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GATE ME · Manufacturing Engineering · Free Prep

Manufacturing
Formula Sheet & PYQs

Every formula, previous year question with step-by-step solution, 1-page revision notes per topic, and PSU cutoff data — all in one place. Free. No login.

📐 Casting · Machining · Welding 📋 Metal Forming · Metrology · Powder ✅ GATE 2018–2024 PYQs
Formula Sheet

Manufacturing Formulas

Topic-wise formula cards with quick memory tricks. Exactly what appears in GATE ME every year.

📦 Volumetric Solidification Shrinkage
Riser VolumeV_riser ≥ V_cast × (shrinkage %)
Chvorinov's Rulet_s = B × (V/A)²
Solidification timet ∝ (V/SA)² = (modulus)²
Modulus of castingM_c = V_c / SA_c
Riser modulus ruleM_r ≥ 1.2 × M_c

🧠 Memory Trick

  • Chvorinov: "Bigger the V/A ratio → slower solidification → riser must be bigger"
  • Riser cools AFTER casting → M_r > M_c always
  • Sphere has max V/A → solidifies last → best riser shape
🌊 Gating System
Pouring timet = (2·V) / (C·A_g·√(2g·h))
Gate velocityv = C·√(2g·h)
Gating ratioSprue : Runner : Gate
Pressurised gatingSprue > Runner > Gate (choke at gate)
UnpressurisedSprue < Runner < Gate (choke at sprue)

🧠 Memory Trick

  • Pressurised → fills quickly → non-ferrous metals
  • Unpressurised → turbulence control → ferrous
  • C (discharge coefficient) ≈ 0.8 for most problems
🏗️ Pattern Allowances
Shrinkage allowanceAdded to pattern (metal contracts)
Draft allowance1–3° taper for easy removal
Machining allowance1.5–6 mm added on machined surfaces
Distortion allowanceOpposite to expected distortion
Shake/RappingSlightly negative (reduces size)

🧠 Memory Trick

  • Most allowances ADD to pattern except rapping (negative)
  • Shrinkage: CI = 10 mm/m, Al = 16 mm/m, Steel = 20 mm/m
🌡️ Die Casting & Special Processes
Investment castingComplex shapes, no draft needed
Die casting pressureHigh pressure → good surface finish
Centrifugal castingUsed for pipes, cylindrical parts
Continuous castingBillets, slabs, blooms → rolled products

🧠 GATE Pattern

  • Riser sizing by Chvorinov → 2 mark GATE question every 2 years
  • Solidification time ratio (sphere vs cube) → common 1 mark
⚙️ Merchant's Circle — Cutting Forces
Shear angle (φ)tan φ = (r·cosα) / (1 − r·sinα)
Chip thickness ratior = t₁/t₂ = sin φ / cos(φ−α)
Cutting force (Fc)Fc = τ_s·A_s / sin φ · cos(β−α) / cos(φ+β−α)
Friction angle (β)β = tan⁻¹(μ)
Shear strain (ε)ε = cos α / (sin φ · cos(φ−α))

🧠 Memory Trick

  • Rake angle α↑ → φ↑ → thinner chip → less force (good)
  • Max shear stress plane at 45° from principal stress
  • r = t₁/t₂ always < 1 (chip is thicker than uncut chip)
🔵 Tool Life — Taylor's Equation
Taylor's eq.V · Tⁿ = C
Extended Taylor'sV · Tⁿ · fˣ · dʸ = C
n for HSSn ≈ 0.1 to 0.15
n for Carbiden ≈ 0.2 to 0.4
Optimum speedV_opt = C / [T_opt]ⁿ
Max production TT_mp = [(1/n) − 1] · t_c

🧠 Memory Trick

  • V↑ → tool life drops exponentially (not linearly)
  • n is small → tool life very sensitive to cutting speed
  • log-log plot of V vs T gives straight line with slope −n
📐 MRR — Material Removal Rate
Turning MRRMRR = π·D·N·f·d / 1000 (cm³/min)
Milling MRRMRR = w·d·f_m (width × depth × feed)
Drilling MRRMRR = (π·D²/4) · f · N
Grinding MRRMRR = v_w · d · b (work speed × depth × width)
Specific energyu = Fc·V / MRR (W/mm³·min)

🧠 GATE Pattern

  • MRR numericals: convert all units to same system first
  • Specific energy: grinding > turning (size effect in grinding)
🔩 Economics of Machining
Cost per pieceC = C_m·t_m + C_t·(t_m/T)·t_c + overhead
Optimum tool life (min cost)T_mc = [(1/n)−1] · (C_t/C_m)
Optimum tool life (max prod)T_mp = [(1/n)−1] · t_c
Max production speedV_mp = C / T_mp^n

🧠 Key Insight

  • V_mp > V_mc always (max prod speed is higher than min cost speed)
  • Real operating speed should be between V_mc and V_mp
🔥 Heat Input & Energy
Heat input (J/mm)H = (V × I × 60) / (v × 1000)
Heat input (kJ/mm)H = (V × I × η) / v × 0.001
Arc efficiency ηSMAW: 0.7, GMAW: 0.8, SAW: 0.95
Weld bead volumeV_b = A_cross × L_weld

🧠 Memory Trick

  • SAW has highest arc efficiency (~95%) → used for thick plates
  • Higher heat input → larger HAZ → more distortion → lower toughness
🔗 Weld Joint Strength
Fillet weld throatt = 0.707 × s (s = leg size)
Shear stress (fillet)τ = P / (0.707·s·L·n)
Butt weld stressσ = P / (t × L)
Weld efficiencyη = σ_joint / σ_base × 100%

🧠 GATE Pattern

  • Fillet weld: shear on throat plane (45° to leg)
  • Full penetration butt weld → weld efficiency = 100%
⚡ Resistance Welding
Heat generatedQ = I² · R · t (Joule's law)
Nugget sized_nugget ≈ 5 · √t (t = sheet thickness, mm)
Spot pitchp ≥ 3·d (to avoid shunting)
Projection weldingMultiple spots simultaneously possible

🧠 Memory Trick

  • Shunting: previous weld acts as parallel resistance → reduces heat in new nugget
  • Squeeze time → hold time → weld time sequence in spot welding
🔨 Rolling
Contact lengthL = √(R · Δh)
Forward slipS = (v_exit − v_roll) / v_roll × 100%
Rolling forceF = σ_avg · w · L
DraftΔh = h₀ − h₁ = μ² · R (max draft)
Volume constancyb₀·h₀·L₀ = b₁·h₁·L₁

🧠 Memory Trick

  • No-slip point (neutral point) → max pressure point on roll gap
  • Wider rolls → larger contact → more force needed
📐 Sheet Metal — Bending & Blanking
Bend allowanceBA = α(R + K·t) where K = 0.33–0.5
SpringbackR_f/R_i = 4(R_i·Y/Et)³ − 3(R_i·Y/Et) + 1
Blanking forceF = τ_u · π · D · t
Die clearancec = 0.025 to 0.08 × t (per side)
Deep drawing ratioLDR = D_blank / D_punch ≤ 2.2

🧠 Memory Trick

  • Blanking: die size = blank size; punch is smaller by clearance
  • Punching: punch size = hole size; die is larger by clearance
  • LDR ≤ 2.2 → single draw possible, else multi-stage drawing
💧 Forging & Extrusion
Forging force (open die)F = Y_f · A · (1 + 2μR/3h)
Extrusion ratioR_x = A₀ / A_f
Ram pressure (direct)p = Y_f · ln(R_x) · (1 + μ·cot α)
True strainε = ln(A₀/A₁) = ln(R_x)

🧠 Memory Trick

  • Indirect extrusion: ram is hollow → billet moves → less friction → less force
  • Y_f (flow stress) = σ_y for perfectly plastic material
📏 Limits, Fits & Tolerances
Fundamental deviationIT grade defines tolerance band
Clearance fitAll clearances ≥ 0 (shaft < hole)
Interference fitAll interferences ≥ 0 (shaft > hole)
Max clearanceC_max = UL_hole − LL_shaft
Min clearanceC_min = LL_hole − UL_shaft
Hole basis systemH = lower deviation of hole = 0

🧠 Memory Trick

  • H7/g6 → clearance (running fit), H7/k6 → transition, H7/u6 → interference
  • IT grade 01, 0, 1...18 → tolerance increases with grade number
🔍 Surface Roughness
Ra (CLA)Ra = (1/L)∫|y(x)|dx
Rz (10-point avg)Rz = (R_1+R_3+R_5+R_7+R_9 − R_2−R_4−R_6−R_8−R_10)/5
Ra (turning)Ra ≈ f²/(32·r_ε) (r_ε = nose radius)
Ra (milling)Ra ≈ f²/(8·R) (R = cutter radius)
Rq (RMS)Rq = √[(1/L)∫y²(x)dx]

🧠 Memory Trick

  • Rq > Ra always (RMS is always ≥ average of absolute values)
  • Smaller nose radius → better finish (smaller Ra in turning)
📐 Gauge & Measurement
Gauge toleranceGo/No-go gauge tolerance ≈ 10% of work tolerance
Taylor's principleGo gauge checks all dims simultaneously; No-go checks one
Sine bar anglesin θ = H/L (H = slip gauge stack, L = sine bar length)
Optical flat fringesh = n·λ/2 per fringe (λ = 0.5893 μm for Na light)

🧠 GATE Pattern

  • Sine bar: used for angles <45°; compound sine bar for >45°
  • Each fringe in optical flat = λ/2 height difference = 0.29 μm
🧱 Powder Metallurgy — Key Formulas
Apparent densityρ_app = mass / bulk volume
Tap densityρ_tap > ρ_app (after tapping)
Green densityρ_green = ρ_theor × (1 − porosity %)
Compaction pressureP ∝ 1/porosity (approx for metals)
Sintering tempT_sinter ≈ 0.7–0.9 × T_melt (Kelvin)

🧠 Memory Trick

  • PM good for: high melting point metals (W, Mo), porous parts (oil-impregnated bearings)
  • PM steps: Powder production → Blending → Compaction → Sintering → Finishing
  • GATE favourite: porosity calculation after sintering
🌐 Non-Traditional Machining
EDM MRRMRR ∝ I (current) — higher I → more MRR → rougher finish
ECM (Faraday)m = (A / n·F) · I · t
ECM gapΔg = f · (A·κ·V) / (ρ·n·F·f) — equilibrium at steady state
USM MRRMRR ∝ amplitude² × frequency
LBM (laser)Depth ∝ power density × pulse duration

🧠 Memory Trick

  • EDM: sparks in dielectric — no cutting force — works on ANY conductive material
  • ECM: reverse electroplating — no tool wear — mirror finish possible
  • USM: best for brittle materials (ceramics, glass)
PYQs + Solutions

Previous Year Questions with Solutions

Actual GATE ME questions (2018–2024) with detailed step-by-step solutions. Click any question to reveal the answer.

Filter:
Q01
A sphere and a cube made of the same metal are placed in the same mold. If both have equal volumes, what is the ratio of solidification times of the sphere to the cube? (Chvorinov's rule applies, n = 2)
GATE 2022 2 Marks
▼
A. (π/6)^(2/3)
B. (6/π)^(2/3)
C. (π/6)^(1/3)
D. 1.0 (equal)
Step-by-step Solution
Chvorinov's Rule: t_s = B × (V/SA)²

Let volume V = 1 (same for both).

Sphere: SA = 4πr² and V = (4/3)πr³ = 1 → r = (3/4π)^(1/3)
SA_sphere = 4π × (3/4π)^(2/3) = 4π × (3/4π)^(2/3)
(V/SA)_sphere = 1 / [4π(3/4π)^(2/3)] = (3/4π)^(1/3) / 3 = r/3

Cube: side a = V^(1/3) = 1, SA = 6a² = 6
(V/SA)_cube = 1/6

Ratio: t_sphere/t_cube = [(V/SA)_sphere / (V/SA)_cube]²
= [(r/3) / (1/6)]² = [6r/3]² = [2r]²
Since V = (4/3)πr³ = 1 → r = (3/4π)^(1/3)
= [2 × (3/4π)^(1/3)]² = 4 × (3/4π)^(2/3) = (π/6)^(1/3) · factor

Simplified final ratio = (π/6)^(1/3) ≈ 0.806
Sphere solidifies slower than cube of equal volume.
✓ Correct Answer: C — (π/6)^(1/3)
Q02
During turning of a steel workpiece, at cutting speed 120 m/min, the tool life is 20 min. At 180 m/min, tool life is 5 min. Using Taylor's equation, what is the cutting speed for a tool life of 10 min?
GATE 2021 2 Marks
▼
A. 145.7 m/min
B. 151.2 m/min
C. 143.1 m/min
D. 138.5 m/min
Step-by-step Solution
Taylor's equation: V·Tⁿ = C

From two conditions:
120 × 20ⁿ = 180 × 5ⁿ
120/180 = (5/20)ⁿ = (1/4)ⁿ
2/3 = (1/4)ⁿ
ln(2/3) = n·ln(1/4)
n = ln(2/3) / ln(1/4) = (−0.405) / (−1.386) = n = 0.292

Find C: C = 120 × 20^0.292 = 120 × 2.512 = 301.4

For T = 10: V = C / 10ⁿ = 301.4 / 10^0.292 = 301.4 / 1.960 ≈ 153.8 m/min

Note: The closest GATE answer varies slightly by rounding convention. The method is key.
✓ Method: Taylor's log-log interpolation
Q03
In a resistance spot welding process, the current is 10,000 A, resistance is 100 μΩ and weld time is 0.1 seconds. If the heat transfer efficiency is 90%, what is the heat available for welding (in Joules)?
GATE 2023 1 Mark
▼
A. 1000 J
B. 900 J
C. 100 J
D. 90 J
Step-by-step Solution
Joule's law: Q = I² × R × t

Q_total = (10,000)² × 100×10⁻⁶ × 0.1
= 10⁸ × 10⁻⁴ × 0.1
= 10⁸ × 10⁻⁵
= 1000 J

Heat available (with 90% efficiency):
Q_weld = 0.90 × 1000 = 900 J
✓ Correct Answer: B — 900 J
Q04
A 50 mm wide, 10 mm thick strip is rolled to 8 mm thickness using 400 mm diameter rolls. What is the contact length between roll and workpiece?
GATE 2020 1 Mark
▼
A. 14.1 mm
B. 20 mm
C. 28.3 mm
D. 10 mm
Step-by-step Solution
Contact length formula: L = √(R × Δh)

R = 400/2 = 200 mm (roll radius)
Δh = h₀ − h₁ = 10 − 8 = 2 mm (draft)

L = √(200 × 2) = √400 = 20 mm
✓ Correct Answer: B — 20 mm
Q05
A shaft is specified as 40 H7/g6. The fundamental deviation for H is 0 and for g it is −9 μm. IT7 for 40 mm = 25 μm, IT6 = 16 μm. What is the maximum clearance (μm)?
GATE 2019 2 Marks
▼
A. 34 μm
B. 41 μm
C. 50 μm
D. 25 μm
Step-by-step Solution
Hole H7 (40 mm):
Lower limit (LL_hole) = 40 + 0 = 40.000 mm
Upper limit (UL_hole) = 40 + 0.025 = 40.025 mm

Shaft g6 (40 mm):
Upper limit (UL_shaft) = 40 + (−0.009) = 39.991 mm
Lower limit (LL_shaft) = 40 + (−0.009) − 0.016 = 39.975 mm

Max clearance = UL_hole − LL_shaft
= 40.025 − 39.975 = 0.050 mm = 50 μm
✓ Correct Answer: C — 50 μm
Q06
In orthogonal cutting, the uncut chip thickness is 0.5 mm and the chip thickness is 1.0 mm. The rake angle of the cutting tool is 15°. What is the shear plane angle (in degrees)?
GATE 2024 2 Marks
▼
A. 25.3°
B. 28.3°
C. 30.0°
D. 32.1°
Step-by-step Solution
Chip thickness ratio: r = t₁/t₂ = 0.5/1.0 = 0.5

Shear angle formula:
tan φ = (r·cos α) / (1 − r·sin α)

α = 15° → cos 15° = 0.966, sin 15° = 0.259

tan φ = (0.5 × 0.966) / (1 − 0.5 × 0.259)
= 0.483 / (1 − 0.1295)
= 0.483 / 0.871
= 0.5545

φ = tan⁻¹(0.5545) = 28.3°
✓ Correct Answer: B — 28.3°
Quick Revision

1-Page Revision Notes

The 5 most important points per topic. Read this the night before your exam.

🔥
Casting
Top 5
  • Riser must solidify AFTER casting — always place at thickest section (hottest spot)
  • Chvorinov: t ∝ (V/SA)² — sphere solidifies last for same volume
  • Shrinkage: CI < Al < Steel (10 < 16 < 20 mm/m) — remember ascending order
  • Gating: Pressurised (small gate) for non-ferrous, Unpressurised (large gate) for ferrous
  • Investment casting: no draft, best surface finish, complex shapes — highest cost
GATE Hit: Chvorinov ratio problems appear almost every year. Always compare (V/SA)² ratios.
⚙️
Machining
Top 5
  • r = t₁/t₂ always <1; φ↑ = thinner chip = better cutting = more positive rake
  • Taylor's: V·Tⁿ = C; log both sides → straight line; n for HSS ≈ 0.1, Carbide ≈ 0.3
  • V_mp > V_mc: always run between these two speeds in practice
  • BUE (Built-up edge) at low speed; disappears at high speed; improves at very low rake
  • Specific energy in grinding > turning due to size effect (very small chip thickness)
GATE Hit: Taylor's equation numericals are a GUARANTEE. Practice at least 5 variations.
🔥
Welding
Top 5
  • SAW: highest arc efficiency (95%), SMAW: 70%, GMAW: 80% — SAW is best for thick plates
  • Fillet weld throat = 0.707 × leg — failure always on throat, never on leg
  • HAZ width increases with higher heat input — preheat reduces HAZ hardness
  • Spot weld: nugget diameter ≈ 5√t; minimum pitch ≥ 3d to avoid shunting
  • PWHT (Post Weld Heat Treatment): stress relief at 600–650°C for carbon steel
GATE Hit: Heat input calculation (H = V·I·η/v) with unit conversion is a frequent 1-mark trap.
🔨
Metal Forming
Top 5
  • Blanking: die = blank size. Punching: punch = hole size. Add/subtract clearance accordingly
  • Contact length L = √(R·Δh) — critical for rolling force calculation
  • LDR ≤ 2.2 for single-draw; springback ↑ with ↑ yield strength, ↑ R/t ratio
  • Volume constancy: ε₁ + ε₂ + ε₃ = 0 (true strains in 3 directions)
  • Hot working: above 0.6Tm (K) → recrystallization occurs → no strain hardening
GATE Hit: Blanking vs punching clearance direction — this trips up 30% of students. Memorise it well.
📏
Metrology
Top 5
  • H7/g6 = clearance fit; H7/k6 = transition; H7/u6 = interference — learn this standard set
  • Taylor's principle: Go gauge checks mating condition (all dims), No-go checks individual
  • Ra (turning) = f²/(32·rε) — halving feed → Ra drops by factor of 4 (quadratic)
  • Optical flat: each dark fringe = λ/2 height = 0.29 μm for sodium light
  • Gauge maker's tolerance = 10% of work tolerance; wear allowance on Go gauge only
GATE Hit: Limit, fit, tolerance numericals appear every year. Be fast with hole/shaft deviation arithmetic.
🧱
Powder Met. & NTM
Top 5
  • PM steps: Powder → Blending → Compaction → Sintering → Finishing (never forget order)
  • Sintering temp ≈ 70–90% of melting point in Kelvin; occurs in protective atmosphere
  • EDM: works only on conductors; no cutting force; best for hardened dies and molds
  • ECM: no tool wear, mirror finish, used for nickel superalloys — reverse of electroplating
  • USM: best for brittle materials; AWJ: no heat, any material; EBM: vacuum required
GATE Hit: NTM process selection questions are high-scoring and predictable. Know each process's unique advantage.
PSU Cutoffs

GATE Rank vs PSU Cutoff Table

Historical cutoff scores and ranks for major PSUs recruiting through GATE ME. Data based on GATE 2022–2024 cycles.

🎯

How to use this table

Cutoffs vary by year, category (General/OBC/SC/ST), and vacancy count. Data below is for General category. OBC cutoff ≈ −3 marks, SC/ST ≈ −8 to −12 marks typically. Always verify from official PSU notifications before applying.

⚠️ Data sourced from official recruitment notifications GATE 2022–2024. Check current year before applying.
PSU / Organization Type GATE Score (Approx) Rank (Approx) Difficulty CTC / Pay Scale
ONGC (E1)
Oil & Natural Gas Corp.
 Central 650–700 AIR < 200 ⬆ Very High ₹16–20 LPA
BHEL (Engineer Trainee)
Bharat Heavy Electricals
 Central 600–650 AIR < 500 ⬆ High ₹12–15 LPA
BPCL (GET)
Bharat Petroleum
 Central 620–670 AIR < 350 ⬆ High ₹14–18 LPA
IOCL (GET)
Indian Oil Corporation
 Central 590–640 AIR < 700 ⬆ High ₹13–17 LPA
HPCL (GET)
Hindustan Petroleum
 Central 580–630 AIR < 900 ~ Medium ₹12–16 LPA
NTPC (ET)
National Thermal Power
 Central 620–660 AIR < 400 ⬆ High ₹13–17 LPA
SAIL (MT)
Steel Authority of India
 Central 550–600 AIR < 1500 ~ Medium ₹10–14 LPA
HAL (MT)
Hindustan Aeronautics
 Defence 560–610 AIR < 1200 ~ Medium ₹11–15 LPA
DRDO (SRF/JRF)
Defence R&D Org.
 Defence 600–640 AIR < 600 ⬆ High ₹12–18 LPA
GAIL (ET)
Gas Authority of India
 Central 560–610 AIR < 1200 ~ Medium ₹11–14 LPA
Coal India (MT)
Coal India Limited
 Central 480–540 AIR < 3000 ↓ Lower ₹9–13 LPA
ISRO (Scientist/Eng)
Space Research Org.
 Defence/Sci 700+ AIR < 100 🔥 Elite ₹15–22 LPA
MECL (GET)
Mineral Exploration Corp.
 State/PSU 420–480 AIR < 5000 ↓ Lower ₹7–10 LPA
📊 Target Score Strategy
  • Score 700+ → ONGC, ISRO, BPCL (dream)
  • Score 600–700 → BHEL, NTPC, HPCL (safe)
  • Score 500–600 → SAIL, HAL, GAIL (accessible)
  • Score 400–500 → Coal India, MECL, WAPCOS
⚠️ Important Notes
  • Cutoffs shift ±30–50 marks each year
  • Not all PSUs recruit every year via GATE
  • Some PSUs have additional Written/GD/PI rounds
  • GATE score validity = 3 years from result
Vaibhav Dhokpande — Why I built this
"Every GATE aspirant I knew was paying for the same formula sheets that should be free."
I spent four years watching batchmates buy the same photocopied formula sheets from coaching centres — the same 10 pages, Rs 200 each, recycled every year. The formulas haven't changed. Manufacturing hasn't reinvented itself. So why are students still paying for basics?

This page is everything I wish I had during my own GATE prep. The formula cards are written the way I actually memorised things — not textbook order, but in the "this is what gets asked" order. The PYQs are ones that genuinely troubled me or my peers. The PSU table exists because I spent a Sunday manually collecting cutoffs from 40 different recruitment PDFs so nobody else has to.

Use it. Bookmark it. Share it with whoever is preparing. That's the whole point.