Engineering Mechanics — GATE Formula Sheet

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01 / 08

Statics & Force Systems

Concurrent & coplanar forces, resultant, equilibrium, Lami's theorem

Theory

Force & Its Characteristics

A force is defined by its magnitude, direction, and point of application (3 characteristics — memorise as M-D-P).

Concurrent forces: all pass through one point
Coplanar forces: lie in the same plane
Collinear forces: act along the same line
Coplanar parallel forces: parallel lines, same plane

Formula

Resultant of Two Forces

R = √(P² + Q² + 2PQ·cosθ)

tanα = (Q·sinθ) / (P + Q·cosθ)

R = resultant magnitude · P, Q = forces · θ = angle between P & Q · α = angle R makes with P

Formula — GATE Favourite

Lami's Theorem

A/sinα = B/sinβ = C/sinγ

Applies when 3 concurrent coplanar forces are in equilibrium.
α = angle opposite to A · β = angle opposite to B · γ = angle opposite to C.
All three angles must sum to 360°.

Theory

Conditions of Equilibrium

For a body to be in static equilibrium, two conditions must be satisfied:

∑F = 0 → Sum of all forces = 0
∑M = 0 → Sum of all moments about any point = 0

In 2D: ΣFx = 0, ΣFy = 0, ΣM = 0 (3 equations, 3 unknowns max for determinate structures).

Formula

Moment of a Force

M = F × d

Varignon's Theorem: M_O(R) = ΣM_O(F_i)

M = moment (N·m) · F = force (N) · d = perpendicular distance from pivot (m)
Varignon's Theorem: Moment of resultant = algebraic sum of moments of components.

Formula

Couple

C = F × d

A couple consists of two equal, opposite, non-collinear forces.
C = couple moment · F = magnitude of each force · d = perpendicular distance between them.
A couple produces pure rotation only — net force = 0.

🧠 Memory Trick

Lami's Theorem — "Opposite Sins"

Each force is divided by the sine of the angle opposite to it (the angle not between the other two forces, but the angle "facing" that force).

Think: "Each force / its Opposite Sin" — and all three ratios are equal.

🎯 GATE Focus — Statics

High-Probability GATE Questions

3-force concurrent equilibrium using Lami's theorem — appears every 2nd year
Resultant of forces on a body + direction angle
Moment about a point from distributed/point loads
Couple equivalence and wrench systems

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02 / 08

Free Body Diagram & Support Reactions

FBD procedure, support types, beam reactions, static determinacy

Theory

FBD — What to Include

A Free Body Diagram isolates a body from its environment. Include:

Applied loads — all external forces/moments
Reaction forces — at supports
Weight — at centre of gravity
Do NOT include internal forces or forces on other bodies

Draw the FBD first before writing any equilibrium equation — non-negotiable in GATE.

Theory

Support Types & Reactions

Roller (1 reaction): perpendicular to surface only
Pin/Hinge (2 reactions): H + V components
Fixed/Cantilever (3 reactions): H + V + moment
Link (1 reaction): along the link axis

Determinacy condition: For beams, if unknowns = 3, it's statically determinate. More → indeterminate.

Formula

Simply Supported Beam — Point Load at 'a' from left

R_A = W·b / L

R_B = W·a / L

W = point load · a = distance from A · b = distance from B · L = total span
Take moments about A to find R_B, then ΣFy = 0 for R_A.

Formula

UDL on Simply Supported Beam

R_A = R_B = w·L / 2

w = UDL intensity (N/m) · L = span (m)
For symmetry, reactions are equal. Total load = wL acts at centre.

🧠 Memory Trick

Reaction from opposite end rule

For a simply supported beam with point load W:

R_A = W × (distance from B) / L

Closer to B means more reaction at A — the reaction is "pulled" towards the near support. Remember: reaction is proportional to how close the load is to the opposite support.

🎯 GATE Focus — FBD

High-Probability GATE Questions

Finding support reactions with multiple loads (point + UDL + moment)
Identifying static determinacy/indeterminacy
Overhanging beam reactions
FBD of pin-connected structures

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03 / 08

Trusses

Method of Joints, Method of Sections, zero-force members, determinacy

Theory

Truss Assumptions

A perfect truss assumes:

All members are two-force members (carry only tension or compression)
Loads applied only at joints
All joints are frictionless pins
All members are straight

This means: no bending in members — purely axial forces only.

Formula — Critical

Determinacy of Truss

m + r = 2j → Determinate (Perfect Truss)

m + r < 2j → Deficient (Mechanism)

m + r > 2j → Redundant (Indeterminate)

m = number of members · r = number of external reactions · j = number of joints

Theory

Zero-Force Members — Inspection Rules

Identify zero-force members without calculation:

Rule 1: At a joint with only 2 members (and no external load), both members carry zero force.
Rule 2: At a joint with 3 members where 2 are collinear and no external load — the 3rd member carries zero force.

Zero-force members add structural stability — don't remove them physically!

Theory

Method of Joints vs Method of Sections

Method of Joints:

Isolate each joint, apply ΣFx = 0, ΣFy = 0
Start at joint with ≤2 unknowns
Best when you need ALL member forces

Method of Sections:

Cut through ≤3 members, take a section, apply ΣFx, ΣFy, ΣM = 0
Best when you need force in ONE specific member (faster in GATE)

Method

Method of Joints — Sign Convention

Assume all members in Tension (+)

If calculated force is positive → member is in tension (being pulled apart)
If negative → member is in compression (being pushed together)
Draw arrows away from joint on FBD for assumed tension.

🧠 Memory Trick

Sections Method — "≤3 and Take Moments"

Cut must pass through ≤3 members (because you have 3 equilibrium equations).

Then take moment about the intersection of the other two unknown members — this eliminates them and gives you the third directly.

Cut → Isolate → Take moment at smart point → Direct answer

🎯 GATE Focus — Trusses

High-Probability GATE Questions

Find force in a specific member using Method of Sections
Identify zero-force members in a given truss
Determine if truss is perfect/deficient/redundant using m + r vs 2j
Pratt vs Warren truss — type identification

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04 / 08

Friction

Dry friction (Coulomb's law), angle of friction, wedge & screw friction, belt friction

Theory

Coulomb's Laws of Dry Friction

Friction force is proportional to normal force: F = μN
Friction is independent of contact area
Static friction ≥ Kinetic friction: μₛ > μₖ
Kinetic friction is independent of velocity (approximately)
Friction acts opposite to impending/actual motion

Formula

Friction Fundamentals

F_max = μₛ × N

tanφ = μ

Angle of repose θ = φ = tan⁻¹(μ)

μₛ = static friction coefficient · N = normal force
φ = angle of friction (= angle of repose for inclined plane)
At angle of repose, body is on verge of sliding.

Formula — GATE Favourite

Wedge Friction

P = W · tan(α + φ) [push up wedge]

P = W · tan(α − φ) [push down wedge]

Self-locking if α ≤ φ

P = applied force · W = load · α = wedge angle · φ = friction angle = tan⁻¹(μ)
Self-locking: wedge stays in place when P is removed.

Formula — GATE Favourite

Belt / Rope Friction (Capstan Equation)

T_tight / T_slack = e^(μθ)

ln(T₁/T₂) = μθ

T₁ = tight side tension · T₂ = slack side tension
μ = coefficient of friction · θ = angle of wrap in radians
For flat belt: same formula. For V-belt: replace μ with μ/sin(β/2)

Formula

Screw (Square Thread) Jack

P = W · tan(α + φ) [lifting]

P = W · tan(φ − α) [lowering]

η = tanα / tan(α + φ)

α = lead angle = tan⁻¹(l/πd) · φ = friction angle · l = lead · d = mean diameter
η = efficiency. Max η when α = 45° − φ/2.

🧠 Memory Trick

Belt Friction: "tight OVER slack = e to the μθ"

T_tight > T_slack always. The ratio = e^(μθ).

"Tight Over Slack = Exponent (mu × theta)"

θ must be in radians. 180° = π radians. 360° = 2π. This is the single most common belt friction mistake in GATE.

🎯 GATE Focus — Friction

High-Probability GATE Questions

Belt friction: find T₁ given T₂, μ, and wrap angle
Self-locking condition for wedge or screw
Efficiency of screw jack
Minimum force to push block up inclined plane with friction

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05 / 08

Kinematics of Particles & Rigid Bodies

Rectilinear, projectile, curvilinear, relative motion, rotation

Formula — Fundamental

Equations of Motion (Constant Acceleration)

v = u + at

s = ut + ½at²

v² = u² + 2as

s = (u + v)t / 2

u = initial velocity · v = final velocity · a = acceleration · s = displacement · t = time

Formula

Projectile Motion

R = u²·sin(2θ) / g

H = u²·sin²θ / (2g)

T = 2u·sinθ / g

R = range · H = max height · T = total time of flight
u = initial velocity · θ = launch angle · g = 9.81 m/s²
Max range when θ = 45° → R_max = u²/g

Formula

Curvilinear Motion — Normal & Tangential

a_t = dv/dt (tangential acceleration)

a_n = v² / ρ (centripetal/normal acceleration)

a = √(a_t² + a_n²)

ρ = radius of curvature · v = speed at the point
a_n always points toward centre of curvature. a_t is tangent to path.

Formula

Angular Kinematics (Rotation)

ω = ω₀ + αt

θ = ω₀t + ½αt²

ω² = ω₀² + 2αθ

v = ωr · a_t = αr · a_n = ω²r

ω = angular velocity (rad/s) · α = angular acceleration (rad/s²) · θ = angular displacement (rad) · r = radius

Formula

Relative Motion

v_B/A = v_B − v_A

v_B = v_A + ω × r_AB

v_B/A = velocity of B relative to A
For rigid body: velocity of B = velocity of A + rotational component (ω × r)
Where r_AB = position vector from A to B.

🧠 Memory Trick

Projectile: "Range = u²sin2θ / g"

SURGE:
Sin(2θ) × U² ÷ G = Range

For maximum range (θ = 45°): sin(90°) = 1, so R_max = u²/g — the simplest formula.

Two angles give same range: θ and (90° − θ). E.g., 30° and 60° give identical range.

🎯 GATE Focus — Kinematics

High-Probability GATE Questions

Velocity & acceleration of a point on a rotating rigid body
Instantaneous centre of velocity for mechanisms
Relative velocity between two links of a mechanism
Projectile range at different angles

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06 / 08

Dynamics — Newton's Laws & Energy Methods

Newton's laws, impulse-momentum, work-energy theorem, collision, D'Alembert's principle

Theory

Newton's Laws of Motion

1st Law (Inertia): Body remains at rest or uniform motion unless acted on by external force
2nd Law (F = ma): Net force = mass × acceleration. Direction of acceleration = direction of net force.
3rd Law (Action-Reaction): Every action has equal and opposite reaction. Forces act on different bodies.

Formula

Work-Energy Theorem

W_net = ΔKE = ½mv₂² − ½mv₁²

KE = ½mv²

PE (gravity) = mgh

Net work done on body = change in kinetic energy
Conservation of energy (no friction): KE₁ + PE₁ = KE₂ + PE₂

Formula

Impulse-Momentum Theorem

F·t = Δ(mv) = m(v₂ − v₁)

Linear Momentum: p = mv

Impulse = change in momentum. Useful when force acts for a short time (impact problems).
Conservation of momentum: if ΣF_ext = 0, then Σ(mv) = constant.

Formula — GATE Favourite

Coefficient of Restitution (Collision)

e = (v₂ − v₁) / (u₁ − u₂) = relative velocity of separation / relative velocity of approach

e = 1 → Perfectly Elastic

e = 0 → Perfectly Inelastic (stick together)

u₁, u₂ = velocities before collision · v₁, v₂ = velocities after
Use with momentum conservation to find v₁, v₂.

Formula

D'Alembert's Principle

ΣF − ma = 0

D'Alembert adds an imaginary inertia force = −ma (opposite to acceleration) to convert a dynamic problem into a static one.
Useful for: Atwood's machine, accelerating frames, connected bodies.

🧠 Memory Trick

Elastic vs Inelastic: "e = 1 for Elastic, e = 0 for Zero-energy"

Elastic (e = 1): Kinetic energy AND momentum conserved. Balls bouncing perfectly.

Inelastic (e = 0): Stick together, momentum conserved, KE is NOT.

For GATE: always write momentum conservation equation first, then use e to write the second equation.

🎯 GATE Focus — Dynamics

High-Probability GATE Questions

Collision problems: find velocity after impact using e + momentum conservation
Connected body problems (Atwood's machine via D'Alembert)
Work done by friction on inclined plane
Velocity after drop using energy conservation

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07 / 08

Moment of Inertia & Centroid

Area MOI, mass MOI, parallel axis theorem, perpendicular axis theorem, centroid of standard shapes

Formula — Core Theorems

Parallel & Perpendicular Axis Theorems

Parallel: I_xx = I_G + Ad²

Perpendicular: I_z = I_x + I_y (for thin laminas)

I_G = MOI about centroidal axis · A = area · d = distance between axes
Perpendicular axis theorem: applies only to planar laminas (2D bodies). z-axis perpendicular to the plane.

Reference Table

Area MOI of Standard Shapes (Centroidal)

Shape	I_xx (centroidal)	I_yy (centroidal)	Centroid from base
Rectangle (b × h)	`bh³/12`	`hb³/12`	h/2 from base
Triangle (base b, height h)	`bh³/36`	`—`	h/3 from base
Circle (radius r)	`πr⁴/4`	`πr⁴/4`	Centre
Semicircle (radius r)	`0.11r⁴`	`πr⁴/8`	4r/3π from base
Hollow Circle (R, r)	`π(R⁴−r⁴)/4`	`π(R⁴−r⁴)/4`	Centre

Formula

Mass Moment of Inertia

I = ∫ r² dm

I = mk² (k = radius of gyration)

Common mass MOIs:
Solid cylinder (about axis): I = ½mR²
Solid sphere: I = 2/5·mR²
Thin rod (about centre): I = mL²/12
Thin rod (about end): I = mL²/3

🧠 Memory Trick

Circle, Rect, Triangle: "4, 12, 36"

Area MOI denominators follow a pattern:
Circle: πr⁴/4
Rectangle: bh³/12
Triangle: bh³/36

"Four, twelve, thirty-six" — each is 3× the previous. Once you know the denominators, the numerators follow from geometry.

🎯 GATE Focus — MOI

High-Probability GATE Questions

MOI of composite sections using parallel axis theorem
Radius of gyration of a section
Centroid of L-section or T-section
Rolling without slipping problems using mass MOI

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08 / 08

Virtual Work & Simple Machines

Principle of virtual work, ideal machines, mechanical advantage, efficiency

Theory

Principle of Virtual Work

For a system in equilibrium, the total virtual work done by all forces during any virtual displacement is zero.

δW = ΣF · δr = 0

Advantages:

No need to find reaction forces at fixed supports
Only active forces (doing virtual work) appear
Powerful for multi-body mechanism analysis

Formula

Mechanical Advantage & Efficiency

MA = Load (W) / Effort (P)

VR = distance moved by Effort / distance moved by Load

η = MA / VR × 100%

For ideal machine (no friction): MA = VR → η = 100%
For real machine: MA < VR → η < 100%
Self-locking condition: η < 50%

Reference

VR of Common Simple Machines

Machine	Velocity Ratio (VR)
Pulley — single movable	`2`
Pulley — n movable pulleys	`2n`
Screw Jack	`2πl / p` (l=arm, p=pitch)
Wheel & Axle	`R / r`
Inclined Plane	`1 / sin θ`
Wedge	`1 / sin α`

🧠 Memory Trick

Virtual Work: "Only Active Forces Count"

Reaction forces at fixed pins and smooth surfaces do NO virtual work — because they are perpendicular to the allowed virtual displacement (or the displacement is zero).

So when writing the virtual work equation, skip all reactions and only include applied loads and weights. This simplifies complex multi-body problems dramatically.

🎯 GATE Focus — Virtual Work & Machines

High-Probability GATE Questions

Find equilibrium position of a mechanism using virtual work
Efficiency of a machine given MA and VR
Self-locking condition for machines
Pulley system VR and effort required

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Quick Reference

Essential Constants & Unit Conversions

Commonly used in GATE Engineering Mechanics calculations

Gravity

g = 9.81 m/s² ≈ 10 m/s² (in GATE)

1 radian

180°/π ≈ 57.3°

π (pi)

3.14159 (use 22/7 for quick calc)

e (Euler's number)

2.71828 (needed for belt friction)

1 N·m = 1 J (Joule)

Work = Force × Displacement

1 kW = 1000 W

Power = Work / Time = F·v

1 rad/s → RPM

N = 60ω / 2π = 30ω/π

Angle of repose

tanθ = μ (slope where object just slides)

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