GATE ME Formula Sheet 2027: Key Formulas by Subject

Q: Can I take a formula sheet into the GATE ME exam?

No. Candidates cannot carry paper, a calculator, a phone or a watch into the exam hall. The exam provides a virtual calculator, rough sheets and a pen. A useful habit is to write a few easily forgotten formulas on the rough sheet in the first five minutes. Confirm the current exam-day rules at gate2027.iitm.ac.in before the exam.

Q: What are the hoop and longitudinal stresses in a thin cylinder?

For a thin cylinder of diameter d, wall thickness t and internal pressure p, the hoop stress is pd/2t and the longitudinal stress is pd/4t, half the hoop value. A thin sphere carries pd/4t in every direction. The results hold only for a thin wall, with t below about d/20.

Q: When does a spring mounting isolate vibration?

Only when the frequency ratio r = ω/ωₙ is greater than √2. Below that, the transmissibility ratio is above 1 and the mount passes on more force than it receives. At r = √2 it equals 1 for every damping ratio. Above √2, extra damping raises the transmitted force, so isolators use soft springs and light damping.

Q: Which GATE ME 2027 topics are new and need formulas?

The 2027 syllabus names several topics for the first time. They include velocity potential and converging-diverging nozzles, transmissibility ratio, PID controllers and transfer functions, belt drives, boiling and condensation, combustion with air-fuel and equivalence ratio, pump characteristics, additive manufacturing, CNC and actuators, programmable logic controllers, work study and six sigma. Each has at least one formula or rule worth knowing.

Q: How should I revise formulas before GATE ME?

Read a compact formula sheet once a day in the final two weeks and once more on the exam morning. For each formula, know the condition it needs and the trap built around it. Write the unit next to every number, use kelvin inside any temperature ratio, carry full precision and round only the final numerical answer.

By MD ANISH AHAMADUpdated 4 Oct 20268 min read

GATE ME Formula Sheet 2027: Key Formulas by Subject

This GATE ME formula sheet collects the formulas mechanical engineering questions lean on most, grouped by subject, with the condition or trap to watch for beside each. It is a free selection from the last-minute sheet in the GATE ME 2027 book. Use it to test your recall, then practise each formula on previous-year questions.

In this guide

Key takeaways
How to read this sheet
Engineering Mathematics
Engineering Mechanics
Mechanics of Materials
Theory of Machines and Vibrations
Machine Design
Fluid Mechanics
Heat Transfer
Thermodynamics and Applications
Materials, Manufacturing and Metrology
Industrial Engineering
General Aptitude essentials
How to use a formula sheet in the exam
Subject-wise formula sheets
Quick revision

Key takeaways

Most numerical questions in GATE ME reuse a small set of standard formulas with new numbers.
Every formula carries a condition, such as a thin wall, an isentropic process or a Biot number below 0.1.
Temperatures inside any ratio, power or exponential must be in kelvin; Celsius is safe only in a difference.
The 2027 syllabus adds topics such as PID control, nozzles, combustion and six sigma, so their formulas belong on your sheet too.
You cannot carry paper into the exam (confirm current rules at gate2027.iitm.ac.in), so formulas must be in memory.

How to read this sheet

SI units are used throughout, with $g = 9.81\ \text{m/s}^2$ . $\ln$ is the natural logarithm, and angles inside formulas are in radians. Subscript 1 is an inlet or initial state; subscript 2 is an exit or final state.

The condition of a formula is what must be true for it to hold: steady flow, small damping, a constant failure rate. The Watch out for column gives that condition or the slip that most often costs marks. A formula used outside its condition gives a confident wrong answer.

Engineering Mathematics

Formula	Watch out for
$\sum \lambda_i = \operatorname{trace} A$ , $\prod \lambda_i = \det A$ ; $A^k$ has $\lambda^k$ , $A^{-1}$ has $1/\lambda$	Triangular matrix: eigenvalues sit on the diagonal
$Ax = b$ has a unique solution iff $\operatorname{rank} A = \operatorname{rank}\,[A \mid b] = n$	Equal ranks below $n$ give infinitely many
$rt - s^2 > 0$ and $r > 0$ : minimum; $r < 0$ : maximum ( $r = f_{xx}$ , $s = f_{xy}$ , $t = f_{yy}$ )	$rt - s^2 < 0$ is a saddle point
$\displaystyle \mathcal{L}\{e^{at}\} = \frac{1}{s - a}$ ; $\mathcal{L}\{f'\} = sF(s) - f(0)$	Final value theorem needs every pole of $sF(s)$ in the left half-plane
$\displaystyle \oint \frac{f(z)}{z - a}\,dz = 2\pi i\,f(a)$	Only for $a$ inside the curve
Poisson $\displaystyle P(k) = \frac{e^{-\lambda}\lambda^k}{k!}$ , mean $=$ variance $= \lambda$ ; Binomial mean $np$ , variance $np(1 - p)$	Sample variance divides by $n - 1$
$\displaystyle x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$	Fails when $f'$ is near zero
Simpson's rule $\displaystyle \frac{h}{3}\big[y_0 + 4(y_1 + y_3 + \cdots) + 2(y_2 + y_4 + \cdots) + y_n\big]$	$n$ must be even; exact for cubics

Engineering Mechanics

Formula	Watch out for
Belt or band $\displaystyle \frac{T_1}{T_2} = e^{\mu\theta}$	$\theta$ in radians; $T_1$ is the tight side
Screw jack $\displaystyle \eta = \frac{\tan\alpha}{\tan(\alpha + \phi)}$ ; self-locking iff $\phi > \alpha$	Self-locking keeps $\eta$ below 50 per cent
Plane truss determinate and rigid when $m = 2j - 3$	Zero-force members at unloaded joints
Rolling down an incline $\displaystyle a = \frac{g\sin\theta}{1 + k^2/r^2}$	Disc $\displaystyle \frac{2}{3}g\sin\theta$ , solid sphere $\displaystyle \frac{5}{7}g\sin\theta$
$\displaystyle I_{\text{rod, centre}} = \frac{mL^2}{12}$ , $\displaystyle I_{\text{rod, end}} = \frac{mL^2}{3}$ ; $I = I_G + md^2$	Parallel axis only from the centroidal axis
$\displaystyle e = \frac{v_2' - v_1'}{u_1 - u_2}$	$e = 1$ elastic, $e = 0$ plastic

Mechanics of Materials

Formula	Watch out for
$E = 2G(1 + \nu) = 3K(1 - 2\nu)$	$0 \le \nu \le 0.5$
$\displaystyle \sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$	Angle $\theta$ on the element is $2\theta$ on Mohr's circle
Thin cylinder: hoop $\displaystyle \frac{pd}{2t}$ , longitudinal $\displaystyle \frac{pd}{4t}$ ; thin sphere $\displaystyle \frac{pd}{4t}$	Thin wall only, $t$ below about $d/20$
$\displaystyle \frac{M}{I} = \frac{\sigma}{y} = \frac{E}{R}$ ; $\displaystyle \frac{T}{J} = \frac{\tau}{r} = \frac{G\theta}{L}$	$\displaystyle I = \frac{\pi d^4}{64}$ for bending, $\displaystyle J = \frac{\pi d^4}{32}$ for torsion
$\delta$ : cantilever end load $\displaystyle \frac{PL^3}{3EI}$ ; simply supported central load $\displaystyle \frac{PL^3}{48EI}$ , UDL $\displaystyle \frac{5wL^4}{384EI}$	$w$ is load per length
$\displaystyle P_{cr} = \frac{\pi^2 EI}{L_e^2}$ ; $\displaystyle L_e = L, 2L, \frac{L}{2}, \frac{L}{\sqrt{2}}$	Pinned, fixed–free, fixed–fixed, fixed–pinned; use the least $I$
$\displaystyle \tau = \frac{VQ}{Ib}$	Rectangle: maximum is 1.5 times the mean

Remember: hoop stress is twice longitudinal stress, and $J$ is twice $I$ for a circle. Halving or doubling the wrong one is the commonest slip in this subject.

Theory of Machines and Vibrations

Formula	Watch out for
Kutzbach $F = 3(n - 1) - 2j_1 - j_2$	$n$ counts the frame
Grashof $s + l \le p + q$ for a fully rotating link	Which link is fixed decides the inversion
Coriolis component $2\omega v$	Only with sliding on a rotating link
Flywheel $\Delta E = I\omega^2 C_s$ , $\displaystyle C_s = \frac{\omega_{\max} - \omega_{\min}}{\omega_{\text{mean}}}$	$\omega$ in rad/s, not rpm
$\omega_n = \sqrt{k/m} = \sqrt{g/\delta_{st}}$ ; $\displaystyle \zeta = \frac{c}{2\sqrt{km}}$ ; $\omega_d = \omega_n\sqrt{1 - \zeta^2}$	Series springs add as reciprocals
$\displaystyle \delta = \ln\frac{x_1}{x_2} = \frac{2\pi\zeta}{\sqrt{1 - \zeta^2}}$	$\zeta \approx \delta/(2\pi)$ only when small
$\displaystyle TR = \frac{\sqrt{1 + (2\zeta r)^2}}{\sqrt{(1 - r^2)^2 + (2\zeta r)^2}}$ , $r = \omega/\omega_n$	New in 2027; isolation only for $r > \sqrt{2}$
PID $\displaystyle G_c(s) = K_p + \frac{K_i}{s} + K_d s$ ; overshoot $e^{-\pi\zeta/\sqrt{1 - \zeta^2}}$	New in 2027; integral action removes the step offset

Machine Design

Formula	Watch out for
Tresca $\sigma_1 - \sigma_3 \le S_y/N$ ; von Mises $\sqrt{\sigma_1^2 - \sigma_1\sigma_2 + \sigma_2^2} \le S_y/N$	Shear yield $0.5S_y$ against $0.577S_y$
Shaft $\displaystyle \tau_{\max} = \frac{16\sqrt{M^2 + T^2}}{\pi d^3}$	Equivalent moment uses $\displaystyle \frac{1}{2}\big[M + \sqrt{M^2 + T^2}\big]$
Goodman $\displaystyle \frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}} = \frac{1}{N}$ ; Soderberg uses $S_y$	Soderberg is the most conservative
$K_f = 1 + q(K_t - 1)$	$0 \le q \le 1$
$L_{10} = (C/P)^p$ million revolutions	$p = 3$ ball, $10/3$ roller
Spring $\displaystyle \delta = \frac{8FD^3 n}{Gd^4}$ ; $\displaystyle K_w = \frac{4C - 1}{4C - 4} + \frac{0.615}{C}$ , $C = D/d$	Wahl factor for stress, not deflection
Clutch, uniform wear $\displaystyle T = \mu W\frac{R_1 + R_2}{2}$ per friction surface	Multiply by the number of surface pairs
Open belt $\displaystyle L = \frac{\pi(D + d)}{2} + 2C + \frac{(D - d)^2}{4C}$	New in 2027; crossed belt uses $(D + d)^2$

This page gives a selection. The book's last-minute sheet covers every topic the 2027 syllabus names, each entry with its condition, plus a worked number recomputed before printing. It is part of the GATE ME 2027 book, alongside 942 questions with worked solutions and 10 full mock tests.

Fluid Mechanics

Formula	Watch out for
$F = \rho g\bar{h}A$ ; centre of pressure at $\displaystyle \bar{h} + \frac{I_G\sin^2\theta}{A\bar{h}}$	$\bar{h}$ is the depth of the centroid
$\displaystyle GM = \frac{I}{V} - BG$	Stable only if $GM > 0$
$\displaystyle \frac{p}{\rho g} + \frac{V^2}{2g} + z = \text{constant}$	Steady, incompressible, inviscid, along a streamline
$\displaystyle \Delta p = \frac{32\mu VL}{D^2}$ ; $\displaystyle f = \frac{64}{Re}$	Laminar only, below $Re \approx 2000$
$\displaystyle h_f = \frac{fLV^2}{2gD}$	$f$ is the Darcy factor
$\displaystyle u = \frac{\partial\psi}{\partial y}$ , $\displaystyle v = -\frac{\partial\psi}{\partial x}$ ; $\vec{V} = \nabla\phi$	New in 2027; $\phi$ exists only for irrotational flow
$\displaystyle \frac{T_0}{T} = 1 + \frac{\gamma - 1}{2}M^2$ ; $\displaystyle \frac{p^*}{p_0} = 0.528$ for $\gamma = 1.4$	New in 2027; a converging nozzle never exceeds $M = 1$
$\displaystyle \frac{dA}{A} = (M^2 - 1)\frac{dV}{V}$	Supersonic flow speeds up in a diverging passage

Heat Transfer

Formula	Watch out for
$\displaystyle R = \frac{L}{kA}$ ; cylinder $\displaystyle \frac{\ln(r_2/r_1)}{2\pi kL}$ ; convection $\displaystyle \frac{1}{hA}$	Series resistances add
Critical radius $r_c = k/h$ (cylinder), $2k/h$ (sphere)	Below $r_c$ , insulation raises heat loss
$\displaystyle m = \sqrt{\frac{hP}{kA_c}}$ ; $q = \sqrt{hPkA_c}\,\theta_b\tanh(mL)$	Insulated tip; correct the length for a convecting tip
$\displaystyle \frac{T - T_\infty}{T_i - T_\infty} = e^{-t/\tau}$ , $\displaystyle \tau = \frac{\rho Vc}{hA_s}$	Lumped only if $Bi < 0.1$
$Nu = 0.023\,Re^{0.8}Pr^{n}$	$n = 0.4$ heating the fluid, $0.3$ cooling
$\displaystyle \Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1/\Delta T_2)}$ ; $\displaystyle \varepsilon = \frac{NTU}{1 + NTU}$ (counterflow, $C_r = 1$ )	$NTU = UA/C_{\min}$
$\lambda_{\max}T = 2898\ \mu\text{m K}$ ; plates $\displaystyle q = \frac{\sigma(T_1^4 - T_2^4)}{1/\varepsilon_1 + 1/\varepsilon_2 - 1}$	Kelvin only

Boiling and condensation, new in 2027, are mostly read off the pool-boiling curve: nucleate boiling gives the highest useful $h$ , and dropwise condensation beats filmwise.

Thermodynamics and Applications

Formula	Watch out for
$Q - W = \Delta U$ ; steady flow $\displaystyle \dot{Q} - \dot{W} = \dot{m}\big[\Delta h + \frac{\Delta V^2}{2} + g\Delta z\big]$	Throttling: $h_2 = h_1$
Polytropic $\displaystyle W = \frac{p_1V_1 - p_2V_2}{n - 1}$ ; isothermal $\displaystyle p_1V_1\ln\frac{V_2}{V_1}$	Absolute pressures
$\displaystyle \Delta s = c_p\ln\frac{T_2}{T_1} - R\ln\frac{p_2}{p_1}$	Kelvin inside the logarithm
$\displaystyle \eta_{\text{Carnot}} = 1 - \frac{T_L}{T_H}$ ; $COP_{HP} = COP_R + 1$	Kelvin
Otto $\displaystyle \eta = 1 - \frac{1}{r^{\gamma - 1}}$ ; Brayton $\displaystyle \eta = 1 - \frac{1}{r_p^{(\gamma - 1)/\gamma}}$	$r$ is a volume ratio, $r_p$ a pressure ratio
$\displaystyle C_xH_y + \left(x + \frac{y}{4}\right)(O_2 + 3.76N_2)$ ; $\displaystyle \phi = \frac{(A/F)_{\text{stoich}}}{(A/F)_{\text{actual}}}$	New in 2027; $\phi < 1$ is lean
$\displaystyle COP = \frac{h_1 - h_4}{h_2 - h_1}$ , with $h_4 = h_3$	Throttle is isenthalpic
$\displaystyle \omega = 0.622\frac{p_v}{p - p_v}$	Use total pressure $p$
Turbine $\displaystyle N_s = \frac{N\sqrt{P}}{H^{5/4}}$ ; pump $\displaystyle N_s = \frac{N\sqrt{Q}}{H^{3/4}}$	Pump characteristics are new in 2027

Trap: a turbine's isentropic efficiency is actual over ideal work, but a compressor's or pump's is ideal over actual. Inverting one gives a wrong answer that still looks plausible.

Materials, Manufacturing and Metrology

Formula	Watch out for
Packing: BCC 0.68, FCC and HCP 0.74	BCC $a = 4r/\sqrt{3}$ , FCC $a = 2\sqrt{2}r$
$\sigma_t = \sigma(1 + e)$ , $\varepsilon_t = \ln(1 + e)$	Valid up to necking only
Chvorinov $\displaystyle t_s = C\left(\frac{V}{A}\right)^2$	Riser must freeze after the casting
Merchant $2\phi + \beta - \alpha = 90^\circ$ ; $\displaystyle \tan\phi = \frac{r\cos\alpha}{1 - r\sin\alpha}$	$r = t/t_c$ is below 1
Taylor $VT^n = C$ ; least-cost life $\displaystyle T = \left(\frac{1}{n} - 1\right)\left(t_c + \frac{C_t}{C_m}\right)$	Maximum production drops the $C_t/C_m$ term
Peak-to-valley roughness $\displaystyle \frac{f^2}{8r}$	$r$ is the nose radius
Tolerance unit $i = 0.45D^{1/3} + 0.001D$	$D$ is a geometric mean, in mm
CNC step $\displaystyle = \frac{\text{pitch} \times \text{step angle}}{360^\circ}$ ; cylinder force $pA$ out, $p(A - a)$ in	New in 2027

Industrial Engineering

Formula	Watch out for
$\displaystyle Q^* = \sqrt{\frac{2DS}{H}}$ ; EPQ $\displaystyle Q^* = \sqrt{\frac{2DS}{H(1 - d/p)}}$	At $Q^*$ ordering cost equals holding cost
Standard time $=$ normal time $\times (1 + \text{allowance})$	Work study is new in 2027
$F_{t+1} = \alpha D_t + (1 - \alpha)F_t$	Larger $\alpha$ responds faster
M/M/1: $\displaystyle L = \frac{\lambda}{\mu - \lambda}$ , $\displaystyle W = \frac{1}{\mu - \lambda}$ ; $L = \lambda W$	Needs $\lambda < \mu$
PERT $\displaystyle t_e = \frac{a + 4m + b}{6}$ , $\displaystyle \sigma = \frac{b - a}{6}$	Add variances, not standard deviations
Series $R = \prod R_i$ ; parallel $R = 1 - \prod(1 - R_i)$ ; $R(t) = e^{-\lambda t}$	Constant failure rate only
$\displaystyle \text{DPMO} = \frac{\text{defects} \times 10^6}{\text{units} \times \text{opportunities}}$	New in 2027

General Aptitude essentials

Average speed is total distance over total time; for two equal distances it is $\displaystyle \frac{2v_1v_2}{v_1 + v_2}$
The minute hand gains $5.5^\circ$ a minute on the hour hand
Change of base: $\displaystyle \log_{a^k} x = \frac{1}{k}\log_a x$
$\max(0, P(A) + P(B) - 1) \le P(A \cap B) \le \min(P(A), P(B))$
$\displaystyle \binom{n}{r} = \frac{n!}{r!\,(n - r)!}$ ; a set of $n$ elements has $2^n$ subsets

The General Aptitude preparation guide covers the verbal and data interpretation side.

How to use a formula sheet in the exam

Read the sheet daily in the final two weeks. For each line, say its condition aloud before you use it. In the exam, write your five most fragile formulas on the rough sheet in the first five minutes; the exam-day rules guide explains what you may and may not carry.

Carry full precision on the GATE virtual calculator and round only the final NAT answer. Negative marks apply to MCQs only, as the marking scheme guide, written for CSE but common to every paper, explains.

In one line: a formula is worth a mark only when you know its units, its condition and its usual trap.

Subject-wise formula sheets

Each of these goes deeper on one part of the GATE ME paper, with a worked example and the trap for every formula:

Quick revision

Convert rpm to rad/s with $\displaystyle \omega = \frac{2\pi N}{60}$ ; MPa is $\text{N/mm}^2$ .
Use kelvin inside any temperature ratio, power or exponential.
Use absolute pressure in the ideal-gas law, isentropic ratios and choking.
$\displaystyle I = \frac{\pi d^4}{64}$ for bending, $\displaystyle J = \frac{\pi d^4}{32}$ for torsion.
Hoop stress $\displaystyle \frac{pd}{2t}$ is twice longitudinal stress $\displaystyle \frac{pd}{4t}$ .
Isolation needs $r > \sqrt{2}$ ; resonance amplitude is $\displaystyle \frac{F_0/k}{2\zeta}$ .
Lumped analysis only when $Bi < 0.1$ .
EOQ $\sqrt{2DS/H}$ ; M/M/1 needs $\lambda < \mu$ .

Frequently asked questions

Can I take a formula sheet into the GATE ME exam?

No. Candidates cannot carry paper, a calculator, a phone or a watch into the exam hall. The exam provides a virtual calculator, rough sheets and a pen. A useful habit is to write a few easily forgotten formulas on the rough sheet in the first five minutes. Confirm the current exam-day rules at gate2027.iitm.ac.in before the exam.

What are the hoop and longitudinal stresses in a thin cylinder?

For a thin cylinder of diameter $d$ , wall thickness $t$ and internal pressure $p$ , the hoop stress is $\displaystyle \frac{pd}{2t}$ and the longitudinal stress is $\displaystyle \frac{pd}{4t}$ , half the hoop value. A thin sphere carries $\displaystyle \frac{pd}{4t}$ in every direction. The results hold only for a thin wall, with $t$ below about $d/20$ .

When does a spring mounting isolate vibration?

Only when the frequency ratio $r = \omega/\omega_n$ is greater than $\sqrt{2}$ . Below that, the transmissibility ratio is above 1 and the mount passes on more force than it receives. At $r = \sqrt{2}$ it equals 1 for every damping ratio. Above $\sqrt{2}$ , extra damping raises the transmitted force, so isolators use soft springs and light damping.

Which GATE ME 2027 topics are new and need formulas?

The 2027 syllabus names several topics for the first time. They include velocity potential and converging-diverging nozzles, transmissibility ratio, PID controllers and transfer functions, belt drives, boiling and condensation, combustion with air-fuel and equivalence ratio, pump characteristics, additive manufacturing, CNC and actuators, programmable logic controllers, work study and six sigma. Each has at least one formula or rule worth knowing.

How should I revise formulas before GATE ME?

Read a compact formula sheet once a day in the final two weeks and once more on the exam morning. For each formula, know the condition it needs and the trap built around it. Write the unit next to every number, use kelvin inside any temperature ratio, carry full precision and round only the final numerical answer.

Sources

Dates, fees and the syllabus are set by the GATE 2027 organising institute and can change. Always confirm at gate2027.iitm.ac.in.

GATE ME Formula Sheet 2027: Key Formulas by Subject

Key takeaways

How to read this sheet

Engineering Mathematics

Engineering Mechanics

Mechanics of Materials

Theory of Machines and Vibrations

Machine Design

Fluid Mechanics

Heat Transfer

Thermodynamics and Applications

Materials, Manufacturing and Metrology

Industrial Engineering

General Aptitude essentials

How to use a formula sheet in the exam

Subject-wise formula sheets

Quick revision

Frequently asked questions

Sources

Keep reading