https://engineers.academy/product/l3-nd-engineering-principles-exam-preparation/ Level 3 Engineering Principles - Dynamic Systems Equations Sheet

1. LEVEL 3 ENGINEERING PRINCIPLES – 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 of Motion
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 for Rotation
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 Equation Variables 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