δp = m(v–u)
Impulse = change in momentum. For variable force: J = ∫F dt = area under F-t graph. SI unit of impulse = N·s = kg·m/s = same as momentum.
Newton's Third Law of Motion (from Aakash PDF — Laws of Motion):
For every action, there is an equal and opposite reaction. If body A exerts force F_AB on body B, then body B exerts F_BA = –F_AB on body A. Action and reaction are: equal in magnitude, opposite in direction, act on DIFFERENT bodies (so they do not cancel each other), of the same type (both contact or both gravitational, etc.).
Equilibrium of Concurrent Forces (from Aakash PDF — Laws of Motion JEE Main):
If three forces P, Q, R act on a body such that it is in equilibrium (F_net = 0), and the forces are concurrent, then by Lami's Theorem:
P/sinα = Q/sinβ = R/sinγ
where α, β, γ are the angles between the forces: α = angle between Q and R (i.e., opposite to P), β = angle between P and R (opposite to Q), γ = angle between P and Q (opposite to R). The three angles satisfy α+β+γ = 360°.
Alternative: For three concurrent forces in equilibrium, they can be represented as three sides of a triangle (triangle law of equilibrium). Download the Free PDF for Laws of Motion for all Newton's law examples and Lami's theorem applications for JEE Main.
Constraint Relations (from Aakash PDF — Laws of Motion JEE Main): In a pulley system, the string is inextensible (fixed length). This imposes a fixed mathematical relation between the displacements, velocities, and accelerations of the ends of the string. These are called constraint relations.
Case I — Fixed Middle End (from Aakash PDF — Laws of Motion):
When the middle end of the string over a pulley is fixed and the two outer masses hang on either side:
x₁ = x₂; v₁ = v₂; a₁ = a₂
(Both masses move identically — their displacements, velocities and accelerations are equal in magnitude.)
Case II — Fixed Side End (from Aakash PDF — Laws of Motion):
When one side end of the string is fixed (attached to ceiling or wall), and a mass hangs from the movable pulley:
x₂ = 2x₁; v₂ = 2v₁; a₂ = 2a₁
(The hanging mass moves at half the speed/acceleration of the string's free end — or equivalently, the mass attached to the movable pulley moves half as fast as the free end.)
Case III — All Three Ends Free to Move (from Aakash PDF — Laws of Motion JEE Main):
When all three ends of the string are free (two masses hang from the ends, and the pulley itself moves):
(x₁+x₂)/2 = x₃; (v₁+v₂)/2 = v₃; a₁+a₂ = 2a₃
Equivalently: a₁+a₂ = 2a₃ (the average acceleration of the two end masses equals the acceleration of the pulley).
Sign convention: downward is taken positive. If any direction is upward, incorporate a negative sign in the corresponding term.
General Constraint Method: Let the string have total length L (constant). Differentiate the geometric sum of all string segments twice with respect to time. The resulting equation is the constraint relation. Download the Free PDF for Laws of Motion for all constraint relation examples for JEE Main.
Atwood's Machine — Stationary Pulley (from Aakash PDF — Laws of Motion JEE Main):
Two masses M₁ and M₂ (M₂ > M₁) hang over a fixed, massless, frictionless pulley. The string is massless and inextensible.
Acceleration: a = (M₂–M₁)g / (M₁+M₂)
Tension: T = 2M₁M₂g / (M₁+M₂)
Note: T lies between M₁g and M₂g (always). If M₁=M₂: a=0, T=M₁g=M₂g (no acceleration). The heavier mass M₂ accelerates down; M₁ accelerates up.
Atwood's Machine — Pulley Moving Upward with Acceleration a₀ (from Aakash PDF — Laws of Motion JEE Main):
When the pulley itself accelerates upward with a₀ (using pseudo force in pulley's frame — replace g by g+a₀):
T = 2M₁M₂(g+a₀) / (M₁+M₂)
Relative acceleration between masses (w.r.t. pulley): aᵣ = (M₂–M₁)(g+a₀)/(M₁+M₂)
Absolute accelerations: a₁ = –(a₀+aᵣ); a₂ = aᵣ–a₀ (taking upward positive).
Inclined Plane Pulley System (from Aakash PDF — Laws of Motion JEE Main):
Mass M₁ on incline of angle α, mass M₂ on incline of angle β, connected by string over pulley at the top:
a = (M₂ sinβ – M₁ sinα)g / (M₁+M₂)
(assuming M₂ sinβ > M₁ sinα for the motion shown, and smooth inclines.)
T = M₁M₂(sinα+sinβ)g / (M₁+M₂)
Mass on a String from Ceiling (from Aakash PDF — Laws of Motion JEE Main):
Mass M hangs from ceiling by string. (a) Stationary or uniform velocity: T = Mg. (b) System accelerates UP with a: T = M(g+a). (c) System accelerates DOWN with a: T = M(g–a). Note: if accelerating down with a=g (free fall), T=0 (apparent weightlessness).
Four standard configurations (from Aakash PDF — Laws of Motion JEE Main): All of these share the same formula structure:
Linear acceleration: a = F_net / (effective mass + I_cm/R²)
For all four standard pulley setups: a = F/(m + I_cm/R²) and T or friction = I_cm·a/R²
Download the Free PDF for Laws of Motion for all pulley system examples for JEE Main.
Machine Gun / Bullet Force (from Aakash PDF — Laws of Motion JEE Main):
A machine gun fires n bullets per second, each of mass m with speed u.
Force required to keep gun stationary: F = nmv (reaction force = rate of change of momentum of bullets)
This is also the recoil force on the gun.
Bullets hitting a wall at speed v (n per second):
(i) Bullets come to rest: F_wall = nmv
(ii) Bullets rebound elastically: F_wall = 2nmv (change in momentum = 2mv per bullet)
Liquid Jet Force (from Aakash PDF — Laws of Motion JEE Main):
Liquid jet of cross-sectional area A moving with speed v hits a wall. Density of liquid = ρ. Mass hitting per second = ρAv.
(i) Force required by pump to move liquid: F = (dm/dt)v = ρAv² (rate of momentum supply)
(ii) Jet hits wall, does NOT rebound: F_wall = ρAv²
(iii) Jet rebounds elastically: F_wall = 2ρAv²
(iv) Oblique impact at angle θ to wall: F_wall = 2ρAv²cosθ (only normal component changes)
Multi-Block Contact Forces (from Aakash PDF — Laws of Motion JEE Main):
Three blocks M₁, M₂, M₃ pushed by force F on a smooth surface (M₁ in front, F on M₁): all blocks accelerate together with:
a = F/(M₁+M₂+M₃)
Contact force between M₁ and M₂ (force F₁ on M₂ from M₁ by Newton's 3rd law):
F₁ = (M₂+M₃)a = (M₂+M₃)F/(M₁+M₂+M₃)
Contact force between M₂ and M₃: F₂ = M₃a = M₃F/(M₁+M₂+M₃)
String Tension at Distance x from Left End (from Aakash PDF — Laws of Motion JEE Main):
For a uniform rod/string of mass M and length L pulled by force F from left end:
Tₓ = Fx/L (tension at distance x from end where F is applied, mass to the right pulls back)
Tₓ = F(L–x)/L (tension at distance x from end where F is NOT applied)
For F₁ applied at left, F₂ applied at right: Tₓ = [F₁x/L] + [F₂(L–x)/L]
Uniform Rope Tension (from Aakash PDF — Laws of Motion JEE Main):
Uniform rope of total mass Mₛ, length L, hanging from ceiling or pulled vertically. Tension at distance x from the free end:
(a) Stationary: Tₓ = Mₛgx/L (weight of rope below the point)
(b) Accelerating upward with a: Tₓ = Mₛx(g+a)/L
(c) Accelerating downward with a: Tₓ = Mₛx(g–a)/L (if a=g: T=0, free fall)
Download the Free PDF for Laws of Motion for all contact force and rope tension examples for JEE Main.
Types of Friction (from Aakash PDF — Laws of Motion JEE Main):
Static friction (fₛ): Friction between two surfaces when there is no relative motion. It is self-adjusting — it adjusts to oppose the tendency of relative motion. Maximum static friction = limiting friction = μₛN.
0 ≤ fₛ ≤ μₛN (static friction can be anywhere from 0 to maximum)
Kinetic friction (fₖ): Friction when there IS relative sliding motion between surfaces.
fₖ = μₖN (fixed value, independent of area and speed)
μₖ < μₛ (kinetic friction coefficient is always less than static)
Angle of Friction and Angle of Repose (from Aakash PDF — Laws of Motion JEE Main):
Consider a block on an inclined plane with friction μₛ. The maximum angle of incline for which the block stays at rest:
Angle of repose: λ = tan⁻¹(μₛ)
At angle of repose: N = mg cosλ; fₛ = mg sinλ; fₛ/N = tanλ = μₛ.
Minimum Force to Move Block on Rough Surface (from Aakash PDF — Laws of Motion JEE Main):
For a block of mass m on a horizontal rough surface (μ), a force F applied at angle θ above horizontal:
For motion: F cosθ > μ(mg – F sinθ) [normal force reduces due to upward component of F]
Normal force: N = mg – F sinθ; Friction: f = μN = μ(mg–F sinθ)
For motion: F cosθ ≥ μ(mg–F sinθ) → F(cosθ + μ sinθ) ≥ μmg
F ≥ μmg/(cosθ + μ sinθ)
This is minimum when (cosθ + μ sinθ) is maximum: d/dθ(cosθ+μ sinθ)=0 → –sinθ+μ cosθ=0 → tanθ=μ → θ = tan⁻¹(μ)
Fmin = mg·μ/√(1+μ²) = mg sinλ at angle θ = tan⁻¹(μ) = λ (angle of friction).
Wedge — Block on Accelerating Wedge (from Aakash PDF — Laws of Motion JEE Main):
Case I: Block m on smooth wedge M, no sliding — both move together with acceleration a. Force F required:
F = (M+m)g tanθ; acceleration a = g tanθ (from pseudo force analysis of block on wedge)
Contact force between block and wedge: R = mg/cosθ
Case II: Minimum F for block to fall freely (contact force between m and M = 0): F = Mg cotθ. At this F: wedge accelerates at g cotθ; block falls vertically with a = g.
Two-Block Systems — Friction (from Aakash PDF — Laws of Motion JEE Main):
Mass M₁ on bottom (on smooth ground), mass M₂ on top. Friction between M₁ and M₂ = μ. Force F on LOWER block M₁:
(a) If F ≤ (M₁+M₂)μg: both blocks move together, a = F/(M₁+M₂), maximum: amax = μg.
(b) If F > (M₁+M₂)μg: M₂ slips backward on M₁. M₂ has acceleration a₂ = μg. M₁ has acceleration a₁ = (F–M₂μg)/M₁.
Force F on UPPER block M₂:
(a) If F ≤ (M₁+M₂)M₂μg/M₁: both move together with amax = M₂μg/M₁.
(b) If F > above: M₂ slips forward on M₁. M₂: a₂ = (F–M₂μg)/M₂; M₁: a₁ = μg (friction from M₂ drives M₁). Download the Free PDF for Laws of Motion for all friction and two-block examples for JEE Main.
Circular Motion Dynamics — General (from Aakash PDF — Laws of Motion JEE Main):
For a body moving in a horizontal circle of radius r at speed v, the net inward (centripetal) force must equal mv²/r. This centripetal force is provided by the real forces on the body (tension, normal force, friction, gravity component).
Neglecting gravity: T = centripetal force = mv²/r = mω²r
Conical Pendulum (from Aakash PDF — Laws of Motion JEE Main):
A bob of mass m attached to a string of length l, swinging in a horizontal circle. The string makes angle θ with vertical. Vertical height h = l cosθ.
Resolving string tension T: T cosθ = mg (vertical equilibrium) and T sinθ = mω²r = mω²l sinθ (centripetal).
From these: T = mg/cosθ and ω² = g/h = g/(l cosθ)
Time period of conical pendulum: τ = 2π/ω = 2π√(h/g) = 2π√(l cosθ/g)
Key result from PDF: For θ to be 90° (string horizontal), T would need to be infinite → string CANNOT be horizontal. Maximum θ < 90°.
T sinθ = mω²l sinθ → T = mω²l (from centripetal equation).
Banked Road — Maximum Speed (from Aakash PDF — Laws of Motion JEE Main):
A vehicle negotiates a curve of radius r on a road banked at angle θ. Coefficient of friction = μ.
Maximum safe speed: vmax = √[rg(μ+tanθ)/(1–μtanθ)]
Special Cases from PDF:
(i) Smooth banked road (μ=0): v = √(rg tanθ) — the optimum speed for zero friction.
(ii) Horizontal rough road (θ=0): vmax = √(μrg) — friction provides all centripetal force.
Minimum speed for banked road: vmin = √[rg(tanθ–μ)/(1+μtanθ)]
Between vmin and vmax, the vehicle can negotiate the curve without slipping. Download the Free PDF for Laws of Motion for all circular motion dynamics examples for JEE Main.
Vertical Circle Setup (from Aakash PDF — Laws of Motion JEE Main):
A particle of mass m is tied to a string of length l, and revolves in a vertical circle. At the lowest point L, it is given speed vL (horizontal). At an arbitrary point P (angle θ from bottom), its speed is vP. At the highest point H, its speed is vH.
Energy Conservation in Vertical Circle (from Aakash PDF — Laws of Motion):
Using conservation of energy from L to P: ½mvL² = ½mvP² + mg(l–l cosθ) = ½mvP² + mgl(1–cosθ)
So: vP² = vL² – 2gl(1–cosθ)
From L to H (height = 2l): vH² = vL² – 4gl
Tension in the String at Any Point P (from Aakash PDF — Laws of Motion JEE Main):
At point P (angle θ from bottom), net force toward center = Tₚ – mg cosθ = mvP²/l:
Tₚ = mvP²/l + mg cosθ [Wait — at angle θ from bottom, component of mg toward center depends on position]
More precisely: at point P, if θ is measured from the BOTTOM (lowest point):
The component of mg along the string toward center = mg cos(π–θ) = –mg cosθ for upper half. Let's use the angle from VERTICAL (from top):
At angle φ from the top (highest point): T – mg cosφ = mv²/l → T = mv²/l + mg cosφ
At highest point H (φ=0, cosφ=1): TH = mvH²/l – mg [mg acts downward = toward center at top]
At lowest point L (φ=π, cosφ=–1): TL = mvL²/l + mg [mg acts downward = away from center at bottom]
Key Result — TL – TH = 6mg (from Aakash PDF — Laws of Motion JEE Main):
TL – TH = (mvL²/l + mg) – (mvH²/l – mg) = m(vL²–vH²)/l + 2mg = m(4gl)/l + 2mg = 4mg + 2mg = 6mg
This result TL–TH = 6mg is always true regardless of the values of vL and vH.
Three conditions for vertical circle (from Aakash PDF — Laws of Motion JEE Main):
Condition 1: If vL ≤ √(2gl): particle oscillates between M and M' (does not reach the top). The string goes slack before reaching H.
Condition 2: If √(2gl) < vL < √(5gl): particle leaves the circular path somewhere between M (horizontal point) and H (top). String goes slack at some point in upper half.
Condition 3: If vL ≥ √(5gl): particle COMPLETES the vertical circle. Minimum speed at top: vH_min = √(gl) (at vH=√gl, TH=0).
Minimum speed at bottom to complete: vL_min = √(5gl)
At vL = √(5gl): vH = √(vL²–4gl) = √(5gl–4gl) = √(gl) ✓
At points M (horizontal, height l): vM² = vL²–2gl. Tension at M: TM = mvM²/l = m(vL²–2gl)/l.
Download the Free PDF for Laws of Motion for all vertical circle examples for JEE Main.
All Laws of Motion formulas from the Aakash Rapid Revision PDF: Newton's 1st Law (inertia, F_net=0↔a=0), Newton's 2nd Law (F=ma=dp/dt, p=mv, impulse J=FΔt=Δp), Newton's 3rd Law (F_AB=–F_BA, action-reaction on different bodies), Lami's theorem (P/sinα=Q/sinβ=R/sinγ for 3-force equilibrium), constraint relations (Case I fixed middle: a₁=a₂; Case II fixed side: a₂=2a₁; Case III all free: a₁+a₂=2a₃), Atwood's machine (a=(M₂–M₁)g/(M₁+M₂); T=2M₁M₂g/(M₁+M₂)), moving pulley (T=2M₁M₂(g+a₀)/(M₁+M₂); aᵣ=(M₂–M₁)(g+a₀)/(M₁+M₂)), inclined plane pulley (a=(M₂sinβ–M₁sinα)g/(M₁+M₂); T=M₁M₂g(sinα+sinβ)/(M₁+M₂)), mass on string (stationary T=Mg; up T=M(g+a); down T=M(g–a); free fall T=0), machine gun force (F=nmv; wall: stop nmv, rebound 2nmv), liquid jet force (ρAv²; rebound 2ρAv²; oblique 2ρAv²cosθ), multi-block contact forces (a=F/ΣM; F₂=M₃a; F₁=(M₂+M₃)a), string tension at x (Tₓ=Fx/L), uniform rope tension (stationary Mₛgx/L; up Mₛx(g+a)/L; down Mₛx(g–a)/L), static friction (fₛ≤μₛN, self-adjusting), kinetic friction (fₖ=μₖN, μₖ<μₛ), angle of repose (λ=tan⁻¹μₛ), minimum force Fmin=mg/√(1+μ²) at θ=tan⁻¹(μ), wedge-block (F=(M+m)g tanθ, a=g tanθ, R=mg/cosθ), min F for free fall (Mg cotθ), two-block systems (F on lower: amax=μg; F on upper: amax=M₂μg/M₁), conical pendulum (T=mg/cosθ; τ=2π√(l cosθ/g)=2π√(h/g); string cannot be horizontal), banked road (vmax=√[rg(μ+tanθ)/(1–μtanθ)]; smooth: √(rg tanθ); horizontal rough: √(μrg)), vertical circle (TH=mvH²/l–mg; TL=mvL²/l+mg; TL–TH=6mg; conditions: vL≤√(2gl) oscillate; √(2gl)
Newton's Second Law F=ma applied through FBD is the single most-used technique in JEE Main Physics. It appears in Laws of Motion questions, but also in Work-Energy (net force × displacement = ΔKE), Rotational Motion (τ=Iα), Electrostatics (qE=ma), and everywhere else. The FBD technique — isolate each body, draw all real forces, apply F=ma — is the universal solving algorithm for JEE Main mechanics.
Atwood's machine formulas a=(M₂–M₁)g/(M₁+M₂) and T=2M₁M₂g/(M₁+M₂) are direct 4-mark JEE Main answers. Every pulley question reduces to identifying the effective driving force and the total inertia: a = (net force down)/(total mass). The tension formula comes from applying F=ma to either block alone: T–M₁g=M₁a → T=M₁(g+a)=M₁g+M₁a=M₁(g+(M₂–M₁)g/(M₁+M₂)) = 2M₁M₂g/(M₁+M₂).
The vertical circle result TL–TH=6mg is the most elegant formula in Laws of Motion for JEE Main. It is always true, regardless of the actual values of vL and vH. Derived purely from Newton's law and energy conservation, it can be used to immediately find TH if TL is given, or vice versa. Combined with the three conditions (vL<√(2gl): oscillates; between √(2gl) and √(5gl): leaves path; ≥√(5gl): completes circle), every vertical circle question in JEE Main is solved.
Banking road formula vmax=√[rg(μ+tanθ)/(1–μtanθ)] covers all road-friction-angle scenarios. The two special cases — smooth bank (μ=0) giving v=√(rg tanθ) and flat road (θ=0) giving v=√(μrg) — are both directly tested. The minimum speed formula vmin=√[rg(tanθ–μ)/(1+μtanθ)] is also testable. Download the Free PDF for Laws of Motion to have all formulas ready.
After working through Laws of Motion using this formula sheet, a student should confidently accomplish the following for JEE Main physics. On Newton's Laws: state and apply all three Newton's laws; draw correct free-body diagrams for any system; apply Lami's theorem for three-force equilibrium; compute impulse and change in momentum.
On constraint relations: identify the type of pulley setup (fixed middle, fixed side, all free); write the correct constraint equation; relate accelerations of connected masses; apply constraint along with Newton's 2nd law to find acceleration and tension in any Atwood-type system including inclined planes and moving pulleys.
On contact forces: compute force on gun/wall from bullets; compute force from liquid jet (normal and oblique impact); find contact forces between blocks in multi-block systems; find tension at any point in a string or uniform rope under gravity and acceleration.
On friction: distinguish static (self-adjusting) from kinetic (fixed at μₖN); find angle of repose; compute minimum force to move block at optimal angle; solve wedge-block problems for force and acceleration; solve two-block problems for maximum F before slipping in both F-on-lower and F-on-upper configurations.
On circular dynamics: apply centripetal force equation for any horizontal circular motion; solve conical pendulum for T, ω, and time period; apply banked road formula for vmax and vmin; solve horizontal rough road for maximum speed. On vertical circle: apply energy conservation to find speed at any point; compute string tension at any point; state and use TL–TH=6mg; identify which of the three conditions applies given vL. Download the Free PDF for Laws of Motion to test all outcomes before your JEE Main exam.
The Aakash Rapid Revision & Formula Bank PDF for Laws of Motion contains all Newton's law results, all pulley system formulas, all friction results, all circular motion dynamics formulas, and vertical circle conditions in one structured JEE Main physics reference.
Laws of Motion is the chapter that transforms kinematics (description of motion) into dynamics (explanation of motion). Newton's three laws — inertia, F=ma, action-reaction — are the complete framework for analysing any mechanical system. The FBD method, constraint relations, and Newton's 2nd law together can solve any pulley, inclined plane, or multi-block problem. Friction converts the ideally smooth world of kinematics into the realistic world of surfaces and resistance.
The four most-tested results in Laws of Motion for JEE Main: (1) Atwood's machine a=(M₂–M₁)g/(M₁+M₂) and T=2M₁M₂g/(M₁+M₂) — memorise both; (2) TL–TH=6mg and vertical circle three conditions — highest difficulty, highest marks; (3) Banked road vmax=√[rg(μ+tanθ)/(1–μtanθ)] — pure formula substitution; (4) Two-block amax=μg (lower) and amax=M₂μg/M₁ (upper) — direct substitution. Use this page and the Free PDF Download for Laws of Motion as your complete JEE Main revision foundation.
In Laws of Motion, Lami's theorem states: if three concurrent coplanar forces P, Q, R act on a body in equilibrium, then P/sinα = Q/sinβ = R/sinγ, where α is the angle between forces Q and R (opposite to P), β is the angle between P and R (opposite to Q), and γ is the angle between P and Q (opposite to R). The three angles satisfy α+β+γ=360°. Lami's theorem is applicable ONLY when exactly three concurrent forces act on a body in equilibrium — it cannot be applied to systems with more than three forces. Application in JEE Main Laws of Motion: a weight hanging from two strings at angles — the tension in each string and the weight satisfy Lami's theorem. Method: (1) Draw FBD showing all three forces meeting at a point. (2) Identify the angle opposite to each force (angle between the other two). (3) Apply P/sinα = Q/sinβ = R/sinγ. Special case: if three forces form an equilateral triangle of forces (each 120° apart), all three forces are equal. If two forces are equal and act symmetrically, the resultant bisects the angle between them.
In Laws of Motion, constraint relations come from the fixed string length. Case I (middle end of string is fixed, two masses hang on sides): x₁=x₂; v₁=v₂; a₁=a₂ — both masses have same magnitude of displacement, velocity, and acceleration. Case II (one side end is fixed, mass hangs from movable pulley): x₂=2x₁; v₂=2v₁; a₂=2a₁ — the mass connected to the movable end moves at twice the speed of the movable end, or the hanging mass moves at half the rate of the string end. Case III (all three ends free, two masses hang from ends, pulley moves): (x₁+x₂)/2=x₃; a₁+a₂=2a₃ — average displacement of end masses equals displacement of pulley. Sign convention: downward positive; if any motion is upward, include a negative sign. General method: write string-length constraint (sum of all string segments = constant), differentiate once for velocity, twice for acceleration. Application in JEE Main Laws of Motion: these constraints, combined with Newton's 2nd law for each mass, give the system of equations to find a and T.
In Laws of Motion, Atwood's machine has masses M₁ and M₂ (M₂>M₁) connected by string over fixed frictionless pulley. FBD of M₁ (up): T–M₁g=M₁a …(1). FBD of M₂ (down): M₂g–T=M₂a …(2). Adding (1)+(2): (M₂–M₁)g=(M₁+M₂)a → a=(M₂–M₁)g/(M₁+M₂). Substitute back: T=M₁(g+a)=M₁g+M₁(M₂–M₁)g/(M₁+M₂)=2M₁M₂g/(M₁+M₂). Check: if M₁=M₂: a=0, T=M₁g=M₂g ✓. Moving pulley (a₀ up): replace g by (g+a₀) in all formulas. T=2M₁M₂(g+a₀)/(M₁+M₂). Inclined plane: M₁ on smooth incline α, M₂ on smooth incline β: driving force = M₂g sinβ–M₁g sinα; total mass = M₁+M₂; so a=(M₂sinβ–M₁sinα)g/(M₁+M₂); T=M₁M₂g(sinα+sinβ)/(M₁+M₂). Mass on string hanging (not Atwood): T=Mg (stationary); T=M(g+a) (accelerating up); T=M(g–a) (accelerating down); T=0 (free fall, a=g). These Laws of Motion Atwood results are tested as direct 4-mark questions in JEE Main.
In Laws of Motion, angle of repose λ = tan⁻¹(μₛ): the maximum angle of an inclined plane for which a block stays at rest. At angle λ: N=mg cosλ, maximum fₛ=μₛN=μₛmg cosλ=mg sinλ (since tanλ=μₛ → sinλ/cosλ=μₛ → μₛ cosλ=sinλ). The block is on the verge of sliding at angle λ. For θ<λ: block stays at rest, friction fₛ=mg sinθ (self-adjusting). For θ=λ: block on verge of sliding. For θ>λ: block slides, fₖ=μₖmg cosθ applies. Minimum force to move block on horizontal surface: apply F at angle θ above horizontal. N=mg–F sinθ (upward component reduces N). For motion: F cosθ≥μ(mg–F sinθ) → F≥μmg/(cosθ+μ sinθ). Minimum F when d/dθ[cosθ+μ sinθ]=0 → tanθ=μ=tanλ → θ=λ. Fmin=μmg/(cosλ+μ sinλ)=μmg/√(1+μ²)×(1/1)=mg sinλ=mg·μ/√(1+μ²). Kinetic friction fₖ=μₖN is always fixed (independent of speed, area of contact, and independent of normal force dependence on velocity). μₖ<μₛ always. These Laws of Motion friction results are tested in JEE Main directly.
In Laws of Motion, conical pendulum: bob of mass m, string of length l, half-angle θ with vertical, horizontal radius r=l sinθ, vertical height h=l cosθ. Forces: tension T along string, weight mg downward. Equations: vertical: T cosθ=mg → T=mg/cosθ. Centripetal: T sinθ=mω²r=mω²l sinθ → T=mω²l. From both: mω²l=mg/cosθ → ω²=g/(l cosθ)=g/h. Time period: τ=2π/ω=2π√(h/g)=2π√(l cosθ/g). Key results: (1) As θ increases (string more horizontal), ω increases, τ decreases. (2) String CANNOT be horizontal: if θ→90°, h→0, T→∞ (infinite tension required) — impossible physically. (3) Time period depends on h=l cosθ (vertical height), not on l or θ separately. (4) Speed of bob: v=ωr=ω·l sinθ=l sinθ√(g/l cosθ)=√(gl sinθ tanθ). (5) If θ is very small (oscillation around vertical): τ≈2π√(l/g) (same as simple pendulum). In JEE Main Laws of Motion: conical pendulum questions ask for time period, tension, or angular velocity given the geometry.
In Laws of Motion, for a vehicle on a banked road (angle θ, coefficient μ, radius r): N cosθ=mg+f sinθ (vertical) and N sinθ+f cosθ=mv²/r (horizontal centripetal). At maximum speed, friction f=μN (pointing down the bank, opposing tendency to slide up). From vertical: N(cosθ–μ sinθ)=mg → N=mg/(cosθ–μ sinθ). Substitute in centripetal: mg(sinθ+μ cosθ)/(cosθ–μ sinθ)=mv²max/r → v²max=rg(sinθ+μ cosθ)/(cosθ–μ sinθ)=rg(tanθ+μ)/(1–μ tanθ). So vmax=√[rg(μ+tanθ)/(1–μtanθ)]. For minimum speed: friction points up the bank (opposing tendency to slide down). vmin=√[rg(tanθ–μ)/(1+μtanθ)]. Special cases in Laws of Motion: smooth bank (μ=0): v=√(rg tanθ) [optimum/design speed]. Horizontal rough road (θ=0): vmax=√(μrg). Safe speed range: vmin≤v≤vmax. If vehicle moves at exactly √(rg tanθ), friction=0 (ideal case). In JEE Main Laws of Motion: banked road questions give r, θ, μ and ask for vmax — direct formula substitution.
In Laws of Motion, vertical circle (string length l, given speed vL at bottom): Three conditions based on vL: Condition 1 (vL ≤ √(2gl)): Particle oscillates — it cannot reach the horizontal position M (height l). The string goes taut and motion reverses before the particle reaches the side. It oscillates back and forth between the bottom and some angle below the horizontal. Condition 2 (√(2gl) < vL < √(5gl)): Particle leaves circular path somewhere in the upper half (between M and H). It has enough energy to go past the horizontal but not enough tension to maintain circular motion at the top. At some angle in the upper half, T=0, string goes slack, particle follows projectile motion after that. Condition 3 (vL ≥ √(5gl)): Particle completes the vertical circle. At minimum (vL=√(5gl)), speed at top vH=√(gl) and TH=0 (just maintaining contact). For vL>√(5gl): TH>0. Key formula TL–TH=6mg is always true regardless of which condition applies (as long as the string is taut). At L: TL=mvL²/l+mg. At H: TH=mvH²/l–mg. At M (side): TM=mvM²/l (string perpendicular to weight, no component along string). These Laws of Motion vertical circle results are tested in JEE Main as multi-part questions.
In Laws of Motion, two-block systems (M₁ on bottom on smooth ground, M₂ on top, friction μ between them): Case A — Force F on LOWER block M₁: For F≤(M₁+M₂)μg: both move together, a=F/(M₁+M₂). Maximum friction: f_max=μM₂g. They move together as long as a≤μg (since for M₂: f≥M₂a is needed, max f=μM₂g → amax=μg). When F exceeds (M₁+M₂)μg: M₁ accelerates at a₁=(F–μM₂g)/M₁ (friction μM₂g backward); M₂ accelerates at a₂=μg (friction μM₂g forward drives M₂). M₂ slips backward on M₁ (stays behind). Case B — Force F on UPPER block M₂: Maximum friction on M₂ from M₁ = μM₁g (reaction to M₂ on M₁: but wait — N₂ on M₁ from M₂ = M₂g; friction from M₁ on M₂ = μM₂g backward). For M₁ to accelerate with M₂, friction from M₂ on M₁ = μM₂g (forward on M₁). For both together: a=F/(M₁+M₂); for M₁ alone: μM₂g=M₁a → amax=M₂μg/M₁. F_max for together=(M₁+M₂)M₂μg/M₁. If F exceeds this: M₂ slides forward on M₁. These Laws of Motion two-block results are tested in JEE Main.
In Laws of Motion, machine gun/bullets: gun fires n bullets per second, each of mass m with velocity u. Force to hold gun stationary = rate of change of momentum of bullets = nmv (recoil force by Newton's 3rd law). Bullets hitting wall: if bullets stop at wall: force on wall = rate of momentum removed = nmv. If bullets rebound elastically: each bullet's momentum changes from +mv to –mv, so Δp=2mv; force = n×2mv = 2nmv. Liquid jet (density ρ, area A, velocity v): mass hitting per second = ρAv. Force on wall (no rebound) = ρAv × v = ρAv². Force on pump to create jet = ρAv² (same — provides kinetic energy rate). Jet rebounds elastically: force = 2ρAv². Oblique impact at angle θ to wall normal: only normal component changes momentum; normal component = v cosθ; force = ρA(v cosθ)×2v cosθ = 2ρAv²cos²θ. Wait — more precisely for oblique: if velocity makes angle θ WITH the wall (not normal): normal component = v cosθ (where θ is with wall), and force = 2ρAv²cos²θ. From PDF: F = 2ρAv²cosθ where θ is angle with NORMAL to wall. These Laws of Motion jet/bullet force formulas appear in JEE Main as 1 direct question per few sessions.
In Laws of Motion, wedge-block: smooth wedge of mass M, inclination angle θ, block of mass m on wedge. Case I (block does not slide relative to wedge — both move together): horizontal force F applied on wedge, system accelerates at a. For block m (FBD in lab frame): horizontal: N sinθ=ma; vertical: N cosθ=mg. So tanθ=a/g → a=g tanθ. F=(M+m)a=(M+m)g tanθ. Contact force N between block and wedge: N sinθ=ma → N=ma/sinθ=mg tanθ/sinθ=mg/cosθ. Case II (minimum F so block falls freely, contact force N=0): if N=0, only gravity acts on block → block falls vertically. For wedge to maintain N=0: wedge must accelerate so that the block appears to fall freely relative to wedge. For the wedge (FBD): net force = F from outside. For N=0 at wedge surface: F acts on wedge at angle θ to normal → for equilibrium of normal force: F cosθ = Mg → F = Mg/cosθ × cosθ = Mg cotθ × sinθ... More precisely: pseudo force on block in wedge frame = ma (horizontal backward). For N=0: gravity mg (down) and pseudo force ma (backward) must give resultant along wedge surface. tanθ = a/g → a=g tanθ. But for block to fall freely: need N=0 → wedge accelerates at a=g/tanθ=g cotθ. F_wedge only = Ma_wedge = Mg cotθ. Block falls with a=g vertically. Contact force N=0.
Laws of Motion – JEE Main Physics Formula Sheet