Appendix A — Formulae

A.1 Rules of Cosine and Sine

\[ c^2 = a^2 + b^2 - 2ab\cos \gamma \] \[ \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} \]

A.2 Linear Motion

\[ \vec{v} = \vec{u} + \vec{a}t \]

\[ \vec{s} = \frac{\vec{u}+\vec{v}}{2}t \]

\[ \vec{s} = \vec{u}t + \frac{1}{2}\vec{a}t^2 \]

\[ \vec{v}^2 = \vec{u}^2 + 2\vec{a} \cdot \vec{s} \]

where:

  • \(\vec{u}\): Initial velocity
  • \(\vec{v}\): Final velocity
  • \(\vec{s}\): Displacement
  • \(\vec{a}\): Acceleration
  • \(t\): Time

A.3 Angular Motion

\[ \omega_2 = \omega_1 \mp \alpha t \]

\[ \theta = \frac{\omega_1+\omega_2}{2}t \]

\[ \theta = \omega_1 t \mp \frac{1}{2}\alpha t^2 \]

\[ \omega_2^2 = \omega_1^2 \mp 2\alpha \theta \]

where:

  • \(\omega_1\): Initial angular velocity (rad/s)

  • \(\omega_2\): Final angular velocity (rad/s)

  • \(\theta\) Angular displacement (rad)

  • \(\alpha\): Angular acceleration (rad/s2)

  • \(t\): Time (s)

A.4 Relation Between Linear and Angular Motion

The relationship between linear and angular motion is described by the following equations:

\(s = r \theta\) (linear displacement \(s\) and angular displacement \(\theta\)).

\(v = r \omega\) (linear velocity \(v\) and angular velocity \(\omega\)),

\(a = r \alpha\) (linear acceleration \(a\) and angular acceleration \(\alpha\)).

A.5 Centre of Gravity

\[ \bar{x} = \frac{ \sum Moments\ of\ Weights}{\sum Weights} \qquad \bar{y} = \frac{ \sum Moments\ of\ Weights}{\sum Weights} \]

A.6 Centroid

\[ \bar{x} = \frac{ \sum {\bar{x}_i}\ A_i}{\sum A_i} \qquad \bar{y} = \frac{ \sum {\bar{y}_i}\ A_i}{\sum A_i} \]

A.7 Parallel Axis Theorem

To find the moment of inertia about an axis parallel to the centroidal axis:

\[ I = I_c + A d^2 \]

A.8 Radius of Gyration

\[ k = \sqrt{\frac{I}{A}} \quad \text{(for area)} \quad \text{or} \quad k = \sqrt{\frac{I}{m}} \quad \text{(for mass)}, \]

where:

  • \(I\): Moment of inertia about the axis

  • \(A\): Area of the cross-section (for area calculations)

  • \(m\): Mass of the body (for mass calculations)

A.8.1 Rectangle (about its centroidal axis)

  • Dimensions: ( b ) (breadth), ( h ) (height)

  • Radius of gyration about the centroidal x-axis: \[ k_x = \sqrt{\frac{I_x}{A}} = \sqrt{\frac{\frac{1}{12} b h^3}{b h}} = \frac{h}{\sqrt{12}} \]

A.8.2 Circle (about its centroidal axis)

  • Radius: ( r )

  • Radius of gyration: \[ k = \sqrt{\frac{I}{A}} = \sqrt{\frac{\frac{\pi r^4}{4}}{\pi r^2}} = \frac{r}{\sqrt{2}} \]

A.9 Beam Calculations

Sum of Horizontal Forces Sum of Vertical Force Sum of Moments
\(\sum F_x = 0\) \(\sum F_y = 0\) \(\sum M = 0\)
Load Type Shear Diagram Shape Moment Diagram Shape
Point Load Rectangular (constant) Triangular
Uniformly Distributed Load (UDL) Triangular Parabolas (second degree)

A.10 Dynamics

A.10.1 Linear momentum

\[ Linear\ momentum = m \vec{v} \]

Where:

  • Linear momentum is in kg·m/s.

  • \(m\) is the mass of the object in kilograms.

  • \(\vec{v}\) is the velocity of the object in meters per second.

A.10.2 Angular momentum

\[ \text{Angular momentum} = I \omega \]

Where:

  • \(I\) is the moment of inertia in \(m^4\).
  • \(\omega\) is the angular velocity in \(rad/s\).

A.10.3 Moment of inertia

\[ I = m k^2 \]

Where:

  • \(I\) is the moment of inertia in \(m^4\).
  • \(m\) is the mass in \(kg\).
  • \(k\) is the radius of gyration in \(m\).

A.10.4 Turning moment

\[ \tau = I \alpha \]

Where:

  • \(\tau\) is the torque in \(Nm\).
  • \(I\) is the moment of inertia in \(m^4\).
  • \(\alpha\): Angular acceleration in \(rad/s^2\).

A.10.5 Power by Torque

\[ P = \tau \cdot \omega \]

Where:

\(P\) is the power in watts (W),

\(\tau\) is the torque in newton-meters (Nm), and

\(\omega\) is the angular velocity in radians per second (rad/s).

A.10.6 Kinetic Energy of Rotation

\[ \text{Rotational K.E.} = \frac{1}{2} I \omega^2 \] Where:

  • \(I\) is the moment of inertia in \(m^4\).

  • \(\omega\) is the angular velocity in radians per second (rad/s).

A.11 Stress and Strain

A.11.1 Stress

\[ \sigma = \frac{F}{A} \]

\[ \tau = \frac{F}{A} \]

Where:

  • \(\sigma\) is the stress (Pa),

  • \(\tau\) is the shearing stress (Pa),

  • \(F\) is the shearing force (N),

  • \(A\) is the cross-sectional area (m\(^2\)).

A.11.2 Strain

\[ \varepsilon = \frac{\Delta L}{L_0} \]

Where:

  • \(\varepsilon\) is the strain (unitless),

  • \(\Delta L\) is the change in length,

  • \(L_0\) is the original length.

A.11.3 Hooke’s Law

\[ E = \frac{\sigma}{\varepsilon} \]

Where:

  • \(\sigma\) is the stress (Pa).
  • \(\varepsilon\) is the strain (unitless).
  • E: Young’s modulus (Pa), a material property (modulus of elasticity).

A.11.4 Factor of Safety (FOS)

\[ \text{FOS} = \frac{\text{Breaking Stress}}{\text{Working Stress}} \]

A.12 Hydrodynamics

A.12.1 Volume Flow

\[ \dot{v} = A \cdot C \]

Where:

\(\dot{v}\): Volume flow rate, \(\mathrm{m^3/s}\)

\(A\): Cross-sectional area of the flow, \(\mathrm{m^2}\)

\(C\): Mean (average) velocity of the fluid, \(\mathrm{m/s}\)

A.12.2 Mass Flow

\[ \dot{m} = \rho \cdot \dot v \]

Where:

\(\dot{m}\): mass flow rate, \(\mathrm{kg/s}\)

\(\rho\): density, \(\mathrm{kg/m^3}\)

\(\dot{v}\): volume flow, \(\mathrm{m^3/s}\)

A.12.3 Specific Weight

\[ \gamma = g \cdot \rho \]

Where:

\(\gamma\): specific weight, \(\mathrm{N/m^3}\)

\(g\): gravitational acceleration, \(\mathrm{m/s^2}\)

\(\rho\): density, \(\mathrm{kg/m^3}\)

A.12.4 Continuity Equation

\[ A_1 \cdot C_1= A_2 \cdot C_2 \]

A.12.5 Energy Equation

\[ Z_1+\frac{C_1^2}{2g}+\frac{P_1}{g\rho_1}=Z_2+\frac{C_2^2}{2g}+\frac{P_2}{g\rho_2} \]

\[ Z_1+\frac{C_1^2}{2g}+\frac{P_1}{\gamma_1}=Z_2+\frac{C_2^2}{2g}+\frac{P_2}{\gamma_2} \]

Each term has units of m, therefore:

  • Potential energy \(Z\) is known as the elevation head.

  • Kinetic energy \(\frac{c^2}{2g}\) is known as the velocity head.

  • Pressure energy \(\frac{P}{\gamma}\) is known as the pressure head.

\[ Total\ Head= Elevation\ Head + Velocity\ Head + Pressure\ Head \]

A.12.6 Bernoulli’s Equation

\[ P_1 + \frac{1}{2} \rho C_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho C_2^2 + \rho g h_2 \]

Where:

  • \(P_1\) and \(P_2\) are the pressures at points 1 and 2, respectively.

  • \(\rho\) is the density of the fluid.

  • \(C_1\) and \(C_2\) are the velocities of the fluid at points 1 and 2, respectively.

  • \(g\) is the acceleration due to gravity.

  • \(h_1\) and \(h_2\) are the heights of the fluid at points 1 and 2, respectively.

Table A.1: Centroids of Common Shapes
Shape Area \(\bar{x}\) \(\bar{y}\)
\(A = b h\) \(b/2\) \(h/2\)
\(\dfrac{bh}{2}\) \(b/3\) \(h/3\)
\(\dfrac{(a+b) h}{2}\) \(\dfrac{a^2 +ab + b^2}{3 (a+b)}\) \(\dfrac{h(2a+b)}{3(a+b)}\)
\(\pi r^2\) \(r\) \(r\)
\(\dfrac{\pi r^2}{2}\) \(r\) \(\dfrac{4 r}{3 \pi}\)
\(\dfrac{\pi r^2}{4}\) \(\dfrac{4 r}{3 \pi}\) \(\dfrac{4 r}{3 \pi}\)

A.13 Second Moments of Common Shapes

Table A.2: Second moments
Shape Second moment (\(I_x\)) Second moment (\(I_y\))
\(I_x = \frac{1}{12} b h^3\) \(I_y = \frac{1}{12} b^3 h\)
\(I_x = \frac{1}{3} b h^3\) \(I_y = \frac{1}{3} b^3 h\)
\(I_x = \frac{\pi}{4} r^4\) \(I_y = \frac{\pi}{4} r^4\)
\(I_x = \frac{\pi}{4} (r_2^4-r_1^4)\) \(I_y = \frac{\pi}{4} (r_2^4-r_1^4)\)
\(I_x = \frac{\pi}{8} r^4\) \(I_y = \frac{\pi}{8} r^4\)
\(I_x = \frac{\pi}{16} r^4\) \(I_y = \frac{\pi}{16} r^4\)
\(I_x = \frac{1}{12} b h^3\)
\(I_x = \frac{1}{36} b h^3\)