1. Purpose and scope
What this reference covers
This article explains basic mechanical quantities and dimensional relationships in original, practical language. It supports understanding and independent checking; it is not a design manual.
Controlling information
Approved engineering calculations, design codes, project specifications, drawings, material data, original equipment manufacturer instructions, inspection requirements, calibrated measurements and competent engineering judgement remain controlling. Unit systems, symbols, tolerancing conventions and acceptance criteria vary by context.
2. Units, dimensions and notation
Quantities, dimensions and units
A quantity describes what is measured; its dimension describes its physical nature; its unit provides the agreed scale. Length has dimension L and may be expressed in metres or millimetres. Area has dimension L², so converting a length unit requires squaring the conversion factor: 1 mm² is 10⁻⁶ m², not 10⁻³ m².
The International System of Units (SI) provides coherent base units. Relevant derived units include the newton (N), pascal (Pa) and watt (W). Prefixes scale units: milli is 10⁻³, kilo is 10³ and mega is 10⁶. Values inserted into an equation must use compatible units.
Open the engineering unit converter.
Notation and checking
Scientific notation keeps very large or small values readable. Significant figures should reflect the input quality; retain internal precision and round the displayed result at the end. A dimensional check can expose an incorrect equation or missed conversion.
- Mass and weight
- Mass is an amount of matter in kilograms; weight is the gravitational force on that mass in newtons.
- Scalar and vector
- A scalar has magnitude only. A vector has magnitude and direction.
- Area and volume
- Area is two-dimensional, measured in square units. Volume is three-dimensional, measured in cubic units.
- Density and specific gravity
- Density is mass per unit volume. Specific gravity is a dimensionless ratio to a defined reference density.
- Absolute, gauge and differential
- An absolute value uses a defined zero reference; a gauge value uses the local reference; a differential value compares two points.
- Value and change
- A value locates a state on a scale. A change is the difference between two states and may require different conversion treatment.
3. Measurement fundamentals
Values and instrument behaviour
- Nominal value
- The identifying or intended value used to describe a feature.
- Measured value
- The value reported from a measurement process.
- Actual size
- The size associated with the physical feature, recognised through measurement and its uncertainty.
- Accuracy
- Closeness of agreement to the relevant reference or true quantity value.
- Precision
- Closeness among repeated indications or measured values under stated conditions.
- Repeatability
- Precision under the same specified measurement conditions over a short interval.
- Resolution
- The smallest displayed or detectable change; it is not a guarantee of accuracy.
- Uncertainty
- A quantified description of the dispersion associated with a measurement result.
- Calibration
- An established relationship between indications and reference values, with associated uncertainties, under specified conditions.
- Traceability
- A property of a result connected to a stated reference through a documented calibration chain, each link contributing uncertainty.
- Range
- The interval over which a measurement system is intended to operate.
- Measurement error
- The measured value minus an appropriate reference value; it is distinct from uncertainty.
Meaningful digits
Additional decimal places increase displayed resolution, not necessarily accuracy, precision or confidence. A measurement decision should consider the method, range, calibration status, traceability and uncertainty appropriate to the required tolerance.
4. Pressure
Force distributed over area
Pressure is normal force distributed over area. Pressure and force are related, but they are not interchangeable because the same force produces a different pressure when the area changes.
Pressure equals force divided by area.
Here p is pressure in pascals, F is normal force in newtons and A is area in square metres. One pascal equals one newton per square metre.
References and fluid head
Absolute pressure uses a zero-pressure reference. Gauge pressure is measured relative to local atmospheric pressure, while differential pressure is the difference between two points. Static pressure is the local thermodynamic pressure considered without adding a directional velocity effect.
Absolute pressure equals gauge pressure plus atmospheric pressure.
In a static fluid of approximately uniform density, a vertical height difference produces a hydrostatic pressure difference:
Pressure difference equals density multiplied by gravitational acceleration and height difference.
ρ is density, g gravitational acceleration and Δh signed vertical height difference. This simplified relationship assumes a static fluid and suitable density treatment. Common units include Pa, kPa and MPa; bar and pounds per square inch (psi) are non-SI units and require explicit conversion, including the squared area unit.
Illustrative calculation
An illustrative 2,000 N normal force distributed over 0.010 m² gives 200,000 Pa, or 200 kPa. These figures demonstrate unit handling; they are not design or acceptance values.
5. Force
Magnitude, direction and equilibrium
Force is a vector. Newton's second-law relationship is F = ma, where m is mass and a acceleration. Multiple forces combine as a vector resultant. A reaction force arises through interaction with a constraint or another body. Static equilibrium requires the vector sum of forces and the sum of moments to be zero.
Weight and moments
Weight is gravitational force: W = mg. Kilograms measure mass; newtons measure force. Kilogram-force is a non-SI force unit defined using standard gravity and must not be confused with kilograms.
An offset force creates a moment about a reference point. Static loading changes slowly enough for inertia to be neglected in the selected model; dynamic loading involves acceleration or time-varying effects and requires suitable analysis.
Illustrative calculation
Using standard gravity, an illustrative mass of 12 kg has weight 12 × 9.80665 = 117.6798 N. The mass remains 12 kg.
6. Flow
Volume, mass and velocity
Volumetric flow rate Qv is volume per unit time. Mass flow rate ṁ is mass per unit time. For average velocity v through cross-sectional area A:
Volumetric flow rate equals cross-sectional area multiplied by average velocity.
Mass flow rate equals density multiplied by volumetric flow rate.
These forms assume that the selected average velocity, area and density represent the same flow condition. Conservation of mass means mass accumulation and all inflows and outflows must balance for the defined control region. In steady flow, the selected properties do not change with time at the observation point; changing flow requires time-dependent treatment.
Units and gallon definitions
Litres per minute, litres per second, cubic metres per hour and cubic metres per second are volumetric rates. Kilograms per second and kilograms per hour are mass rates. A US gallon and an Imperial gallon are different volumes and must be identified explicitly.
Illustrative calculation
An illustrative area of 0.002 m² with an average velocity of 1.5 m/s gives 0.003 m³/s, equal to 3 L/s. This does not determine a required section size or permissible operating condition.
7. Temperature
Temperature values and differences
Temperature describes thermal state. Kelvin is the SI thermodynamic-temperature unit and begins at absolute zero. A Celsius value converts to kelvin by adding 273.15. Fahrenheit uses a different scale and offset.
A temperature difference has no scale offset: a change of 1 K equals a change of 1 °C, while a change of 1 °C equals 1.8 °F. Therefore a temperature reading and a temperature change must not use the same affine conversion blindly.
Open the temperature conversion calculator.
Linear thermal expansion
For a simplified small temperature range:
Change in length equals coefficient of linear thermal expansion multiplied by original length and temperature change.
ΔL is length change, α the coefficient of linear thermal expansion, L₀ original length and ΔT temperature change. The coefficient depends on material condition and temperature range and must come from approved material data.
Open the linear thermal expansion calculator.
Illustrative calculation
Twenty degrees Celsius is 293.15 K. A 10 °C temperature difference is a 10 K or 18 °F difference.
8. Torque
Turning effect of force
Torque is the turning effect of force about an axis. It depends on force magnitude, distance from the axis and the angle between the force and lever arm:
Torque equals force multiplied by lever arm and the sine of the force angle.
τ is torque, F force, r lever-arm length and θ the included angle. Rotation direction requires a stated sign convention. Torque is not force; work additionally requires angular displacement, and power is the rate of doing work.
Open the torque, force and lever arm calculator.
Torque, speed and power
Power equals torque multiplied by angular speed. Angular speed equals two pi multiplied by revolutions per minute divided by sixty.
P is power in watts, ω angular speed in radians per second and n rotational speed in revolutions per minute.
Open the torque, power and rotational speed calculator.
Illustrative calculation
An illustrative perpendicular force of 100 N at 0.25 m gives 25 N·m. At 1,200 r/min, 25 N·m corresponds to approximately 3.142 kW in the ideal equation.
9. Fits and tolerances
Limits and deviations
The basic or nominal size identifies the intended size. Upper and lower deviations are algebraic differences from that basic size. Upper and lower limits are the resulting maximum and minimum permissible sizes. Tolerance is the difference between the limits. A unilateral tolerance lies on one side of the nominal value; a bilateral tolerance extends on both sides.
Allowance is the intentional difference between mating features at maximum-material condition, interpreted under the applicable specification. Limit dimensions must be used for worst-case assessment; a nominal dimension alone does not establish fit.
Fit types and basis systems
A clearance fit always leaves clearance across the specified limits. An interference fit always produces interference. A transition fit can produce either, depending on actual sizes. Hole-basis and shaft-basis systems hold the selected basic feature relationship while the mating tolerance position changes.
ISO 286 defines an ISO code system for tolerances and fits. Applicable classifications, tolerance grades and drawing rules must come from the controlling drawing, specification and recognised standard; no standards table is reproduced here.
Illustrative limits
Illustrative hole limits 20.020–20.040 mm and shaft limits 19.980–20.000 mm give 0.020 mm minimum clearance and 0.060 mm maximum clearance. The figures demonstrate limit arithmetic only and are not recommended manufacturing values.
10. Clearances and interference
Worst-case relationships
minimum clearance = minimum hole size − maximum shaft size
maximum clearance = maximum hole size − minimum shaft size
A negative clearance represents interference. Minimum interference is the magnitude of the interference closest to zero; maximum interference is the greatest overlap. The applicable definition of allowance and maximum-material condition must follow the controlling specification.
Open the worst-case tolerance stack calculator.
Temperature and nominal size
Different materials and temperatures can change both feature sizes and therefore the available clearance. The simplified expansion relationship is useful for sensitivity checks only. Actual assessment requires appropriate material data, temperature distribution, reference temperature and geometry.
Illustrative transition
Illustrative hole limits 30.000–30.020 mm and shaft limits 30.010–30.030 mm give −0.030 mm minimum clearance and +0.010 mm maximum clearance. Because the range crosses zero, the mathematical classification is transition.
11. Alignment fundamentals
Geometrical references
Alignment compares geometry to stated reference axes, centrelines, planes or datums. Offset misalignment is a lateral displacement between references; angular misalignment is a change in direction. Parallelism, perpendicularity, straightness and flatness describe different geometrical characteristics. Coaxiality and eccentricity concern relationships between axes or centres.
Radial runout is observed in a radial direction during rotation about a reference axis; axial runout is observed along the axis. Runout can reveal combined geometrical effects, but it is not interchangeable with alignment.
Angle from offset
Angle equals the inverse tangent of offset divided by reference length.
Reference length materially affects the result. For small angles, the offset ratio in millimetres per metre is numerically close to milliradians, but the exact trigonometric value should be retained when the approximation is not adequate. Static measurements may also require correction for expected dimensional change before comparison.
Illustrative calculation
An illustrative offset of 0.20 mm across 1,000 mm gives an exact angle of approximately 0.01146° or 0.200 mrad. No acceptance tolerance is implied.
12. Practical calculators
Engineering tools
The calculation interfaces are grouped in one dedicated Tools workspace so this page can remain focused on principles, definitions and worked examples.
Open a calculator directly:
- Units and conversion
- Engineering unit converter
- Pressure and force
- Pressure, force and area; force, mass, acceleration and weight
- Flow and temperature
- Flow, area and velocity; temperature conversion; linear thermal expansion
- Torque and power
- Torque, force and lever arm; torque, power and rotational speed
- Fits, tolerances and alignment
- Limits and fit; Worst-case tolerance stack; Alignment angle and offset
13. Formula and symbol reference
Symbols used in this article
| Symbol | Meaning | Coherent SI unit |
|---|---|---|
| p, Δp | Pressure and pressure difference | Pa |
| F, W | Force and weight | N |
| m | Mass | kg |
| a, g | Acceleration and gravitational acceleration | m/s² |
| A | Area | m² |
| Qv | Volumetric flow rate | m³/s |
| ṁ | Mass flow rate | kg/s |
| ρ | Density | kg/m³ |
| v | Average velocity | m/s |
| T, ΔT | Temperature and temperature difference | K |
| α | Coefficient of linear thermal expansion | K⁻¹ |
| L₀, ΔL | Original length and length change | m |
| τ | Torque | N·m |
| P | Power | W |
| ω | Angular speed | rad/s |
| n | Rotational speed | r/min |
| θ | Angle | rad |
Symbol discipline
Qv is used for volumetric flow so it is not confused with heat or energy notation. Symbols must always be read with their definition, sign convention and unit context.
14. Limitations and appropriate use
What the calculators can support
The calculators support education, unit checking, sensitivity checks and transparent arithmetic. They work locally without third-party services, telemetry or network access.
What they cannot approve
They are not suitable as final evidence for engineering design, pressure-boundary verification, lifting calculations, equipment sizing, relief-system design, structural acceptance, bolting specifications, machinery alignment acceptance, material selection, or regulatory or contractual compliance.
The applicable drawing, specification, approved calculation, material data, inspection requirement, calibrated measurement and competent engineering review remain controlling. Simplified equations are valid only when their assumptions match the physical question.