LEVEL 3 ENGINEERING PRINCIPLES – DYNAMIC SYSTEMS EQUATIONS Linear Equations of Motion Subject
Equation
Variables and Units
Displacement
đ?’” = đ?’—đ?’•
s = displacement in meters (m)
(đ?’– + đ?’—) đ?’• đ?&#x;?
đ?’”=
đ?&#x;? đ?’” = đ?’–đ?’• + đ?’‚đ?’•đ?&#x;? đ?&#x;? Velocity
đ?’—=
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)
đ?’” đ?’•
đ?’— = đ?’– + đ?’‚đ?’• Velocity 2
đ?’—đ?&#x;? = đ?’–đ?&#x;? + đ?&#x;?đ?’‚đ?’”
Acceleration
đ?’‚=
đ?’—−đ?’– đ?’•
Newton’s Second Law Equation Subject Sum of Forces Acting on Object
Equation đ?‘ = đ?’Žđ?’‚,
đ?’˜đ?’‰đ?’†đ?’? đ?’‚ ≠đ?&#x;Ž
Variables and Units F = force in Newtons (N) m = mass in kilograms (kg) a = acceleration in meters per second squared (m/s2)
Angular Equations of Motion Subject
Equation
Variables and Units
Angular Displacement
θ = ωt
θ = angular displacement in radians (rads)
1 θ = đ?œ” t + Îąt 2 Angular Velocity
ω = đ?œ” + Îąt
Angular Velocity 2
ω = đ?œ” + 2ιθ
Angular Acceleration
Îą=
ω = final angular velocity in radians per second (rads/s) ωo = initial angular velocity in radians per second (rads/s) ι = angular acceleration in radians per second squared (rad/s2) t = time in seconds (s)
Ď‰âˆ’đ?‘¤ t
Moment of Inertia Equations Shape
Equation
Variables and Units
Solid Cylinder (about polar axis)
1 I = MR 2
I = moment of inertia in kilogram meters squared (kgm2) M = mass in kilograms (kg)
Hoop (about polar axis)
I = MR
Hollow Cylinder (about polar axis)
1 I = M(R − r ) 2
Pin Ended Rod (about end)
1 I = ML 3
R = outside radius in meters (m) r = inside radius in meters (m) L = length in meters (m)
Newton’s Laws for Rotation Subject
Equation
Torque
T = IÎą
đ?‘Ž =đ?œ” đ?‘&#x; Centrifugal / Centripetal Acceleration
đ?‘Ž =
đ?‘Ł đ?‘&#x;
đ??š = đ?‘šđ?œ” đ?‘&#x; Centrifugal / Centripetal Acceleration
đ??š =
Variables and Units T = torque in Newton meters (Nm) I = moment of inertia in kilogram meters squared (kgm2) ι = 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)
đ?‘šđ?‘Ł đ?‘&#x;
Momentum Equations Subject
Equation
Variables and Units
Momentum
đ?‘´ = đ?’Žđ?’—
M = momentum in kilogram meters per second (kgm/s)
Momentum (Conservation of Momentum)
đ?’Žđ?’– = đ?’Žđ?’—
m = mass in kilograms (kg)
đ?’Žđ?&#x;? đ?’–đ?&#x;? + đ?’Žđ?&#x;? đ?’–đ?&#x;? = đ?’Žđ?&#x;? đ?’—đ?&#x;? + đ?’Žđ?&#x;? đ?’—đ?&#x;? when body 1 and 2 collide before moving off in separate directions đ?’Žđ?&#x;? đ?’–đ?&#x;? + đ?’Žđ?&#x;? đ?’–đ?&#x;? = đ?’Ž(đ?&#x;?
đ?&#x;?) đ?’—(đ?&#x;? đ?&#x;?)
when body 1 and 2 collide before moving off together (coupled)
v = final velocity in meters per second (m/s) u = initial velocity in meters per second (m/s)
Energy Equations Subject
Equation
Potential Energy
đ?‘Źđ?‘ˇ = đ?’Žđ?’ˆđ?’‰
Kinetic Energy
đ?&#x;? đ?‘Źđ?‘˛ = đ?’Žđ?’—đ?&#x;? đ?&#x;?
Work
đ?‘ž = đ?‘đ?’…
Power
� �= � � �= �
Variables and Units 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)
Angular Energy Equations Subject Kinetic Energy
Work
Equation
E =
1 Iω 2
W = Tθ
Variables and Units 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)
Power
P = Tω
θ = angular displacement (rads) P = power in Watts (W)
W P= t
t = time in seconds (s)