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Level 3 engineering principles dynamic systems equations sheet
- 1. LEVEL 3 ENGINEERINGPRINCIPLES – DYNAMIC SYSTEMS EQUATIONS Linear Equations of Motion Subject Equation Variables and Units Displacement 𝒔 = 𝒗𝒕 s = displacement in meters (m) v = final velocity in meters per second (m/s) t = time in seconds (s) u = initial velocity in meters per second (m/s) a = uniform acceleration in meters per second squared (m/s2) 𝒔 = (𝒖 + 𝒗) 𝟐 𝒕 𝒔 = 𝒖𝒕 + 𝟏 𝟐 𝒂𝒕 𝟐 Velocity 𝒗 = 𝒔 𝒕 𝒗 = 𝒖 + 𝒂𝒕 Velocity 2 𝒗 𝟐 = 𝒖 𝟐 + 𝟐𝒂𝒔 Acceleration 𝒂 = 𝒗 − 𝒖 𝒕 Newton’s Second Law Equation Subject Equation Variables and Units Sum of Forces Acting on Object 𝑭 = 𝒎𝒂, 𝒘𝒉𝒆𝒏 𝒂 ≠ 𝟎 F = force in Newtons (N) m = mass in kilograms (kg) a = acceleration in meters per second squared (m/s2)
- 2. Angular Equations ofMotion Subject Equation Variables and Units Angular Displacement θ = ωt θ = angular displacement in radians (rads) ω = final angular velocity in radians per second (rads/s) ωo = initial angular velocity in radians per second (rads/s) α = angular acceleration in radians per second squared (rad/s2) t = time in seconds (s) θ = 𝜔 t + 1 2 αt Angular Velocity ω = 𝜔 + αt Angular Velocity 2 ω = 𝜔 + 2αθ Angular Acceleration α = ω − 𝑤 t Moment of Inertia Equations Shape Equation Variables and Units Solid Cylinder (about polar axis) I = 1 2 MR I = moment of inertia in kilogram meters squared (kgm2) M = mass in kilograms (kg) R = outside radius in meters (m) r = inside radius in meters (m) L = length in meters (m) Hoop (about polar axis) I = MR Hollow Cylinder (about polar axis) I = 1 2 M(R − r ) Pin Ended Rod (about end) I = 1 3 ML
- 3. Newton’s Laws forRotation Subject Equation Variables and Units Torque T = Iα T = torque in Newton meters (Nm) I = moment of inertia in kilogram meters squared (kgm2 ) α = angular acceleration in radians per second squared (rad/s2) ac = centrifugal / centripetal acceleration in meters per second squared (m/s2) ω = angular velocity in radians per second (rads/s) r = radius of motion in meters (m) v = instantaneous linear velocity in meters per second (m/s) Fc = centrifugal / centripetal force in meters per second squared (m/s2) m = mass in kilograms (kg) Centrifugal / Centripetal Acceleration 𝑎 = 𝜔 𝑟 𝑎 = 𝑣 𝑟 Centrifugal / Centripetal Acceleration 𝐹 = 𝑚𝜔 𝑟 𝐹 = 𝑚𝑣 𝑟 Momentum Equations Subject Equation Variables and Units Momentum 𝑴 = 𝒎𝒗 M = momentum in kilogram meters per second (kgm/s) m = mass in kilograms (kg) v = final velocity in meters per second (m/s) u = initial velocity in meters per second (m/s) Momentum (Conservation of Momentum) 𝒎𝒖 = 𝒎𝒗 𝒎 𝟏 𝒖 𝟏 + 𝒎 𝟐 𝒖 𝟐 = 𝒎 𝟏 𝒗 𝟏 + 𝒎 𝟐 𝒗 𝟐 when body 1 and 2 collide before moving off in separate directions 𝒎 𝟏 𝒖 𝟏 + 𝒎 𝟐 𝒖 𝟐 = 𝒎(𝟏 𝟐) 𝒗(𝟏 𝟐) when body 1 and 2 collide before moving off together (coupled)
- 4. Energy Equations Subject EquationVariables and Units Potential Energy 𝑬 𝑷 = 𝒎𝒈𝒉 EP = potential energy in Joules (J) EK = kinetic energy in Joules (J) m = mass in kilograms (kg) g = gravitational acceleration in meters per second squared (m/s2) h = height in meters (m) v = velocity in meters per second (m/s) W = work (energy) in Joules (J) F = force in Newtons (N) d = distance in meters (m) P = power in Watts (W) t = time in seconds (s) Kinetic Energy 𝑬 𝑲 = 𝟏 𝟐 𝒎𝒗 𝟐 Work 𝑾 = 𝑭𝒅 Power 𝑷 = 𝑾 𝒕 𝑷 = 𝑬 𝒕 Angular Energy Equations Subject Equation Variables and Units Kinetic Energy E = 1 2 Iω EK = kinetic energy in Joules (J) I = moment of inertia in kilogram meters squared (kgm2) ω = angular velocity (rads/s) W = work in joules (J) T = torque in Newton meters (Nm) θ = angular displacement (rads) P = power in Watts (W) t = time in seconds (s) Work W = Tθ Power P = Tω P = W t