50 Essential Mechanical Engineering Formulas

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Strength of materials (1–10)
Thermodynamics (11–20)
Fluid mechanics (21–30)
Manufacturing (31–40)
Machine design and heat transfer (41–50)

This is the quick-reference sheet: fifty formulas that cover most hand calculations in mechanical design, from a stress check on a bracket to the power of a pump. Each table gives the formula, what it is used for and the SI units of the result. The full guides linked under each table explain every symbol, list the assumptions and work through an example.

Free PDF: 50 essential formulasSix A4 pages: every formula with a sketch, its symbols, SI units and assumptions.Download PDF

One shaft, many formulas. The motor's torque T and speed N set the power, the gear load bends the shaft between its bearings, the torque twists it, and the bearings carry the reactions.

Before you use any of these

Use consistent units: base SI (N, m, Pa, J, W, K), or N, mm and MPa for stresses. Use absolute temperature (K) and absolute pressure in gas formulas. Each formula has assumptions; the full guides list them.

Strength of materials (1–10)

#	Formula	Used for	Result in
1	$\sigma = F/A$	Tensile or compressive stress in a rod, bolt or column	Pa
2	$\sigma = E\varepsilon$ , $\varepsilon = \delta/L$	Hooke’s law: linking stress and strain	Pa
3	$\delta = FL/AE$	Stretch or shortening of a bar	m
4	$\tau = F/nA$	Shear in pins, rivets and keys	Pa
5	$\sigma = E\alpha\Delta T$	Stress in a part that cannot expand	Pa
6	$\sigma_b = My/I$	Bending stress in beams and brackets	Pa
7	$\tau = Tr/J$	Shear stress in a twisted shaft	Pa
8	$\theta = TL/GJ$	Angle of twist of a shaft	rad
9	$\delta = FL^3/48EI$	Deflection of a simply supported beam, central load	m
10	$P_{cr} = \pi^2EI/(KL)^2$	Buckling load of a slender column	N

Full explanations and examples: strength of materials formulas.

Thermodynamics (11–20)

#	Formula	Used for	Result in
11	$pV = mRT$	Mass, pressure or temperature of a gas	—
12	$Q - W = \Delta U$	First law for a closed system	J
13	$Q = mc\Delta T$	Heat to warm or cool a part or a fluid	J
14	$W = p(V_2 - V_1)$	Work at constant pressure	J
15	$W = mRT\ln(V_2/V_1)$	Work in an isothermal process	J
16	$T_2/T_1 = (p_2/p_1)^{(\gamma-1)/\gamma}$	Temperature after adiabatic compression	K
17	$\dot Q - \dot W = \dot m(\Delta h + \Delta V^2/2 + g\Delta z)$	Energy balance on turbines, compressors, nozzles	W
18	$\eta = 1 - T_L/T_H$	Carnot (maximum) efficiency	—
19	$COP_R = Q_L/W$	Performance of a refrigerator or heat pump	—
20	$\Delta S = mc\ln(T_2/T_1)$	Entropy change of a solid or liquid	J/K

Full explanations and examples: thermodynamics formulas.

Fluid mechanics (21–30)

#	Formula	Used for	Result in
21	$p = \rho g h$	Pressure at a depth in a liquid	Pa
22	$F_2 = F_1 A_2/A_1$	Force from a hydraulic press or cylinder	N
23	$F_B = \rho_f g V$	Buoyant force	N
24	$Q = A_1V_1 = A_2V_2$	Continuity: velocity change in a duct	m³/s
25	$p + \tfrac{1}{2}\rho V^2 + \rho g z = \text{const}$	Bernoulli: pressure and velocity along a flow	Pa
26	$Re = \rho VD/\mu$	Laminar or turbulent flow	—
27	$h_f = f(L/D)(V^2/2g)$	Friction loss in a duct	m
28	$f = 64/Re$	Friction factor in laminar flow	—
29	$P = \rho g Q H/\eta$	Pump power	W
30	$F_D = \tfrac{1}{2}\rho V^2 C_D A$	Drag force on a body	N

Full explanations and examples: fluid mechanics formulas.

Manufacturing (31–40)

#	Formula	Used for	Result in
31	$N = 1000V/\pi D$	Spindle speed from cutting speed	rev/min
32	$t = L/fN$	Turning time per pass	min
33	$MRR = 1000Vfd$	Material removal rate in turning	mm³/min
34	$v_f = f_z z N$	Table feed in milling	mm/min
35	$MRR = a_e a_p v_f$	Material removal rate in milling	mm³/min
36	$t = (L + 0.3D + c)/fN$	Drilling time	min
37	$P = F_c V/60$ , $F_c = k_c f d$	Cutting force and power	W
38	$F = L t \tau_s$	Blanking and punching force	N
39	$BA = \theta(\pi/180)(R + Kt)$	Sheet-metal bend allowance	mm
40	$VT^n = C$	Taylor tool life	min

Full explanations and examples: manufacturing calculations.

Machine design and heat transfer (41–50)

#	Formula	Used for	Result in
41	$P = 2\pi N T/60$	Power from torque and speed	W
42	$i = N_1/N_2 = z_2/z_1$	Gear ratio	—
43	$N_1 D_1 = N_2 D_2$	Speed ratio of a belt drive	—
44	$F = kx$	Spring force	N
45	$\tau = 8FDK_w/\pi d^3$	Shear stress in a helical spring	Pa
46	$\sigma = F/A_t$	Tensile stress in a bolt (on the stress area)	Pa
47	$L_{10} = (C/P)^3$	Rating life of a ball bearing	10⁶ rev
48	$\dot Q = kA\Delta T/L$	Heat conducted through a wall	W
49	$\dot Q = hA\Delta T$	Heat carried away by convection	W
50	$\eta = 1 - 1/r^{\gamma - 1}$	Ideal efficiency of a petrol (Otto) engine	—

These are explained with diagrams and examples in how gears work, bearings and shafts, belt and chain drives, springs and clutches, keys, couplings and fasteners, heat transfer fundamentals and Otto and Diesel cycles.

Worked example: one shaft, four formulas

A motor drives a 30 mm solid steel shaft at 1450 rev/min and delivers 5.5 kW. Find the torque, the shaft’s shear stress and the speed of the output gear if the 20-tooth pinion drives a 60-tooth gear.

T = \frac{60P}{2\pi N} = \frac{60 \times 5500}{2\pi \times 1450} = 36.2\ \text{N·m}

\tau = \frac{16T}{\pi d^3} = \frac{16 \times 36.2}{\pi \times 0.030^3} = 6.8\ \text{MPa}

N_2 = N_1\,\frac{z_1}{z_2} = 1450 \times \frac{20}{60} = 483\ \text{rev/min}, \qquad T_2 \approx 3 \times 36.2 = 109\ \text{N·m}

AnswerT = 36.2 N·m, τ = 6.8 MPa, output 483 rev/min at about 109 N·m (before gear losses)

The low shear stress shows that this shaft’s diameter is set by stiffness, keyways and bearing sizes, not by strength, which is common for small drive shafts.

Key points

Formulas 1–10 check whether a part is strong and stiff enough.
Formulas 11–30 cover energy, heat and flow in engines, pumps and ducts.
Formulas 31–40 plan machining and sheet-metal work.
Formulas 41–50 size drives, springs, bolts and bearings and estimate heat flow.

50 essential mechanical engineering formulas with units and applications

Strength of materials (1–10)

Thermodynamics (11–20)

Fluid mechanics (21–30)

Manufacturing (31–40)

Machine design and heat transfer (41–50)

Read next

Strength of materials formulas with units and examples

Thermodynamics formulas with units and examples

Fluid mechanics formulas with units and examples

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