ABSTRACT
Formula Formula symbols Units
Stress = applied force cross-sectional area
σ = F A
Pa
Strain = change in length original length
ε = x L
Young’s modulus of elasticity = stress strain
E = σ ε
Pa
Stiffness = force extension
k = F δ
N/m
Modulus of rigidity = shear stress shear strain
G = τ γ
Pa
Thermal strain = coefficient of linear expansion × temperature rise ε = αT
Thermal stress in compound bar σ1 = (α1 − α2)E1E2A2T (A1E1 + A2E2) Pa
Ultimate tensile strength = maximum load original cross-sectional area
Pa
Moment = force × perpendicular distance M = Fd N m stress
distance from neutral axis = bending moment
second moment of area
= Young’s modulus radius of curvature
σ
y = M
I = E
R N/m3
Torque = force × perpendicular distance T = Fd N m Power = torque × angular velocity P = T ω = 2πnT W Horsepower 1 hp = 745.7 W Torque = moment of inertia × angular acceleration T = Iα N m shear stress
radius = torque
polar second moment of area
= (rigidity)(angle of twist) length
τ
r = T
J = Gθ
L N/m3
Average velocity = distance travelled time taken
v = s t
m/s
Acceleration = change in velocity time taken
a = v − u t
Linear velocity v = ωr m/s (Continued )
Formula Formula symbols Units
Angular velocity ω = θ t
= 2πn rad/s Linear acceleration a = rα m/s2
Relationships between initial velocity u, final velocity v, displacement s, time t and constant acceleration a
v2 = v1 + at s = ut + 12at2 v2 = u2 + 2as
m/s m
Relationships between initial angular velocity ω1, final angular velocity ω2, angle θ , time t and angular acceleration a
ω2 = ω1 + αt θ = ω1t + 12αt2 ω22 = ω21 + 2αθ
rad/s rad
Momentum = mass × velocity kg m/s Impulse = applied force× time = change in momentum kg m/s Force = mass × acceleration F = ma N Weight = mass × gravitational field W = mg N Centripetal acceleration a = v
r m/s2
Centripetal force F = mv 2
r N
Density = mass volume
ρ = m V
Work done = force × distance moved W = Fs J Efficiency = useful output energy
input energy
Power = energy used (or work done) time taken
= force × velocity P = E
t = Fv W
Potential energy = weight × change in height Ep = mgh J kinetic energy = 12 × mass × (speed)2 Ek = 12 mv2 J kinetic energy of rotation = 12 × moment of inertia × (angular velocity)2 Ek = 12Iω2
J
Frictional force = coefficient of friction × normal force F = μN N Angle of repose, θ , on an inclined plane tan θ = μ Efficiency of screw jack η = tan θ
tan(λ + θ) SHM periodic time T = 2π
√ displacement acceleration
T = 2π √
y
a s
T = 2π √
mass
stiffness T = 2π
√ m
k s
simple pendulum T = 2π √
L
g s
Formula Formula symbols Units
compound pendulum T = 2π √
(k2G + h2) gh
s
Force ratio = load effort
Movement ratio = distance moved by effort distance moved by load
Efficiency = force ratio movement ratio
Kelvin temperature = degrees Celsius + 273 Quantity of heat energy = mass×specific heat capacity × change in temperature
Q = mc(t2 − t1)
New length = original length + expansion L2 = L1[1 + α(t2 − t1)] m New surface area = original surface area + increase in A2 = A1[1 + β(t2 − t1)] m2
area
New volume = original volume + increase in volume V2 = V1[1 + γ (t2 − t1)] m3
Pressure = force area
p = F A
Pa
= density × gravitational acceleration × height p = ρgh Pa 1 bar = 105Pa
Absolute pressure = gauge pressure + atmospheric pressure
Metacentric height, GM GM = Px W
cot θ m
Bernoulli’s equation P1
ρ + v
+ gz1 = P2 ρ
+ v 2 2 2
+ g(z2 + hf ) Coefficient of discharge Cd = Cv × Cc Characteristic gas equation
T1 = p2V2
T2 = k
pV = mRT
In Figure F1, shaded area = R 2
2 (α − sinα) a
Shape Position of axis Second moment of area, I
Radius of gyration, k
Rectangle length d breadth b
(1) Coinciding with b
(2) Coinciding with d
(3) Through centroid, parallel to b
(4) Through centroid, parallel to d
d√ 3 b√ 3
d√ 12
b√ 12
Triangle Perpendicular height h base b
(1) Coinciding with b
(2) Through centroid, parallel to base
(3) Through vertex, parallel to base
h√ 6 h√ 18
h√ 2
Circle radius r diameter d
(1) Through centre, perpendicular to plane (i.e. polar axis)
(2) Coinciding with diameter
(3) About a tangent
2 or
32 πr4
4 or
4 or
r√ 2 r
2 √ 5 2
r
Semicircle radius r
Coinciding with diameter πr4
8 r
