a = (v_f - v_i) / t, where v_f is final velocity, v_i is initial velocity, and t is time.

Meters per second squared (m/s²).

Yes, negative acceleration indicates slowing down or deceleration.

Yes, it helps determine vehicle acceleration or braking rates.

Yes, it can compute acceleration due to gravity for objects in free fall.

Acceleration cannot be calculated with zero time; division by zero is undefined.

Yes, conversion options are available for both unit systems.

Yes, ideal for physics education, automotive, and dynamics analysis.

Yes, it finds average acceleration over a given time interval.

It estimates the velocity of an arrow based on bow draw weight, draw length, and arrow mass.

By estimating kinetic energy from bow efficiency and converting it to velocity using arrow mass.

Feet per second (fps) or meters per second (m/s).

Yes, heavier arrows travel slower but carry more energy.

Efficiency determines how much of the stored energy converts into arrow motion.

Yes, it works for compound, recurve, and traditional bows.

String type, brace height, draw length, and environmental conditions.

It’s recommended to verify actual speed and compare with calculator predictions.

Yes, understanding arrow speed helps in tuning and predicting arrow flight.

Yes, ideal for optimizing hunting setups and target shooting performance.

A dimensionless number that describes a projectile’s aerodynamic efficiency, combining sectional density and drag form factor.

BC = Sectional Density (SD) ÷ form factor (i).

SD = mass ÷ diameter²; relates mass to cross-sectional area.

A coefficient representing drag relative to a standard projectile shape.

Units cancel out so BC is dimensionless; mass and diameter must be consistent.

It helps predict bullet drop, wind drift, and retained velocity over distance.

Yes, if unit-conversion is built in before computing BC.

Higher BC (e.g. 0.5-1.0+) indicates better aerodynamic efficiency.

No, BC is fixed for a bullet shape and mass but may differ by drag model (G1, G7).

Yes, it helps reloaders evaluate bullet performance and ballistic trajectories.

It estimates how far a car will travel when launched from a ramp using projectile motion formulas.

You need to input car speed, ramp angle, and height from which the car jumps.

It uses physics equations for projectile motion to compute time, range, and trajectory.

Yes, it’s ideal for stunt planning, racing simulations, and vehicle dynamics analysis.

No, the basic model assumes no air resistance for simplicity.

Use consistent units — for example, meters for distance and meters per second for speed.

Yes, many versions also compute vertical velocity at impact.

It’s theoretically accurate, but real conditions like drag and tire friction affect results.

Yes, it’s perfect for learning projectile motion and real-world applications.

Yes, it’s optimized for mobile and desktop use.

The total momentum of an isolated system remains constant during interactions (no external impulse).

Masses and velocities (scalars for 1D or vector components for 2D), and collision type (elastic/inelastic).

Momentum p = m × v (vector form p = m·v⃗).

Yes — enter velocity components (vx, vy) and the tool solves momentum per axis.

Only in elastic collisions; inelastic collisions dissipate kinetic energy but conserve momentum.

A number (0≤e≤1) that quantifies relative speed after/before along the impact line; e=1 elastic, e=0 perfectly inelastic.

Yes — for two-body problems it returns final velocities using conservation laws and e if provided.

Yes — it shows intermediate steps, checks conservation, and explains energy loss for inelastic cases.

Use consistent units (SI: kg for mass, m/s for velocity) to get momentum in kg·m/s.

The basic conservation applies only if external impulses are negligible; otherwise include external impulse terms.

Density is mass per unit volume, expressed as ρ = m/V.

Divide an object's mass by its volume using the formula ρ = m/V.

The SI unit is kilograms per cubic meter (kg/m³).

Yes, g/cm³ is commonly used for solids and liquids.

Yes, rearrange the formula: mass = density × volume.

Yes, volume = mass ÷ density.

At 4°C, water’s density is approximately 1000 kg/m³ or 1 g/cm³.

Temperature and pressure can change a material’s density, especially gases.

Yes, if you know the gas mass and occupied volume.

No, density measures compactness; weight depends on gravity.

Displacement is the vector that shows the shortest path between initial and final positions.

Subtract initial coordinates from final coordinates: Δx = x_f - x_i, Δy = y_f - y_i.

Distance is total path length; displacement is straight-line vector from start to end.

Yes, it computes magnitude and direction of the displacement vector.

Yes, simply enter x coordinates; the displacement is Δx = x_f - x_i.

Meters, centimeters, feet, or inches; ensure consistency between initial and final positions.

For 2D, θ = arctangent(Δy / Δx) gives angle from x-axis.

Yes, negative values indicate direction along the chosen coordinate axis.

Displacement is a vector; magnitude is scalar.

Yes, it helps students understand vectors, motion, and kinematics concepts.

It calculates range, height, and flight time of an object launched at an angle.

Initial velocity, launch angle, and gravity are the main required inputs.

No, it assumes ideal projectile motion without air resistance.

Some versions include visual plots of the projectile’s path.

Because it provides the optimal balance between vertical and horizontal velocity.

Use meters, seconds, and meters per second for consistent results.

Yes, it can estimate the speed when the projectile hits the ground.

Yes, it helps approximate bullet or object trajectories in ideal conditions.

Yes, it’s designed for easy educational and practical use.

Yes, it’s fully responsive and mobile-friendly.

It calculates system efficiency, entropy change, and energy conversion between heat reservoirs.

You need the temperatures of the hot and cold reservoirs, usually in Kelvin.

It uses the Carnot efficiency formula: η = 1 - (Tc/Th).

Yes, but it’s internally converted to Kelvin for accurate results.

Yes, it can calculate entropy change based on energy transfer values.

Yes, it helps analyze the coefficient of performance (COP) in cooling systems.

It provides ideal (theoretical) efficiency; actual systems are lower.

It’s the maximum theoretical efficiency achievable between two temperatures.

Engineers, physics students, and researchers studying thermodynamic cycles.

Yes, it’s optimized for both desktop and mobile devices.

It calculates the modulus of rigidity, showing how resistant a material is to shear deformation.

Shear stress (τ) and shear strain (γ).

G = τ / γ, where G is the shear modulus.

It’s measured in pascals (Pa) or gigapascals (GPa).

Yes, it’s widely used to test materials like steel, aluminum, and copper.

Yes, torsional stress-strain values can be used to compute G.

Yes, shear strain (γ) is dimensionless, representing deformation ratio.

Yes, higher temperatures usually reduce G in most materials.

Yes, it’s perfect for lab experiments and engineering coursework.

Yes, it works smoothly on both desktop and mobile devices.

It calculates parameters like displacement, velocity, acceleration, and time for oscillating systems.

Amplitude, angular frequency, phase angle, and time are needed.

It uses x = A sin(ωt + φ) and related SHM equations.

Yes, it computes both using derivatives of the displacement function.

No, it assumes ideal SHM without damping or friction.

Yes, it works for small-angle simple pendulum motion.

Displacement in meters, time in seconds, and angular frequency in rad/s.

Yes, it’s perfect for studying periodic and oscillatory systems.

Some versions include visual plots of displacement vs time.

Yes, it works smoothly on both desktop and mobile devices.

It calculates material stress using the applied force and cross-sectional area.

Stress (σ) = Force (F) / Area (A).

Stress is measured in pascals (Pa) or newtons per square meter (N/m²).

Yes, it works for both tensile and compressive load cases.

Yes, if you input shear force and area parallel to the applied load.

Mechanical engineers, civil engineers, and physics students.

Applied force in newtons and area in square meters.

Some versions include automatic unit conversion (e.g., mm² to m²).

Yes, it provides ideal stress values based on accurate input data.

Yes, it’s responsive and easy to use on both desktop and mobile.

It is the increase in size of a material when its temperature rises.

ΔL = α × L₀ × ΔT is used to find the change in length.

α is the coefficient of thermal expansion for a specific material.

Yes, it supports linear, area, and volumetric expansion calculations.

You can use metric or imperial units like meters, inches, or °C.

Yes, a negative ΔT results in contraction instead of expansion.

It provides accurate results if correct coefficients and units are entered.

Yes, engineers use it to predict material movement due to heat.

Common material α values are available in engineering handbooks.

Yes, it’s a free online tool for students and professionals.

It states that the work done on an object equals the change in its kinetic energy.

Work = Force × Distance × cos(θ).

Kinetic energy is energy an object has due to its motion, given by ½mv².

It’s stored energy due to position, calculated as mgh.

Both are measured in Joules (J).

Yes, when the force and motion are in opposite directions.

Force, mass, displacement, and direction influence it.

Energy is conserved in isolated systems with no external loss.

Yes, power can be derived as Work divided by Time.

Yes, it’s designed for physics learning and engineering use.

It is the ratio of stress to strain that defines a material’s stiffness or elasticity.

E = Stress / Strain.

Pascals (Pa), Megapascals (MPa), Gigapascals (GPa), or psi.

Stress is the force applied per unit area, measured in Pascals.

Strain is the ratio of change in length to the original length, a dimensionless quantity.

It indicates the material is stiffer and less likely to deform under stress.

Steel, diamond, and tungsten have high modulus values.

Yes, it works for both tensile and compressive stress conditions.

Yes, they are often used interchangeably in engineering.

Yes, it provides precise and professional results for academic or industrial use.

It is the velocity of one object as observed from another moving object.

V_AB = V_A – V_B for 1D, and √(V_A² + V_B² – 2V_A V_B cosθ) for 2D.

Yes, negative relative velocity indicates motion in opposite directions.

Meters per second (m/s), kilometers per hour (km/h), miles per hour (mph).

Yes, especially for 2D motion to determine the vector direction of relative velocity.

Yes, you can enter the angle between two velocities for 2D calculations.

Physics students, engineers, and anyone analyzing relative motion.

Yes, it can calculate relative velocity between cars, boats, planes, etc.

Yes, if instantaneous velocities of the objects are provided.

Yes, it’s a free online tool for students and professionals.

Velocity is the rate of change of position with respect to time and has both magnitude and direction.

Velocity = Distance / Time or v = d / t.

Meters per second (m/s), kilometers per hour (km/h), or miles per hour (mph).

No, velocity includes direction; speed is only magnitude.

Yes, negative velocity indicates motion in the opposite direction.

Distance traveled and time taken are needed.

Yes, by inputting distance over a very small time interval.

Physics students, engineers, mechanics, and drivers.

Yes, it can convert between m/s, km/h, and mph.

Yes, it’s a free online velocity calculator.

Torque is the rotational force that causes an object to rotate around an axis.

Torque = Force × Distance from pivot point.

Newton-meter (N·m), pound-foot (lb·ft), and kilogram-centimeter (kg·cm).

It is the perpendicular distance between the pivot and the force line.

Yes, depending on the rotation direction (clockwise or counterclockwise).

No, torque measures rotational force; power measures work per time.

Yes, use motor torque and speed to calculate power output.

Force magnitude and lever arm length determine torque size.

Yes, maximum torque occurs when the force is perpendicular to the lever.

Yes, it’s a free online tool for physics and engineering calculations.

Provide any two or three known kinematic variables from initial velocity (u), final velocity (v), acceleration (a), time (t), or displacement (s); select units and compute.

Yes — treat acceleration as g (9.8 m/s² downward) and enter initial velocity/time/displacement as required; you can change g if needed.

The tool switches to constant-velocity formulas (v=u, s=ut) and will flag division-by-zero cases if you attempt formulas needing nonzero a.

Quadratic kinematic equations can yield two mathematical roots; the calculator shows both and explains physical validity (positive time, context).

Yes — inputs are converted to SI for calculation and then displayed in the units you selected; you can change output units in settings.

Every result includes step-by-step algebraic substitution and intermediate calculations to help learning and verification.

Yes — copy results, download a CSV of the inputs/outputs, or export a printable summary with the worked steps.

Calculations use double-precision arithmetic; displayed values are rounded sensibly but full precision is preserved in exports.

Basic version focuses on linear kinematics; use the converter option to translate angular velocity/acceleration to linear equivalents for radius-based problems.

Yes — it offers both quick numeric answers for engineers and step-by-step derivations for students, plus unit controls and export features.

Uniform motion occurs when an object moves at a constant speed in a straight line without acceleration.

It uses the basic equation s = v × t, where s is distance, v is speed, and t is time.

Yes, time = distance ÷ speed. Just input your values and the calculator finds it instantly.

Yes, it automatically converts between metric and imperial units like km/h, m/s, and mph.

Negative values are invalid; the calculator will prompt you to enter positive quantities.

Yes, for constant velocity motion, the average speed equals the constant speed itself.

Yes, uniform motion implies constant velocity in both magnitude and direction.

No, it only handles linear uniform motion, not rotational or angular systems.

Yes, results can be copied or printed for reports or academic records.

Students, teachers, and engineers can use it for quick, accurate motion calculations.

Non-uniform motion occurs when an object's velocity changes with time due to variable acceleration or direction.

It uses integration and differentiation of velocity-time or acceleration-time data to find distance and acceleration.

In uniform motion, speed is constant. In non-uniform motion, speed or direction changes over time.

Yes, you can input discrete time-velocity or time-acceleration data points for numerical computation.

The calculator uses trapezoidal or Simpson’s numerical methods for integration and differentiation.

Yes, positive acceleration increases speed, while negative acceleration (deceleration) reduces it.

Yes, it can plot velocity-time and acceleration-time graphs for clearer understanding.

Yes, the tool supports automatic conversion between metric and imperial units.

Yes, you can copy, print, or download results in CSV format for analysis or reports.

Students, teachers, and engineers analyzing variable motion in physics or mechanics can use it effectively.

Projectile motion occurs when an object is launched into the air and moves under gravity along a curved path.

It uses standard projectile motion equations to find time of flight, range, and maximum height.

Yes, you can input an initial height to get more accurate results for elevated launches.

The calculator uses g = 9.81 m/s² by default, but it can be changed for other planets or conditions.

Yes, when the angle is 0°, it computes motion as a horizontal projectile.

Yes, it displays both Vx = u cos θ and Vy = u sin θ components.

No, this calculator assumes ideal conditions without air resistance.

Inputs and outputs can be in m/s, km/h, or ft/s for velocity and meters or feet for distance.

Yes, results can be copied or printed for physics lab reports.

Students, teachers, and engineers studying physics, ballistics, or sports trajectories can all use it effectively.

Angular velocity (ω) measures rotational speed — the rate of change of angle with time, usually in radians per second.

Use ω (rad/s) = RPM × 2π / 60.

Linear (tangential) speed v = ω × r, where r is radius.

ω = 2π × f, where f is frequency in hertz (cycles per second).

Period T = 1 / f.

Yes — it converts between rad/s and deg/s (1 rad/s ≈ 57.2958 deg/s).

Angular acceleration (α) is the rate of change of angular velocity: α = Δω / Δt.

RPM (revolutions per minute) measures rotational speed but must be converted to rad/s to get angular velocity.

Yes — useful for motors, gears, wheels, turbines, and any rotating machinery when you need conversions or linear speed from rotation.

Yes — it preserves sign for direction; positive/negative ω indicates rotation sense depending on your sign convention.

Angular acceleration (α) is the rate of change of angular velocity over time, usually in rad/s².

Use α = Δω / Δt, where Δω is (ω_final − ω_initial) and Δt is the time interval.

Yes — for rigid bodies α = τ / I, where τ is net torque and I is moment of inertia.

Common units include rad/s², deg/s², and RPM/s (revolutions per minute per second); the tool converts automatically.

If constant angular deceleration α is known, stopping time t = −ω₀ / α (use negative α for deceleration).

Yes — using θ = ω₀t + ½αt² for constant α; it can compute θ when ω₀, α, and t are known.

Δt = 0 is invalid (division by zero). Negative Δt is treated as reversed time direction; the calculator will warn and ask for valid positive intervals.

Yes — RPM and RPM/s are converted to rad/s and rad/s² internally for calculations.

Yes — sign indicates direction; positive/negative α depends on your rotation sign convention (speeding up vs slowing down).

Mechanical engineers, robotics designers, physics students, and lab users analyzing rotational acceleration and torque systems will find it useful.

Centripetal force is the inward force required to keep an object moving in a circular path; mathematically F_c = m v^2 / r.

Use a_c = v^2 / r or a_c = ω^2 r depending on whether you have linear or angular speed.

Yes — RPM converts to angular velocity ω via ω = 2π·RPM / 60 and is used in F_c = mω^2 r.

Rearrange the formula: r = m v^2 / F_c to solve for radius.

Use SI units for accuracy: kilograms (kg), meters (m), seconds (s) or provide other units — the tool auto-converts.

No — this calculates ideal centripetal force only. Effects like friction, banking angle, or air resistance are outside its scope.

High force often comes from high speed and small radius (F∝v²/r). Check inputs and unit consistency; the tool flags unrealistic magnitudes.

Yes — for an object on a string in horizontal circular motion, tension ≈ centripetal force if no other vertical forces act.

The calculator reports magnitude of centripetal force; direction is always inward toward the circle center (normal to velocity).

Students, lab technicians, mechanical engineers, and anyone designing rotating systems or analyzing circular motion can use it.

Simple harmonic motion (SHM) is periodic oscillatory motion where acceleration is directly proportional to displacement and opposite in direction.

Use x = A sin(ωt + φ), where A is amplitude, ω is angular frequency, and φ is phase angle.

For a spring system, ω = √(k/m); for a pendulum, ω = √(g/L).

The time period is T = 2π√(m/k) for a spring or T = 2π√(L/g) for a pendulum.

Velocity is v = ω√(A² - x²), the rate of change of displacement.

Acceleration is a = -ω²x, directed toward the equilibrium position.

Total energy E = ½kA² remains constant; it converts between potential and kinetic forms.

Yes, select the pendulum mode to calculate period and frequency using length and gravity.

For ideal SHM, frequency depends only on system parameters (k, m, or L, g), not amplitude.

Students, physicists, and engineers studying vibration, acoustics, or mechanical resonance use SHM calculators.

Newton’s second law states that force equals mass multiplied by acceleration: F = m·a.

Use F = m·a; multiply the mass (m) by acceleration (a).

Use a = F/m; divide the force by the mass.

Use m = F/a; divide the force by the acceleration.

Force in N, kN, or lbf; mass in kg; acceleration in m/s² or g.

Yes, it is ideal for physics labs and homework exercises.

Yes, negative values indicate deceleration or force opposite to motion.

Yes, the calculator shows formulas and substitution steps.

Yes, the tool automatically converts units like N↔kN and m/s²↔g.

Students, teachers, engineers, and physics enthusiasts can all use it.

The work-energy principle states that the net work done on an object equals its change in kinetic energy: W = ΔKE.

KE = ½ m v², where m is mass and v is velocity.

PE = m g h, where m is mass, g is gravity, and h is height.

W = F · d · cosθ, where F is force, d is displacement, and θ is the angle between them.

Yes — TE = KE + PE for ideal systems with no energy loss.

This calculator handles constant force; variable forces require integration.

Mass in kg, distance in m, force in N, energy in J, acceleration in m/s².

Yes — using W = ΔKE → v = √(2W/m + v_i²) for initial velocity v_i.

Yes — h = PE / (m g).

Physics students, teachers, engineers, and anyone studying motion or mechanical energy can use it.

Momentum is a measure of motion, p = m·v, where m is mass and v is velocity.

Impulse is the product of force and time: J = F·Δt, causing a change in momentum.

Use F = J / Δt.

Use v = p / m.

Yes, Δp = p_final − p_initial or Δp = J from impulse-momentum theorem.

Mass in kg, velocity in m/s, force in N, impulse in N·s.

Yes, m = p / v.

Yes, the calculator shows formulas and substitution steps.

Yes, you can compute momentum change and resulting velocity during collisions.

Students, teachers, and engineers studying linear motion, impulses, or collisions.

Total momentum of a system remains constant in the absence of external forces: p_initial = p_final.

v_f = (m1 v1 + m2 v2) / (m1 + m2) when bodies stick together.

Use both momentum and kinetic energy conservation equations to solve for unknown velocities.

Yes, Δp = p_final − p_initial for each body.

Mass in kg, velocity in m/s, momentum in kg·m/s.

Yes, the calculator is designed for two-body elastic and inelastic collisions.

Yes, if velocities and other mass are known, you can solve for unknown mass.

Yes, formulas and substitution steps are shown.

Physics students, teachers, lab technicians, and engineers analyzing collisions.

Yes, the user selects collision type, and appropriate formulas are applied.

Kinetic energy is the energy of motion, calculated as KE = ½ m v².

Potential energy is the energy of position, calculated as PE = m g h.

TE = KE + PE, the sum of kinetic and potential energy in a system.

Yes, v = √(2 KE / m).

Yes, h = PE / (m g).

Mass in kg, velocity in m/s, height in m, gravity in m/s², energy in Joules (J).

Yes, formulas and calculations are shown step-by-step.

Yes, m = 2 KE / v² for kinetic energy or m = PE / (g h) for potential energy.

Students, teachers, engineers, and physics enthusiasts studying energy and motion.

Yes, it calculates TE = KE + PE to check energy consistency.

Power is the rate at which work is done or energy is transferred, measured in watts (W).

Use P = W / t for work over time or P = F · v for force times velocity.

Use P = V · I or P = I² R or P = V² / R for circuits.

Mechanical power in watts (W), work in joules (J), electrical power in W, voltage in volts, current in amps, resistance in ohms.

Yes, 1 horsepower ≈ 746 W.

Yes, P = W / t.

Yes, formulas and substitution steps are shown.

Students, engineers, technicians, and physics enthusiasts analyzing power and energy.

Yes, P = I² R or P = V² / R is supported for resistive electrical loads.

Yes, between watts, kilowatts, and horsepower, as well as common mechanical and electrical units.

Torque is the rotational equivalent of force, calculated as T = F × r.

Multiply the force applied perpendicular to the lever arm by the distance from the pivot: T = F × r.

F = T / r, where T is torque and r is lever arm distance.

r = T / F, given torque and applied force.

Torque in N·m, kg·m, lbf·ft; force in N, kgf; distance in meters or feet.

Yes, it can compute applied torque using force and lever arm length.

Yes, formulas and calculations are displayed step-by-step.

Students, engineers, mechanics, and anyone studying rotational systems.

Yes, it supports conversions between N·m, kg·m, lbf·ft, and others.

Yes, it is suitable for physics, engineering, and real-world mechanical applications.

Angular momentum L is a measure of rotational motion, L = I × ω.

Multiply the moment of inertia by the angular velocity: L = I × ω.

I = L / ω, given angular momentum and angular velocity.

ω = L / I, given angular momentum and moment of inertia.

Angular momentum in kg·m²/s, moment of inertia in kg·m², angular velocity in rad/s.

Yes, presets for discs, rods, and spheres are included.

Yes, formulas and calculations are displayed step-by-step.

Physics students, teachers, engineers, and mechanics studying rotational motion.

Yes, it can be used to analyze conservation of angular momentum problems.

Yes, unit conversion options are included for angular momentum and inertia.

The gyroscopic effect is the tendency of a spinning object to resist changes in its orientation due to angular momentum.

L = I × ω, where I is moment of inertia and ω is angular velocity.

τ = r × F, where r is lever arm distance and F is perpendicular force applied.

Ω = τ / L, where τ is torque and L is angular momentum.

Moment of inertia in kg·m², angular velocity in rad/s, torque in N·m, precession rate in rad/s.

Yes, it is suitable for gyroscopes, rotors, flywheels, and spinning machinery.

Yes, formulas and calculations are shown step-by-step.

Students, teachers, engineers, and mechanics studying rotational dynamics.

Yes, unit conversion options are included for torque, angular momentum, and precession rate.

Yes, the calculator can be used for mechanical stabilization and rotational system analysis.

A slider-crank mechanism converts rotational motion of a crank into linear motion of a slider.

Use x = r cos θ2 + √(l² - (r sin θ2)²) with crank length, connecting rod length, and crank angle.

θ3 = arcsin(r sin θ2 / l) gives the angle of the connecting rod relative to the slider.

Yes, if crank angular velocity ω2 is known, v = r ω2 sin θ3 / sin(θ3 - θ2).

Lengths in meters, angles in degrees/radians, velocities in m/s or rad/s.

Yes, it is suitable for engines, pumps, and other slider-crank applications.

Yes, formulas and substitutions are displayed step-by-step.

Mechanical engineering students, teachers, engineers, and designers of crank-slider systems.

Yes, optional plotting of slider displacement over crank rotation is available.

Yes, inverse kinematics can be applied if needed, though this calculator primarily solves forward kinematics.

A quick return mechanism is a crank-slider system where the return stroke occurs faster than the forward stroke, commonly used in shaper machines.

Forward stroke time t_forward = θ_forward / ω, where θ_forward is crank rotation for forward stroke and ω is angular velocity.

Return stroke time t_return = θ_return / ω, where θ_return is crank rotation for return stroke.

Velocity ratio VR = t_forward / t_return, representing how much faster the return stroke is.

Yes, displacement x can be calculated using crank length, stroke length, and crank angles.

Lengths in meters, angles in degrees/radians, time in seconds, velocities in m/s.

Yes, formulas and substitutions are displayed step-by-step.

Mechanical engineering students, educators, engineers, and machinists studying quick return mechanisms.

Yes, slider displacement vs time can be plotted.

Yes, input ω to compute forward and return stroke times and velocity ratio.

A Geneva drive is a mechanical mechanism that converts continuous rotation into intermittent rotary motion using a slotted wheel.

Step angle θ_step = 360° / number of slots (n) on the Geneva wheel.

Driven wheel rotation per crank turn equals the step angle of the Geneva wheel.

ω_driven = ω_crank × θ_step / θ_crank, where θ_crank is the crank rotation.

Angles in degrees or radians, angular velocity in rad/s.

Yes, the calculator supports any number of slots for step angle and rotation computation.

Yes, formulas and substitutions are displayed step-by-step.

Mechanical engineering students, educators, engineers, and designers of Geneva mechanisms.

Yes, optional plotting of wheel rotation vs time is available.

Yes, it can determine the dwell time between intermittent rotations based on crank motion.

The gear ratio is the ratio of the number of teeth on the driven gear to the driver gear or the ratio of input speed to output speed.

Output speed ω_driven = ω_driver / gear ratio (GR).

Output torque T_driven = T_driver × gear ratio (GR).

RPM or rad/s for speed, N·m for torque, number of teeth for gears.

Yes, it can compute gear ratios, speed, and torque for sun, planet, and ring gears.

Yes, all formulas and substitutions are displayed step-by-step.

Mechanical engineers, students, designers, and hobbyists working with gears.

Yes, users can determine teeth numbers for a specific gear ratio.

Yes, output torque is computed based on gear ratio and input torque.

Yes, speed can be converted between RPM and rad/s.

Beam deflection is the displacement of a beam under loads due to bending moments.

Use the moment-curvature relation: d²y/dx² = M(x)/EI, integrating with boundary conditions.

Lengths in meters or mm, forces in N, modulus E in Pa or MPa, angles in radians.

Yes, the calculator computes y_max at midspan or free end depending on beam type.

Yes, slope θ can be calculated at supports or free ends based on boundary conditions.

Yes, simply supported, cantilever, and overhanging beams are supported.

Yes, both point loads and uniform distributed loads can be analyzed.

Yes, all formulas and integration steps are displayed clearly.

Civil and mechanical engineers, students, educators, and designers analyzing beams.

Yes, optional plotting of deflection along the beam length is available.

Bending stress is the internal stress developed in a beam due to bending moments acting on the cross-section.

σ = M × y / I, where M is bending moment, y is distance from neutral axis, and I is moment of inertia.

Depends on beam type and load: for simply supported point load, M_max = P × L / 4; for cantilever point load, M_max = P × L; for UDL, M_max = w × L² / 8.

S = I / y, a measure of the beam's strength to resist bending.

N, m, mm, Pa or MPa for stress, N·m for moment.

Yes, it supports rectangular, circular, and I-section beams.

Yes, formulas and substitutions are displayed step-by-step.

Structural and mechanical engineers, students, educators analyzing beam strength.

Yes, it can compute bending stress for both point and distributed loads.

Yes, optional plotting of bending moment along the beam is available.

A cantilever beam is fixed at one end and free at the other, supporting loads along its length or at the free end.

For a point load at free end: M_max = P × L; for uniform load: M_max = w × L² / 2.

σ_max = M_max × y / I, where M_max is maximum bending moment, y is distance from neutral axis, and I is moment of inertia.

For point load: y_max = P × L³ / (3 × E × I); for UDL: y_max = w × L⁴ / (8 × E × I).

For point load: θ_max = P × L² / (2 × E × I); for UDL: θ_max = w × L³ / (6 × E × I).

Lengths in m or mm, forces in N, modulus E in Pa/MPa, stress in Pa/MPa.

Yes, it supports rectangular, circular, and I-section beams.

Yes, formulas and calculations are displayed step-by-step.

Yes, optional plotting of deflection and bending stress along the beam is available.

Structural and mechanical engineers, students, educators, and designers analyzing cantilever beams.

A simply supported beam rests on two supports and is free to rotate at the ends, commonly used in bridges and floors.

For point load at midspan: M_max = P × L / 4; for uniform load: M_max = w × L² / 8.

σ_max = M_max × y / I, where y is distance from neutral axis, I is moment of inertia.

Point load: y_max = P × L³ / (48 × E × I); Uniform load: y_max = 5 × w × L⁴ / (384 × E × I).

Point load: θ_max = P × L² / (16 × E × I); Uniform load: θ_max = w × L³ / (24 × E × I).

Lengths in meters or mm, forces in N, modulus E in Pa/MPa, stress in Pa/MPa.

Yes, it supports rectangular, circular, and I-section beams.

Yes, formulas and calculations are displayed step-by-step.

Yes, optional plotting along the beam length is available.

Structural and mechanical engineers, students, educators, and designers analyzing simply supported beams.

A truss is a framework of members connected at joints, used in bridges, roofs, and towers.

Use the Method of Joints or Method of Sections based on equilibrium equations.

Apply ΣFx = 0, ΣFy = 0, ΣM = 0 to the entire truss.

Yes, planar trusses including triangular, Pratt, Warren, and custom geometries.

Yes, positive force indicates tension and negative force indicates compression.

Forces in N or kN, lengths in meters or mm.

Yes, it shows equilibrium equations for joints and sections.

Yes, point loads on any joint can be applied.

Civil/structural engineers, mechanical engineers, students, and educators.

Yes, optional color-coded diagram of member forces and supports is available.

Shear force is the internal force along the beam cross-section caused by external loads, causing sliding between sections.

Bending moment is the rotational force that causes the beam to bend under loads.

Use equilibrium equations ΣFy=0 at each section to compute V(x) along the beam.

Bending moment M(x) = ΣM about the section; can be computed using reactions and applied loads.

Yes, simply supported, cantilever, and overhanging beams are supported.

Yes, both point loads and uniform distributed loads are supported.

Yes, optional Shear Force Diagram (SFD) and Bending Moment Diagram (BMD) are generated.

Yes, if beam cross-section and material properties are provided, σ = M × y / I.

Forces in N or kN, lengths in meters or mm, stress in Pa or MPa.

Civil engineers, structural engineers, mechanical engineers, students, and educators analyzing beams.

Axial load is a force applied along the longitudinal axis of a member, causing tension or compression.

Axial stress σ = P / A, where P is axial load and A is cross-sectional area.

Axial strain ε = σ / E, where σ is stress and E is Young’s modulus of the material.

ΔL = PL / (A × E), where P is axial load, L is member length, A is area, E is Young’s modulus.

Forces in N or kN, lengths in meters or mm, stress in Pa or MPa.

Yes, rectangular, circular, or custom cross-sectional areas are supported.

Yes, both tensile and compressive axial loads are supported.

Yes, all formulas and calculations are displayed clearly.

Structural engineers, mechanical engineers, students, and educators analyzing axial members.

Yes, optional safety factor and maximum allowable load can be calculated if material limits are provided.

Torsion is the twisting of a shaft due to an applied torque, producing shear stress and angle of twist along its length.

τ_max = T × r / J, where T is torque, r is outer radius, and J is polar moment of inertia.

θ = T × L / (G × J), where L is shaft length, G is shear modulus, and J is polar moment of inertia.

Yes, it supports solid and hollow circular shafts.

Torque in N·m, lengths in mm or m, stress in Pa or MPa.

J is a geometric property of the shaft cross-section, used to calculate shear stress and angle of twist.

Yes, you can compute the diameter for a given torque and allowable shear stress.

Mechanical and structural engineers, students, and educators designing shafts under torsion.

Yes, formulas and intermediate calculations are shown clearly.

It indicates rotational deformation of the shaft, critical for machinery alignment and performance.

Column buckling is the sideways bending of a slender column under axial compressive load.

Using Euler’s formula: P_cr = π²EI / (KL)², where L is effective length, E is Young’s modulus, I is moment of inertia, and K is end condition factor.

Slenderness ratio λ = L_eff / r, where r is radius of gyration, indicating column susceptibility to buckling.

Pinned-pinned, fixed-fixed, fixed-free (cantilever), pinned-fixed.

σ_cr = P_cr / A, where A is cross-sectional area.

Yes, rectangular, circular, I-section, and custom cross-sections are supported.

Forces in N or kN, lengths in meters or mm, stress in Pa or MPa.

Yes, FS = P_cr / P_applied can be calculated.

Civil engineers, structural engineers, mechanical engineers, students, and educators.

It predicts the critical load at which a slender column will buckle, ensuring safe structural design.

FOS is the ratio of material or structure strength to applied load or stress to ensure safe operation.

FOS = Strength / Applied Load, where strength can be yield or ultimate stress.

Tension, compression, bending, or combined loads.

Yes, both ultimate and yield stresses can be used for FOS calculation.

N, kN, Pa, MPa for loads and stresses.

Typically FOS ≥ 2 is considered safe, but it depends on application and design codes.

Yes, optionally it can recommend increasing strength or reducing load if FOS is low.

Mechanical engineers, civil engineers, structural engineers, students, and educators.

Yes, all formulas and intermediate steps are displayed.

FOS ensures that components or structures can safely withstand applied loads without failure.

A spur gear is a cylindrical gear with straight teeth mounted on parallel shafts for power transmission.

Gear ratio i = z2 / z1, where z1 is pinion teeth and z2 is gear teeth.

Module m = PCD / z, the ratio of pitch circle diameter to number of teeth, representing gear size.

PCD = module × number of teeth, measured in mm.

Center distance a = (PCD1 + PCD2) / 2 for two meshing gears.

Yes, it computes missing teeth, PCD, and center distance for pinion and gear pairs.

All lengths in millimeters (mm).

Yes, the calculator can verify compatibility with standard module sizes.

Mechanical engineers, design engineers, students, and educators designing spur gears.

Gear ratio determines the speed and torque transmitted between shafts, critical for mechanical systems design.

A helical gear has angled teeth that engage gradually, providing smooth, quiet, and efficient torque transmission.

Gear ratio i = z2 / z1, where z1 is pinion teeth and z2 is gear teeth.

Module m = PCD / (z × cos β), representing the size of teeth in a helical gear.

Helix angle β is the angle of teeth with respect to the gear axis, affecting smoothness and load sharing.

PCD = m × z / cos β for helical gears.

Center distance a = (PCD1 + PCD2) / 2 for two meshing gears.

All lengths in millimeters (mm).

Yes, it computes missing teeth, PCD, center distance, and helix angle.

Mechanical engineers, design engineers, students, and educators designing helical gears.

The helix angle ensures smooth engagement, reduces noise, and improves load distribution in helical gears.

A worm gear is a gear system with a worm (screw) meshing with a worm wheel, used for high speed reduction in compact spaces.

Gear ratio i = z2 / number of worm starts, where z2 is worm wheel teeth.

Module m = pitch diameter / number of teeth, representing gear size.

Lead angle λ is the angle of the worm thread relative to the gear axis, affecting smooth engagement and efficiency.

PD = module × number of teeth for the worm wheel or worm.

Center distance a = (PD_worm + PD_wormwheel) / 2.

All lengths in millimeters (mm).

Yes, the calculator supports single-start and multi-start worms.

Mechanical engineers, design engineers, students, and educators designing worm gear systems.

Worm gears allow high speed reduction and torque transmission in a compact design with smooth operation.

A bevel gear is a gear that transmits motion and torque between intersecting shafts, usually at 90° angles.

Gear ratio i = z2 / z1, where z1 is pinion teeth and z2 is gear teeth.

Module m = PCD / z, representing the size of teeth in the bevel gear.

PCD = module × number of teeth.

Center distance a = (PCD1 + PCD2) / 2.

Yes, the calculator can compute gear dimensions for intersecting shafts at various angles.

All lengths in millimeters (mm).

Mechanical engineers, design engineers, students, and educators designing bevel gears.

Yes, all formulas and intermediate steps are displayed.

Bevel gears transmit motion and torque between intersecting shafts efficiently, often at 90° angles.

Bearing load is the force a bearing must support, including radial and axial components.

For ball bearings: P = XFr + YFa, where X and Y depend on bearing type and load ratio.

Radial load (Fr) acts perpendicular to the bearing shaft axis.

Axial load (Fa) acts along the bearing shaft axis.

L10 = (C / P)^p × 10^6 revolutions, where C is dynamic load rating and p depends on bearing type.

Forces in N or kN, dimensions in mm, rotational speed in RPM.

Yes, it supports ball, cylindrical, tapered, and spherical roller bearings.

Mechanical engineers, maintenance engineers, students, and educators working with bearings.

It ensures the bearing is properly sized to avoid premature failure under operating conditions.

Yes, formulas and intermediate steps are displayed for clarity and verification.

Shaft design involves calculating stresses and dimensions to safely transmit torque and bending loads.

Bending stress σb = 32 Mb / (π d^3), where Mb is bending moment and d is shaft diameter.

Torsional stress τ = 16 T / (π d^3), where T is torque and d is shaft diameter.

Combined stress is the resultant stress considering bending, torsion, and axial loads, often using Von Mises criteria.

Use combined stress formula and allowable stress, including factor of safety, to solve for d.

Torque in N·m, forces in N or kN, stress in MPa, diameter in mm, RPM for rotation.

Yes, axial load can be included in combined stress calculations.

Mechanical engineers, design engineers, students, and educators designing rotating shafts.

Proper shaft design prevents failure, ensures reliability, and maintains safety under working loads.

Yes, all formulas and intermediate steps are displayed for clarity and verification.

Thread stress is the stress experienced by the threaded portion of a bolt, screw, or stud under load.

Tensile stress σ = F / At, where F is axial load and At is tensile stress area of the thread.

Shear stress τ = F / As, where As is the shear area of the threaded section.

Bearing stress σb = F / (d × L), where d is bolt diameter and L is thread engagement length.

FOS = allowable stress / applied stress, for tensile, shear, or bearing stress.

Forces in N or kN, dimensions in mm, stresses in MPa.

Yes, it supports coarse, fine, and metric thread pitches.

Mechanical engineers, design engineers, students, and educators designing threaded connections.

Thread stress ensures bolts and screws are safe under applied loads, preventing failure.

Yes, all formulas and intermediate steps are displayed for clarity and verification.

A key is a mechanical element that connects a shaft and hub, transmitting torque and preventing relative rotation.

Shear stress τ = T / (w × L × radius factor), where T is torque, w is key width, L is key length.

Compressive stress σc = T / (h × L × face width), where h is key height and L is key length.

Key width, height, and length are based on shaft diameter, torque, and standard dimension tables.

Yes, it computes dimensions and stresses for rectangular, square, and Woodruff keys.

Dimensions in mm, torque in N·m, stresses in MPa.

Mechanical engineers, design engineers, students, and educators designing shaft-hub assemblies.

Proper key design prevents failure, ensures torque transmission, and maintains mechanical safety.

Yes, the calculator computes maximum torque that the key can safely transmit.

Yes, all formulas and intermediate steps are displayed for clarity and verification.

A coupling is a mechanical device that connects two shafts to transmit torque and accommodate misalignment.

d = √(16 T / (π τ)), where T is torque and τ is allowable shear stress of the shaft material.

Rigid and flexible couplings.

Shear stress τ = T / (polar moment of area × radius factor), depending on shaft and coupling design.

Bending stress σ = M × c / I, where M is bending moment, c is distance from neutral axis, and I is moment of inertia.

Maximum angular, parallel, or axial deviation a flexible coupling can accommodate without failure.

Torque in N·m, dimensions in mm, stress in MPa, shaft speed in RPM.

Mechanical engineers, design engineers, students, and educators designing shaft couplings.

Proper coupling design ensures safe torque transmission, reduces vibration, and prevents shaft or coupling failure.

Yes, all formulas and intermediate steps are displayed for clarity and verification.

A flywheel is a rotating mechanical device that stores kinetic energy to smooth out torque fluctuations.

Mass m = Volume × Material density, where volume is determined from required inertia and diameter.

I = E / (0.5 × ω²), where E is kinetic energy and ω is angular velocity.

Solid, rim, and disc-type flywheels.

Kinetic energy E = 0.5 × I × ω².

Energy in J, mass in kg, torque in N·m, dimensions in mm, angular velocity in rad/s.

Mechanical engineers, design engineers, students, and educators designing rotating machinery.

Proper flywheel design ensures energy storage, torque smoothing, and safe operation of rotating machinery.

Yes, torque T = I × α if angular acceleration is known.

Yes, all formulas and intermediate steps are displayed for clarity and verification.

Stress is the internal resistance offered by a material to an applied force, calculated as force per unit area.

Strain is the deformation per unit length of a material under stress.

Axial stress σ = F / A, where F is applied force and A is cross-sectional area.

Shear stress τ = V / A, where V is the applied shear force and A is cross-sectional area.

Axial strain ε = σ / E, where σ is axial stress and E is Young's modulus of the material.

Deformation δ = ε × L0, where L0 is the original length of the member.

Combined stress considers multiple types of stress (axial, shear) using criteria like Von Mises or principal stress.

Force in N or kN, length in mm or m, stress in MPa or GPa.

Mechanical engineers, civil engineers, design engineers, and students analyzing stress and strain in members.

Yes, formulas and intermediate steps are displayed for clarity and verification.

Poisson's ratio is the negative ratio of lateral (transverse) strain to longitudinal (axial) strain in a material under load.

ν = –ε_lateral / ε_longitudinal, where ε_lateral is transverse strain and ε_longitudinal is axial strain.

Lateral strain is the deformation perpendicular to the applied load.

Longitudinal strain is the deformation along the direction of the applied load.

Strain is unitless (mm/mm or m/m). Stress can be in Pa or N/m².

Yes, ε_lateral = –ν × ε_longitudinal.

Mechanical engineers, civil engineers, design engineers, and students analyzing material deformation.

It helps predict lateral contraction or expansion in materials under axial load, crucial for mechanical and structural design.

Yes, for elastic materials like metals, polymers, and composites within the elastic limit.

Yes, all formulas and intermediate steps are displayed for clarity and verification.

Hooke's Law states that stress in an elastic material is proportional to strain within the elastic limit, σ = E × ε.

Axial stress σ = F / A, where F is applied load and A is cross-sectional area.

Axial strain ε = δ / L0, or ε = σ / E if Young's modulus is known.

Elongation δ = ε × L0, where L0 is the original length of the member.

E = σ / ε, using known stress and strain values.

Force in N or kN, length in mm or m, stress in MPa or GPa.

Mechanical engineers, civil engineers, design engineers, and students analyzing elastic material behavior.

It allows calculation of deformation and stress in materials within the elastic limit, ensuring safe design.

Yes, for metals, polymers, composites, and other materials within the elastic limit.

Yes, all formulas and intermediate steps are displayed for clarity and verification.

Modulus of elasticity (Young's modulus) is the ratio of stress to strain in the linear elastic region of a material.

E = σ / ε, where σ is axial stress and ε is axial strain.

Axial stress σ = F / A, where F is applied load and A is cross-sectional area.

Axial strain ε = δ / L0, where δ is deformation and L0 is original length.

Deformation δ = ε × L0.

Force in N or kN, length in mm or m, stress in MPa or GPa.

Mechanical engineers, civil engineers, design engineers, and students analyzing elastic material behavior.

It helps predict material stiffness and deformation under load, essential for mechanical and structural design.

Yes, for metals, polymers, composites, and other materials within the elastic limit.

Yes, all formulas and intermediate steps are displayed for clarity and verification.

Modulus of rigidity (shear modulus, G) is the ratio of shear stress to shear strain in the elastic region of a material.

G = τ / γ, where τ is shear stress and γ is shear strain.

Shear stress τ = F / A, where F is applied shear force and A is cross-sectional area.

Shear strain γ = θ × (radius / L0) or from measured angular deformation.

θ = γ × (L0 / radius), where L0 is length and radius is shaft radius.

Force in N or kN, length in mm or m, stress in Pa, MPa, or GPa.

Mechanical engineers, civil engineers, design engineers, and students analyzing torsion or shear in materials.

It helps predict material stiffness under shear and torsional loads, crucial for mechanical and structural design.

Yes, for metals, polymers, composites, and other materials within the elastic limit.

Yes, all formulas and intermediate steps are displayed for clarity and verification.

Yield strength is the maximum stress a material can withstand before permanent deformation occurs.

Actual stress σ_actual = F / A, where F is applied load and A is cross-sectional area.

Factor of safety FS = σ_y / σ_actual, where σ_y is material yield strength.

Allowable stress σ_allow = σ_y / FS, where FS is factor of safety.

Force in N or kN, length in mm or m, stress in MPa or GPa.

Mechanical engineers, civil engineers, design engineers, and students analyzing material safety.

It ensures materials can handle applied loads safely without permanent deformation.

Yes, for metals, alloys, and other structural materials within their elastic limit.

Yes, the calculator supports both axial and bending stress calculations.

Yes, all formulas and intermediate steps are displayed for clarity and verification.

Fatigue life is the number of cycles a material or component can withstand under repeated or fluctuating loading before failure.

Alternating stress σ_a = (σ_max – σ_min) / 2.

Mean stress σ_m = (σ_max + σ_min) / 2.

The endurance limit is the maximum stress a material can withstand for infinite cycles without failure.

Force in N or kN, length in mm or m, stress in Pa, MPa, or GPa.

Mechanical engineers, civil engineers, design engineers, and students analyzing cyclic loading.

It predicts component life under repeated stress, preventing unexpected failure in mechanical systems.

Yes, the calculator can include stress concentration factors (K_t) for notches or holes.

S-N curve method, Goodman diagram, and Modified Goodman method are used for calculation.

Yes, all formulas and intermediate steps are displayed for clarity and verification.

Fracture toughness (K_IC) is a material property that indicates resistance to crack propagation.

K = Y × σ × √(πa), where σ is applied stress, a is crack length, and Y is geometry factor.

Critical stress σ_c can be determined from K_IC = Y × σ_c × √(πa).

Critical crack length a_c = (K_IC / (Y × σ))² / π.

Force in N or kN, length in mm or m, stress in Pa, MPa, or GPa.

Mechanical engineers, civil engineers, design engineers, and students analyzing fracture mechanics.

It predicts material resistance to crack growth, preventing catastrophic failure.

Yes, the calculator supports geometry factors (Y) for cracks or notches.

Yes, it can be used for both brittle and ductile materials within elastic limits.

Yes, all formulas and intermediate steps are displayed for clarity and verification.

Creep rate is the rate at which a material deforms over time under constant stress and temperature.

Total creep strain ε_total = ε_0 + ε_creep(t), where ε_0 is initial strain and ε_creep(t) is strain over time.

Force in N or kN, length in mm or m, stress in Pa, MPa, GPa, time in seconds or hours.

Mechanical engineers, civil engineers, design engineers, and students analyzing high-temperature material behavior.

It predicts long-term deformation and prevents failure of components under sustained stress at elevated temperatures.

Material creep constants (A, n, m), initial strain, and temperature-dependent parameters.

Yes, it supports power-law (Norton), exponential, and other common creep models.

Yes, by analyzing total strain and material limits, it can estimate creep failure time.

Yes, all formulas and intermediate steps are displayed for clarity and verification.

Yes, for steel, nickel alloys, titanium alloys, and other materials operating at high temperatures.

Hardness conversion is the process of converting material hardness values from one scale to another, such as Rockwell, Brinell, Vickers, or Shore.

Rockwell (HRA, HRB, HRC), Brinell (BHN), Vickers (HV), and Shore (HS) scales are supported.

The calculator uses empirical formulas and standard tables to convert Rockwell values to Brinell equivalents.

Yes, the calculator provides step-by-step conversion from Vickers to Rockwell hardness.

Yes, some conversions are more accurate when material type (steel, aluminum, etc.) is specified.

Mechanical engineers, metallurgists, material scientists, and students evaluating material hardness.

It ensures accurate comparison and selection of materials for engineering and industrial applications.

Yes, Shore hardness values for plastics and elastomers can be converted to other scales.

Yes, all formulas and intermediate steps are displayed for clarity.

Yes, it helps ensure proper material hardness selection and verification in quality control processes.

It states that energy can neither be created nor destroyed, only transferred as heat or work, causing a change in internal energy.

ΔU = Q – W, where Q is heat added and W is work done by the system.

Work done W is energy transferred by the system due to volume change or mechanical processes.

Heat Q is energy transferred into or out of the system due to temperature difference.

Energy in J or kJ, pressure in Pa, volume in m³, temperature in °C or K.

Mechanical, chemical, and thermal engineers, as well as students analyzing thermodynamic systems.

Yes, it supports isothermal, isobaric, isochoric, and adiabatic processes.

Yes, W = PΔV for processes where volume changes under pressure.

Yes, all formulas and intermediate steps are displayed for clarity.

It ensures accurate energy balance calculations for engines, compressors, turbines, and other systems.

It states that entropy of an isolated system always increases in irreversible processes, and no heat engine can be 100% efficient.

For a reversible process, ΔS = Q_rev / T, where Q_rev is heat transferred reversibly and T is temperature.

For Carnot engine, η = 1 – T_C / T_H, where T_H and T_C are hot and cold reservoir temperatures.

Irreversibility or lost work is the energy that cannot be converted to work due to entropy generation.

Coefficient of performance (COP) measures the efficiency of a refrigerator or heat pump: COP = Q_C / W.

Energy in J or kJ, temperature in K or °C, pressure in Pa, volume in m³.

Mechanical, chemical, and thermal engineers, as well as students analyzing entropy and efficiency.

Yes, the calculator supports both reversible and irreversible process analysis.

Yes, all formulas and intermediate steps are displayed for clarity.

It ensures accurate energy and entropy analysis for engines, turbines, compressors, and refrigeration systems.

The Carnot cycle is an ideal reversible thermodynamic cycle representing the maximum efficiency a heat engine can achieve.

η = 1 – T_C / T_H, where T_H is hot reservoir temperature and T_C is cold reservoir temperature in Kelvin.

Yes, convert Celsius to Kelvin by adding 273.15 before using the formula.

Work output W = η × Q_H, where Q_H is heat input to the engine.

Heat rejected Q_C = Q_H – W.

Temperature in K or °C, energy in J or kJ.

Mechanical and thermal engineers, students, and researchers analyzing ideal engines.

No, Carnot efficiency assumes an ideal reversible engine without losses.

It represents the maximum possible efficiency any heat engine can achieve between two reservoirs.

Yes, all formulas and intermediate steps are displayed for clarity and verification.

The Rankine cycle is an ideal thermodynamic cycle for steam power plants, consisting of boiler, turbine, condenser, and pump.

η = (W_turbine – W_pump) / Q_in, where W_turbine is turbine work, W_pump is pump work, and Q_in is heat added in the boiler.

Boiler pressure, condenser pressure, turbine inlet temperature, and optional superheat or reheat data.

Yes, the calculator allows superheated or reheated steam to improve turbine work and efficiency.

Pressure in kPa or MPa, temperature in °C or K, energy in kJ/kg or MW.

Mechanical and thermal engineers, power plant engineers, and students analyzing steam cycles.

Yes, step-by-step calculations for turbine work and pump work are provided.

Yes, net work output = W_turbine – W_pump.

It determines how effectively a steam power plant converts heat into mechanical work.

Yes, all formulas and intermediate steps are displayed for clarity and verification.

The Otto cycle is an idealized thermodynamic cycle for spark-ignition engines, representing the working of gasoline engines.

η = 1 – 1 / r^(γ – 1), where r is compression ratio and γ is specific heat ratio of the working gas.

Compression ratio (r) is the ratio of the cylinder volume at bottom dead center to the volume at top dead center.

Specific heat ratio (γ) is the ratio of specific heat at constant pressure (Cp) to specific heat at constant volume (Cv).

Yes, W_net = Q_in – Q_out, using heat added and rejected.

Energy in kJ/kg, temperature in °C or K, compression ratio is dimensionless.

Mechanical engineers, automotive engineers, students, and researchers analyzing engine efficiency.

Yes, all formulas and intermediate steps are displayed for clarity.

It determines how effectively a spark-ignition engine converts heat into mechanical work.

No, it assumes an ideal Otto cycle without friction or other losses.

The Diesel cycle is an idealized thermodynamic cycle for compression-ignition engines, used in Diesel engines.

η = 1 – (1 / r^(γ – 1)) × ((ρ^γ – 1) / (γ (ρ – 1))), where r is compression ratio, ρ is cutoff ratio, and γ is specific heat ratio.

Compression ratio (r) is the ratio of cylinder volume at bottom dead center to top dead center.

Cutoff ratio (ρ) is the ratio of cylinder volume at the end of combustion to the volume at the start of combustion.

Yes, W_net = Q_in – Q_out, using heat added and rejected.

Energy in kJ/kg, temperature in °C or K, ratios are dimensionless.

Mechanical engineers, automotive engineers, students, and researchers analyzing Diesel engine efficiency.

Yes, all formulas and intermediate steps are displayed for clarity.

It determines how effectively a compression-ignition engine converts heat into mechanical work.

No, it assumes an ideal Diesel cycle without friction, heat loss, or other losses.

The Brayton cycle is an ideal thermodynamic cycle for gas turbines, consisting of isentropic compression, constant-pressure heat addition, isentropic expansion, and constant-pressure heat rejection.

η = 1 – 1 / r_p^((γ – 1)/γ), where r_p is the compressor pressure ratio and γ is the specific heat ratio.

The ratio of compressor outlet pressure to inlet pressure in the gas turbine.

Specific heat ratio (γ) is the ratio of specific heat at constant pressure (Cp) to specific heat at constant volume (Cv) for the working gas.

Yes, W_net = W_turbine – W_compressor, using turbine and compressor work.

Energy in kJ/kg, temperature in °C or K, pressure ratio is dimensionless.

Mechanical and aerospace engineers, students, and researchers analyzing gas turbine efficiency.

Yes, all formulas and intermediate steps are displayed for clarity.

It determines how effectively a gas turbine or jet engine converts heat into mechanical work.

No, it assumes an ideal Brayton cycle without friction, heat loss, or other inefficiencies.

Coefficient of Performance (COP) is the ratio of refrigeration effect to work input in a cooling system.

COP = Q_L / W, where Q_L is the refrigeration effect and W is the work input. For ideal Carnot cycle, COP = T_E / (T_C - T_E).

The evaporator temperature (T_E) is the temperature at which the refrigerant absorbs heat from the space to be cooled.

The condenser temperature (T_C) is the temperature at which the refrigerant rejects heat to the surroundings.

Yes, W = Q_L / COP.

Yes, Q_H = Q_L + W.

Temperature in °C or K, energy in kJ or kW.

Mechanical engineers, HVAC engineers, students, and researchers analyzing refrigeration and air conditioning systems.

Yes, all formulas and intermediate steps are displayed for clarity.

COP measures the efficiency of a refrigeration system, indicating how much cooling effect is obtained per unit of work input.

Effectiveness (ε) is the ratio of actual heat transfer to the maximum possible heat transfer in a heat exchanger.

ε = Q_actual / Q_max = Q / (C_min × (T_h,in – T_c,in)), where C_min is the smaller heat capacity rate.

NTU (Number of Transfer Units) is a dimensionless parameter used to determine the effectiveness of a heat exchanger.

C_min is the smaller heat capacity rate and C_max is the larger heat capacity rate of the fluids in the heat exchanger.

Yes, outlet temperatures are computed using the heat transfer rate and heat capacity rates of the fluids.

Counterflow, parallel flow, and crossflow heat exchangers are supported.

Temperature in °C or K, energy in kW, kJ, or MW.

Mechanical engineers, chemical engineers, HVAC engineers, and students analyzing heat exchanger performance.

Yes, all formulas and intermediate steps are displayed for clarity.

Effectiveness measures how efficiently a heat exchanger transfers heat relative to the maximum possible transfer.

Thermal expansion is the increase in size of materials due to temperature rise.

ΔL = α × L_0 × ΔT, where α is the linear expansion coefficient, L_0 is initial length, and ΔT is temperature change.

ΔA = 2α × A_0 × ΔT, where α is the linear expansion coefficient and A_0 is the initial area.

ΔV = 3α × V_0 × ΔT, where α is the linear expansion coefficient and V_0 is the initial volume.

Length in m, cm, mm; temperature in °C or K; volume in m³ or cm³.

Metals like aluminum expand more than steel due to higher thermal expansion coefficients.

Yes, materials contract when temperature decreases, ΔT < 0.

Mechanical, civil, and materials engineers, designers, and students analyzing thermal effects.

Yes, formulas and intermediate steps are displayed for clarity.

It is crucial for designing structures and machinery to prevent stress, deformation, or failure due to temperature changes.

Enthalpy (H) is the total heat content of a system, including internal energy and energy to displace surroundings at constant pressure.

ΔH = m × c_p × ΔT, where m is mass, c_p is specific heat, and ΔT is temperature change.

Yes, Q = ΔH.

Enthalpy in kJ or kJ/kg, temperature in °C or K, mass in kg.

Yes, phase changes involve latent heat, which must be added to calculate ΔH.

Mechanical engineers, chemical engineers, thermodynamics students, and researchers analyzing heat transfer and energy changes.

Yes, all formulas and intermediate steps are displayed for clarity.

Yes, it works for solids, liquids, and gases, including superheated steam.

It helps determine energy changes and heat transfer in thermodynamic systems and processes.

Include latent heat: ΔH = m × L_f (fusion) or L_v (vaporization) in addition to sensible heat.

Flow rate is the volume or mass of fluid passing through a cross-section per unit time.

Q = A × v, where A is the cross-sectional area and v is the fluid velocity.

ṁ = ρ × Q, where ρ is fluid density and Q is volumetric flow rate.

Yes, v = Q / A.

Volumetric flow in m³/s or L/s, velocity in m/s, area in m², mass flow in kg/s.

Yes, the calculator supports both gases and liquids, including air, water, and steam.

Mechanical, civil, chemical engineers, HVAC technicians, and students analyzing fluid flow.

Yes, all formulas and intermediate steps are displayed for clarity.

Flow rate determines the capacity and efficiency of fluid transport systems in engineering applications.

Yes, by providing the cross-sectional area and velocity, it can calculate flow in open channels or conduits.

Force is the product of mass and acceleration acting on an object, measured in Newtons (N).

F = m × a, where m is mass in kg and a is acceleration in m/s².

Weight W = m × g, where g is the acceleration due to gravity (≈9.81 m/s²).

Mass in kg, acceleration in m/s², force in N.

Yes, using the mass and acceleration of the object.

Yes, all formulas and intermediate steps are displayed for clarity.

Physics students, mechanical engineers, civil engineers, and researchers analyzing forces.

Force will be zero if acceleration is zero, regardless of mass.

Force will be zero if mass is zero, regardless of acceleration.

Force determines how objects move, accelerate, or remain in equilibrium in mechanical systems.

Gear ratio is the ratio of the number of teeth of the driven gear to the driving gear.

Gear Ratio = N_driven / N_driver, where N represents the number of teeth.

RPM_out = RPM_in / Gear Ratio.

T_out = T_in × Gear Ratio.

Yes, this calculator supports all common gear types.

Mechanical engineers, automotive designers, and students analyzing gear systems.

Yes, all formulas and intermediate steps are displayed for clarity.

It produces an error; driver teeth must be greater than zero.

Yes, multiply individual gear ratios to find overall ratio.

It determines the speed and torque relationship between input and output shafts in a mechanical system.

Gravitational force is the attractive force between two masses, proportional to their masses and inversely proportional to the square of the distance.

F = G × (m₁ × m₂) / r², where G = 6.674×10⁻¹¹ N·m²/kg².

Mass in kg, distance in meters, gravitational force in Newtons (N).

G = 6.674×10⁻¹¹ N·m²/kg², a universal constant for Newton’s law of gravitation.

Yes, input planetary masses and distance between centers to compute gravitational attraction.

Yes, it calculates gravitational force between any two masses.

Physics students, mechanical engineers, astronomers, and researchers studying gravity.

Yes, formulas and intermediate steps are displayed for clarity.

It produces an error since division by zero is undefined.

It governs planetary motion, satellite orbits, and interactions between all masses in the universe.