Engineering Mechanics formula sheet: printable

Engineering Mechanics-1 Newton's Second Law

F = m a

Force equals mass times acceleration.

Engineering Mechanics-2 Weight (W = mg)

W = m g

Weight of a mass at Earth's surface, using standard gravity g = 9.80665 m/s² (exact by definition).

Engineering Mechanics-3 Pressure (P = F/A)

P = \frac{F}{A}

Pressure as perpendicular force spread over an area.

Engineering Mechanics-4 Torque

\tau = r F \sin\theta

Turning effect of a force applied at distance r from a pivot, at angle θ to the lever arm.

Engineering Mechanics-5 Weight Component Along an Incline (mg sin θ)

F_{\parallel} = m g \sin\theta

Component of an object's weight acting down the slope of an incline: the force that drives it…

Engineering Mechanics-6 Normal Force on an Incline (N = mg cos θ)

N = m g \cos\theta

Normal force pressing a resting mass into an incline, equal to the component of its weight…

Engineering Mechanics-7 Maximum Static Friction (f = μₛN)

f_{s,\max} = \mu_s N

Largest static friction force available before an object breaks loose and slides, from the static…

Engineering Mechanics-8 Kinetic Friction Force (f = μₖN)

f_k = \mu_k N

Friction force resisting a sliding surface, equal to the coefficient of kinetic friction times the…

Engineering Mechanics-9 Angle of Repose (μ = tan θ)

\mu_s = \tan\theta

Steepest angle a surface can be tilted before an object slides, where the coefficient of static…

Engineering Mechanics-10 Torque with a Lever Arm (τ = rF sin θ)

\tau = r F \sin\theta

Torque produced by a force applied at distance r from the pivot, at angle θ to the lever.

Engineering Mechanics-11 Mechanical Advantage of a Lever

MA = \frac{d_e}{d_l}

Mechanical advantage of a lever as the ratio of effort arm to load arm, showing how much the lever…

Engineering Mechanics-12 Pulley System Effort Force

F = \frac{W}{n}

Effort force needed to lift a load with a pulley system, divided down by the number of rope…

Engineering Mechanics-13 Machine Efficiency

\eta = \frac{W_{out}}{W_{in}}

Efficiency of a machine as useful work out divided by work in, with the shortfall lost to friction,…

Engineering Mechanics-14 Support Reaction: Simple Beam, Off-Centre Point Load

R_A = \frac{P (L - a)}{L}

Reaction at the near support of a simply supported beam with one point load at distance a from it.

Engineering Mechanics-15 Max Moment: Simple Beam, Off-Centre Point Load

M = \frac{P a (L - a)}{L}

Maximum bending moment in a simply supported beam with a single point load at distance a from one…

Engineering Mechanics-16 Support Reaction: Simple Beam, Uniform Load (R = wL/2)

R = \frac{w L}{2}

Reaction at each support of a simply supported beam under a uniformly distributed load: half the…

Engineering Mechanics-17 Beam Reaction by Moment Equilibrium: Two Point Loads

R_A = \frac{P_1 (L - a_1) + P_2 (L - a_2)}{L}

Reaction at support A of a simply supported beam carrying two point loads, from ΣM = 0 taken about…

Engineering Mechanics-18 Final Velocity (Uniform Acceleration)

v = v_0 + a t

Final velocity after accelerating uniformly from an initial velocity for a given time.

Engineering Mechanics-19 Displacement (Uniform Acceleration)

d = v_0 t + \tfrac{1}{2} a t^2

Distance travelled under constant acceleration, starting from an initial velocity, over a time t.

Engineering Mechanics-20 Velocity-Displacement Relation (v² = v₀² + 2ad)

v^2 = v_0^2 + 2 a d

Links initial and final speeds to acceleration and displacement without involving time.

Engineering Mechanics-21 Acceleration Down a Frictionless Incline

a = g \sin\theta

Acceleration of an object sliding freely down a frictionless incline, set only by gravity and the…

Engineering Mechanics-22 Acceleration Down an Incline with Friction

a = g\left(\sin\theta - \mu_k \cos\theta\right)

Acceleration of an object sliding down an incline once kinetic friction opposes the motion, from…

Engineering Mechanics-23 Rope Tension When Lifting a Mass

T = m\left(g + a\right)

Tension in a rope lifting a mass with an upward acceleration, exceeding the static weight by the…

Engineering Mechanics-24 Atwood Machine Acceleration

a = \frac{\left(m_1 - m_2\right) g}{m_1 + m_2}

Acceleration of an Atwood machine: two masses joined by a rope over a frictionless pulley, driven…

Engineering Mechanics-25 Drag Force (F = ½CdρAv²)

F_D = \tfrac{1}{2} C_d \rho A v^{2}

Aerodynamic drag on a body moving through a fluid, growing with the square of speed and with…

Engineering Mechanics-26 Terminal Velocity

v_t = \sqrt{\frac{2 m g}{\rho A C_d}}

Steady falling speed at which drag balances weight, from mass, air density, frontal area, and the…

Engineering Mechanics-27 Work (W = Fd cos θ)

W = F d \cos\theta

Work done by a constant force acting at an angle to the displacement.

Engineering Mechanics-28 Kinetic Energy

E_k = \tfrac{1}{2} m v^{2}

Energy of a mass in motion.

Engineering Mechanics-29 Gravitational Potential Energy (U = mgh)

U = m g h

Energy stored by raising a mass to height h near Earth's surface, with g = 9.80665 m/s².

Engineering Mechanics-30 Work–Energy Theorem

W = \tfrac{1}{2} m \left(v^{2} - v_0^{2}\right)

Net work done on an object equals its change in kinetic energy, linking force and distance to a…

Engineering Mechanics-31 Hooke's Law

F = k x

Restoring force of an ideal spring, proportional to its displacement from rest.

Engineering Mechanics-32 Elastic Potential Energy

U = \tfrac{1}{2} k x^{2}

Energy stored in an ideal spring displaced x from its rest length.

Engineering Mechanics-33 Power (P = W/t)

P = \frac{W}{t}

Average power as work or energy delivered per unit time.

Engineering Mechanics-34 Power from Force and Velocity (P = Fv)

P = F v

Instantaneous power delivered by a force parallel to the velocity.

Engineering Mechanics-35 Linear Momentum (p = mv)

p = m v

Momentum as the product of an object's mass and velocity.

Engineering Mechanics-36 Impulse (J = FΔt)

J = F \, \Delta t

Impulse delivered by an average force acting over a contact time, equal to the change in momentum.

Engineering Mechanics-37 Conservation of Momentum (Two Bodies)

m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2

Conservation of linear momentum in a two-body collision, solving any one mass or velocity from the…

Engineering Mechanics-38 Perfectly Inelastic Collision

v = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}

Common velocity of two bodies that stick together after a perfectly inelastic collision, from…

Engineering Mechanics-39 Coefficient of Restitution

e = \frac{v_2 - v_1}{u_1 - u_2}

Ratio of separation speed to approach speed in a collision, measuring how much of the relative…

Engineering Mechanics-40 Angular Velocity (ω = θ/t)

\omega = \frac{\theta}{t}

Average angular velocity: the angle swept divided by the time taken.

Engineering Mechanics-41 Angular Acceleration

\alpha = \frac{\omega - \omega_0}{t}

Average angular acceleration: the change in angular velocity divided by the time taken.

Engineering Mechanics-42 Linear Speed from Rotation (v = ωr)

v = \omega r

A point at radius r on a rotating body moves with linear speed ωr.

Engineering Mechanics-43 Angular Velocity from Period

\omega = \frac{2\pi}{T}

One full revolution is 2π radians, so angular velocity is 2π divided by the period.

Engineering Mechanics-44 Speed in Circular Motion (v = 2πr/T)

v = \frac{2\pi r}{T}

Speed of an object in uniform circular motion: one circumference (2πr, with π ≈ 3.14159265) per…

Engineering Mechanics-45 Newton's Second Law for Rotation (τ = Iα)

\tau = I \alpha

Net torque equals moment of inertia times angular acceleration: F = ma for spinning things.

Engineering Mechanics-46 Moment of Inertia: Solid Disk

I = \tfrac{1}{2} m r^{2}

Rotational inertia of a uniform solid disk or cylinder about its central axis.

Engineering Mechanics-47 Moment of Inertia: Point Mass

I = m r^{2}

Rotational inertia of a compact mass circling at radius r from the axis.

Engineering Mechanics-48 Centripetal Acceleration (a = v²/r)

a_c = \frac{v^2}{r}

Inward acceleration of an object moving in a circle at constant speed.

Engineering Mechanics-49 Centripetal Force (F = mv²/r)

F_c = \frac{m v^2}{r}

Net inward force required to keep a mass moving in a circle at constant speed.

Engineering Mechanics-50 Centripetal Acceleration (a = ω²r)

a_c = \omega^{2} r

Centripetal acceleration written in terms of angular velocity rather than linear speed.

Engineering Mechanics-51 Rotational Kinetic Energy

KE_{rot} = \tfrac{1}{2} I \omega^{2}

Kinetic energy stored in rotation: half the moment of inertia times angular velocity squared.

Engineering Mechanics-52 Angular Momentum (L = Iω)

L = I \omega

Angular momentum of a rotating body: moment of inertia times angular velocity.

Engineering Mechanics-53 Rotational Power (P = τω)

P = \tau \omega

Mechanical power delivered by a torque turning at angular velocity ω.

Engineering Mechanics-54 Shaft Torque from Power and Angular Speed

T = \frac{P}{\omega}

The torque in a rotating shaft: transmitted power divided by angular speed, with ω in radians per…

Engineering Mechanics-55 Undamped Natural Frequency

f_n = \frac{1}{2\pi} \sqrt{\frac{k}{m}}

The frequency a mass on a spring returns to when nothing is driving it: stiffness over mass,…

Engineering Mechanics-56 Natural Frequency from Static Deflection

f_n = \frac{1}{2\pi} \sqrt{\frac{g}{\delta_{st}}}

The natural frequency read off the amount the mount sinks under the weight it carries: no spring…