Level 3 Engineering Principles - Dynamic Systems Equations Sheet

1. LEVEL 3 ENGINEERING PRINCIPLES – DYNAMIC SYSTEMS EQUATIONS Linear Equations of Motion Subject Displacement Equation ? = ?? ? =(? + ?) Variables and Units 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) ? =? Velocity ? ? = ? + ?? ??= ??+ ??? ? =? − ? Velocity 2 Acceleration ? Newton’s Second Law Equation Subject Sum of Forces Acting on Object Equation Variables and Units F = force in Newtons (N) ?? = ??, ???? ? ≠ ? m = mass in kilograms (kg) a = acceleration in meters per second squared (m/s2)

2. Angular Equations of Motion Subject Angular Displacement Equation θ = ωt Variables and Units θ = angular displacement in radians (rads) ω = final angular velocity in radians per second (rads/s) θ = ??t +1 2αt? ωo = initial angular velocity in radians per second (rads/s) Angular Velocity ω = ??+ αt α = angular acceleration in radians per second squared (rad/s2) ?+ 2αθ t = time in seconds (s) Angular Velocity 2 ω?= ?? α =ω − ?? Angular Acceleration t Moment of Inertia Equations Shape Equation Variables and Units I = moment of inertia in kilogram meters squared (kgm2) I =1 Solid Cylinder (about polar axis) 2MR? M = mass in kilograms (kg) Hoop (about polar axis) I = MR? R = outside radius in meters (m) r = inside radius in meters (m) I =1 Hollow Cylinder (about polar axis) 2M(R?− r?) L = length in meters (m) I =1 Pin Ended Rod (about end) 3ML?

3. Newton’s Laws for Rotation Subject Equation T = Iα Variables and Units 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) Torque ??= ??? Centrifugal / Centripetal Acceleration ??=?? ? ??= ???? Centrifugal / Centripetal Acceleration ??=??? ? Momentum Equations Subject Momentum Equation Variables and Units M = momentum in kilogram meters per second (kgm/s) ? = ?? m = mass in kilograms (kg) ?? = ?? Momentum (Conservation of Momentum) v = final velocity in meters per second (m/s) ????+ ????= ????+ ???? u = initial velocity in meters per second (m/s) when body 1 and 2 collide before moving off in separate directions ????+ ????= ?(???)?(???) when body 1 and 2 collide before moving off together (coupled)

4. Energy Equations Subject Equation Variables and Units EP = potential energy in Joules (J) ??= ??? Potential Energy EK = kinetic energy in Joules (J) m = mass in kilograms (kg) ??= ? Kinetic Energy ???? g = gravitational acceleration in meters per second squared (m/s2) h = height in meters (m) ? = ?? Work Power 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) Angular Energy Equations Subject Kinetic Energy Equation Variables and Units EK = kinetic energy in Joules (J) E?= 1 2Iω? I = moment of inertia in kilogram meters squared (kgm2) ω = angular velocity (rads/s) Work W = Tθ W = work in joules (J) T = torque in Newton meters (Nm) Power P = Tω θ = angular displacement (rads) P = power in Watts (W) t = time in seconds (s) P =W t