ENGR5004 must memorise

Everything you need in your head for the exam: formulas that are not on the formula sheet, and the definitions examiners ask for. Print it, or cover the right-hand side and test yourself.

Formulas not on the formula sheet

Needed in past-paper solutions

Static equilibrium $\sum F_x = 0$ , $\sum F_y = 0$ , $\sum M = 0$ ; roller gives one force, pin gives two	used in 10 questions, e.g. Statics Progress Test 2024 Q1, ENGR273 2026 Q1, ENGR216 2025 Q A1
Static equilibrium of the beam: $\sum F_x = 0$ , $\sum F_y = 0$ , $\sum M_A = 0$ ; a fixed support gives two forces and a moment	used in 4 questions, e.g. Dynamics L7 Worked example 1, ENGR216 2023 Statics Q A1, ENGR216 2022 Statics Q A1
Relative velocity $\mathbf{v}_B = \mathbf{v}_A + \boldsymbol\omega\times\mathbf{r}_{B/A}$	used in 4 questions, e.g. ENGR273 2026 Q3, ENGR273 2026 Q4, ENGR216 2024 Dynamics Q A2
Relative acceleration $\mathbf{a}_B = \mathbf{a}_A + \boldsymbol\alpha\times\mathbf{r}_{B/A} - \omega^2\mathbf{r}_{B/A}$	used in 4 questions, e.g. ENGR273 2026 Q3, ENGR273 2026 Q4, ENGR216 2024 Dynamics Q A2
Elastic curve $EI\,\dfrac{d^2y}{dx^2} = M(x)$ for the curvature argument in (d)	used in 4 questions, e.g. ENGR216 2024 Statics Q A1, ENGR216 2023 Statics Q A2, ENGR216 2021 Statics Q A2
Elastic curve: $EI\dfrac{d^2y}{dx^2} = M(x)$ , $\theta = \dfrac{dy}{dx}$	used in 4 questions, e.g. Tutorial 2 Problem 2, Old spec Statics Session 11 Example 3, Old spec Statics Session 12 Example 1
Polar acceleration: $a_r = \ddot r - r\dot\theta^2$ , $a_\theta = r\ddot\theta + 2\dot r\dot\theta$	used in 3 questions, e.g. Dynamics Session 4 Worked example 2, Dynamics L9 Worked example 1, Dynamics L9 Worked example 2
Kinetic friction: $F = \mu_k N$ , opposing the sliding	used in 3 questions, e.g. Dynamics L7 Worked example 2, Dynamics L14 Worked example 1, Dynamics L16 Worked example 1
Equations of motion: $\sum F_r = m a_r$ , $\sum F_\theta = m a_\theta$	used in 3 questions, e.g. Dynamics L9 Worked example 1, Dynamics L9 Worked example 2, ENGR216 2024 Dynamics Q A1
Relative velocity on a rigid body: $\mathbf v_B = \mathbf v_A + \boldsymbol\omega \times \mathbf r_{B/A}$	used in 3 questions, e.g. Dynamics L11 Worked example 1, Dynamics L11 Worked example 3, ENGR216 2021 Dynamics Q A3
n-t components: $a_t = \dot v$ , $a_n = v^2/\rho$	used in 2 questions, e.g. Dynamics Session 4 Worked example 2, Dynamics L5 Worked example 2
Newton's second law: $\sum F = m a$ for each body	used in 2 questions, e.g. Dynamics L7 Worked example 2, ENGR216 2022 Dynamics Q A2
n-t equations of motion: $\sum F_n = m\dfrac{v^2}{\rho}$ , $\sum F_b = 0$	used in 2 questions, e.g. Dynamics L8 Worked example 3, Dynamics L8 Worked example 4
Point on a body rotating about a fixed axis: $v = \omega r$ , $a_t = \alpha r$ , $a_n = \omega^2 r$	used in 2 questions, e.g. Dynamics L10 Worked example, Dynamics L13 Worked example 3
Instantaneous centre of zero velocity: intersection of the normals to two known velocity directions; $v = \omega\,r_{/IC}$	used in 2 questions, e.g. Dynamics L12 Worked example 1, ENGR273 2026 Q4
Relative acceleration: $\mathbf a_B = \mathbf a_A + \boldsymbol\alpha \times \mathbf r_{B/A} - \omega^2\mathbf r_{B/A}$	used in 2 questions, e.g. Dynamics L13 Worked example 1, ENGR216 2021 Dynamics Q A3
Rigid-body translation: $\sum F_x = m(a_G)_x$ , $\sum F_y = m(a_G)_y$ , $\sum M_G = 0$	used in 2 questions, e.g. Dynamics L14 Worked example 1, Dynamics L14 Worked example 2
Fixed-axis rotation: $\sum F_n = m\omega^2 r_G$ , $\sum F_t = m\alpha r_G$ , $\sum M_O = I_O\alpha$	used in 2 questions, e.g. Dynamics L15 Worked example 1, Dynamics L15 Worked example 2
Principle of work and energy: $T_1 + \sum U_{1-2} = T_2$	used in 2 questions, e.g. Dynamics L17 Worked example 1, Dynamics L17 Worked example 2
Parallel-axis theorem $I = I_c + A d^2$ (or build the I-section by subtracting rectangles)	used in 2 questions, e.g. ENGR273 2026 Q2, Tutorial 2 Problem 1
Unit-vector cross products: $\mathbf{k}\times\mathbf{i} = \mathbf{j}$ , $\mathbf{k}\times\mathbf{j} = -\mathbf{i}$	used in 2 questions, e.g. ENGR216 2025 Q B2, ENGR216 2024 Dynamics Q A2
Elastic curve $EI\,\dfrac{d^2y}{dx^2} = M(x)$ , slope $\theta = \dfrac{dy}{dx}$	used in 2 questions, e.g. ENGR216 2024 Statics Q A2, ENGR216 2023 Statics Q A1
Polar velocity $\mathbf{v} = \dot r\,\mathbf{u}_r + r\dot\theta\,\mathbf{u}_\theta$	used in 2 questions, e.g. ENGR216 2024 Dynamics Q A1, ENGR216 2023 Dynamics Q B1
Polar acceleration $\mathbf{a} = (\ddot r - r\dot\theta^2)\,\mathbf{u}_r + (r\ddot\theta + 2\dot r\dot\theta)\,\mathbf{u}_\theta$	used in 2 questions, e.g. ENGR216 2024 Dynamics Q A1, ENGR216 2023 Dynamics Q B1
Equilibrium of a rigid body: $\sum F_y = 0$ , $\sum M_A = 0$ ; a couple has the same moment about every point	used in 2 questions, e.g. ENGR216 2023 Statics Q A2, Old spec Statics Session 12 Example 1
Principal stresses: $\sigma_{1,2} = \dfrac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\dfrac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$ , $\tan 2\theta_p = \dfrac{2\tau_{xy}}{\sigma_x - \sigma_y}$	used in 2 questions, e.g. ENGR216 2021 Statics Q A1, ENGR216 2018 Q A2
Rolling without slipping: $v_O = \omega r$ , $a_O = \alpha r$ ; contact point has zero velocity	used in 2 questions, e.g. ENGR216 2021 Dynamics Q A3, ENGR216 2019 Q B2
Boundary conditions for a simply supported beam: $y = 0$ at each support	used in 2 questions, e.g. ENGR216 2019 Q A2, Old spec Statics Session 11 Example 3
Torsion $\tau = \dfrac{T c}{J}$ , $J = \dfrac{\pi R^4}{2}$	used in 2 questions, e.g. ENGR216 2018 Q A2, Tutorial 3 Problem 2
Compatibility for a bar between rigid supports: total change in length $= 0$ ; method of superposition (release a redundant support, then restore it)	used in 2 questions, e.g. ENGR216 2017 Q A1, Statics Session 2 Problem 1
Axial deformation $\delta = \dfrac{FL}{AE}$	used in 2 questions, e.g. ENGR216 2017 Q A3, Statics Session 2 Practice Problem B
Generalised Hooke's law: $\epsilon_x = \dfrac{\sigma_x}{E} - \dfrac{\nu\sigma_y}{E} - \dfrac{\nu\sigma_z}{E}$ (and cyclic)	used in 2 questions, e.g. Tutorial 1 Problem 2, Statics Session 3 Example 2
Maximum transverse shear stress, rectangle: $\tau_{max} = \dfrac{3V}{2A}$	used in 2 questions, e.g. Tutorial 1 Problem 4, Old spec Statics Session 9 Problem 2
Stress transformation: $\sigma_{x'} = \dfrac{\sigma_x+\sigma_y}{2} + \dfrac{\sigma_x-\sigma_y}{2}\cos2\theta + \tau_{xy}\sin2\theta$ , $\tau_{x'y'} = -\dfrac{\sigma_x-\sigma_y}{2}\sin2\theta + \tau_{xy}\cos2\theta$	used in 2 questions, e.g. Tutorial 3 Problem 1, Old spec Statics Session 15 Example
Principal directions: $\tan2\theta_p = \dfrac{2\tau_{xy}}{\sigma_x-\sigma_y}$ ; principal stresses $\sigma_{max,min} = \dfrac{\sigma_x+\sigma_y}{2} \pm \sqrt{\left(\dfrac{\sigma_x-\sigma_y}{2}\right)^2 + \tau_{xy}^2}$	used in 2 questions, e.g. Tutorial 3 Problem 1, Old spec Statics Session 16 Example
Maximum shear: $\tan2\theta_s = -\dfrac{\sigma_x-\sigma_y}{2\tau_{xy}}$ , $\tau_{max} = \sqrt{\left(\dfrac{\sigma_x-\sigma_y}{2}\right)^2 + \tau_{xy}^2}$ , normal stress $\sigma' = \dfrac{\sigma_x+\sigma_y}{2}$	used in 2 questions, e.g. Tutorial 3 Problem 1, Old spec Statics Session 16 Example
Elastic curve: $\dfrac{d^2y}{dx^2} = \dfrac{M(x)}{EI}$ , slope $\theta = \dfrac{dy}{dx}$	used in 2 questions, e.g. Old spec Statics Session 11 Example 1, Old spec Statics Session 11 Example 2
Rectilinear motion: $v = \dfrac{ds}{dt}$ , $a = \dfrac{dv}{dt} = \dfrac{d^2s}{dt^2}$	used in 1 question, e.g. Dynamics Session 1 Example 1
$a = \dfrac{dv}{dt}$ , so for $a = a(v)$ : $\displaystyle\int_{v_0}^{v}\frac{dv}{a(v)} = \int_{t_0}^{t} dt$	used in 1 question, e.g. Dynamics Session 1 Example 2
$a\,ds = v\,dv$ , so for $a = a(v)$ : $\displaystyle\int_{v_0}^{v}\frac{v\,dv}{a(v)} = \int_{s_0}^{s} ds$	used in 1 question, e.g. Dynamics Session 1 Example 2
Constant acceleration: $v = v_0 + a_c t$ , $s = s_0 + v_0 t + \tfrac12 a_c t^2$	used in 1 question, e.g. Dynamics Session 2 Example 1
Projectile (no drag): $a_x = 0$ , $a_y = -g$	used in 1 question, e.g. Dynamics Session 2 Example 1
Cartesian components: $v_x = \displaystyle\int a_x\,dt$ , $x = \displaystyle\int v_x\,dt$ (and the same for $y$ , $z$ )	used in 1 question, e.g. Dynamics Session 2 Example 2
Normal-tangential components: $\mathbf a = \dot v\,\mathbf u_t + \dfrac{v^2}{\rho}\mathbf u_n$ , $a = \sqrt{a_t^2 + a_n^2}$	used in 1 question, e.g. Dynamics Session 3 Worked example 1
Normal-tangential components: $a_t = \dot v$ , $a_n = \dfrac{v^2}{\rho}$ , $a = \sqrt{a_t^2 + a_n^2}$	used in 1 question, e.g. Dynamics Session 3 Worked example 2
Radius of curvature of $y = f(x)$ : $\rho = \dfrac{\left[1 + (dy/dx)^2\right]^{3/2}}{\left\|d^2y/dx^2\right\|}$	used in 1 question, e.g. Dynamics Session 3 Worked example 2
Polar velocity: $\mathbf v = \dot r\,\mathbf u_r + r\dot\theta\,\mathbf u_\theta$	used in 1 question, e.g. Dynamics Session 4 Worked example 1
Polar acceleration: $\mathbf a = (\ddot r - r\dot\theta^2)\mathbf u_r + (r\ddot\theta + 2\dot r\dot\theta)\mathbf u_\theta$	used in 1 question, e.g. Dynamics Session 4 Worked example 1
Polar velocity: $v_r = \dot r$ , $v_\theta = r\dot\theta$	used in 1 question, e.g. Dynamics Session 4 Worked example 2
Relative velocity (translating axes): $\mathbf v_B = \mathbf v_A + \mathbf v_{B/A}$	used in 1 question, e.g. Dynamics L5 Worked example 1
Law of cosines $c^2 = a^2 + b^2 - 2ab\cos C$ and law of sines $\dfrac{a}{\sin A} = \dfrac{b}{\sin B}$	used in 1 question, e.g. Dynamics L5 Worked example 1
Relative motion (translating axes): $\mathbf v_{B/A} = \mathbf v_B - \mathbf v_A$ , $\mathbf a_{B/A} = \mathbf a_B - \mathbf a_A$	used in 1 question, e.g. Dynamics L5 Worked example 2
Newton's second law: $\sum F_x = m a_x$ , $\sum F_y = m a_y$	used in 1 question, e.g. Dynamics L6 Worked example 1
Linear spring: $F_s = k s$	used in 1 question, e.g. Dynamics L6 Worked example 1
$a\,ds = v\,dv$ when $a = a(s)$	used in 1 question, e.g. Dynamics L6 Worked example 1
Equation of motion for a system of particles: $\sum \mathbf F_{ext} = m_{tot}\mathbf a_G$ (internal forces cancel)	used in 1 question, e.g. Dynamics L6 Worked example 2
$a = dv/dt$ , so $v = \int a\,dt$ when $a = a(t)$	used in 1 question, e.g. Dynamics L6 Worked example 2
Newton's second law for the crate: $\sum F_y = m a_y$	used in 1 question, e.g. Dynamics L7 Worked example 1
Dependent motion: write the cord length in position coordinates and differentiate	used in 1 question, e.g. Dynamics L7 Worked example 2
Constant acceleration: $v^2 = v_0^2 + 2a_c(s - s_0)$	used in 1 question, e.g. Dynamics L7 Worked example 2
Equations of motion in n-t components: $\sum F_n = m\dfrac{v^2}{\rho}$ , $\sum F_t = m\dot v$	used in 1 question, e.g. Dynamics L8 Worked example 1
n-t equation of motion: $\sum F_n = m\dfrac{v^2}{\rho}$	used in 1 question, e.g. Dynamics L8 Worked example 2
Friction at impending slip: $F = \mu_s N$ , opposing the impending motion	used in 1 question, e.g. Dynamics L8 Worked example 3
Chain rule for $r = f(\theta)$ : $\dot r = f'(\theta)\dot\theta$ , $\ddot r = f''(\theta)\dot\theta^2 + f'(\theta)\ddot\theta$	used in 1 question, e.g. Dynamics L9 Worked example 1
Spring force: $F_s = k(\ell - \ell_0)$	used in 1 question, e.g. Dynamics L9 Worked example 2
Angular motion: $\omega = d\theta/dt$ , $\alpha = d\omega/dt$	used in 1 question, e.g. Dynamics L10 Worked example
$\omega\mathbf k \times (x\mathbf i + y\mathbf j) = -\omega y\,\mathbf i + \omega x\,\mathbf j$	used in 1 question, e.g. Dynamics L11 Worked example 1
Fixed-axis rotation: $\mathbf v_A = \boldsymbol\omega_{OA} \times \mathbf r_{A/O}$	used in 1 question, e.g. Dynamics L11 Worked example 2
Relative velocity: $\mathbf v_B = \mathbf v_A + \boldsymbol\omega_{AB} \times \mathbf r_{B/A}$	used in 1 question, e.g. Dynamics L11 Worked example 2
Meshing gears: the contact points have the same velocity (no slip)	used in 1 question, e.g. Dynamics L11 Worked example 3
Fixed-axis rotation: $v_B = \omega_{AB}\,r_{B/A}$ , perpendicular to AB	used in 1 question, e.g. Dynamics L12 Worked example 1
Rolling without slipping: the contact point is the IC, $v_C = \omega r$	used in 1 question, e.g. Dynamics L12 Worked example 2
$\mathbf v_P = \boldsymbol\omega \times \mathbf r_{P/IC}$	used in 1 question, e.g. Dynamics L12 Worked example 2
Instantaneous centre of zero velocity: intersection of the normals to $\mathbf v_B$ and $\mathbf v_C$ ; $v = \omega\,r_{/IC}$	used in 1 question, e.g. Dynamics L12 Worked example 3
Fixed-axis rotation: $v = \omega r$ , perpendicular to the link	used in 1 question, e.g. Dynamics L12 Worked example 3
Instantaneous centre: $\omega = v_A/r_{A/IC}$	used in 1 question, e.g. Dynamics L13 Worked example 1
Rolling without slipping: $a_O = \alpha r$ (centre moves on a straight line)	used in 1 question, e.g. Dynamics L13 Worked example 2
Relative acceleration: $\mathbf a_A = \mathbf a_O + \boldsymbol\alpha \times \mathbf r_{A/O} - \omega^2\mathbf r_{A/O}$	used in 1 question, e.g. Dynamics L13 Worked example 2
$\mathbf v_C = \mathbf v_B + \boldsymbol\omega_{BC} \times \mathbf r_{C/B}$ , $\mathbf a_C = \mathbf a_B + \boldsymbol\alpha_{BC} \times \mathbf r_{C/B} - \omega_{BC}^2\mathbf r_{C/B}$	used in 1 question, e.g. Dynamics L13 Worked example 3
Parallel-axis theorem: $I_O = I_G + m r_G^2$ ; slender rod $I_G = \tfrac{1}{12}ml^2$	used in 1 question, e.g. Dynamics L15 Worked example 1
Parallel-axis theorem $I_O = I_G + md^2$ ; sphere $I_G = \tfrac25 mr^2$ , slender rod $I_G = \tfrac{1}{12}ml^2$	used in 1 question, e.g. Dynamics L15 Worked example 2
General plane motion: $\sum F_x = m(a_G)_x$ , $\sum F_y = m(a_G)_y$ , $\sum M_G = I_G\alpha$	used in 1 question, e.g. Dynamics L16 Worked example 1
Radius of gyration: $I_G = m k_G^2$	used in 1 question, e.g. Dynamics L16 Worked example 1
Kinetic energy in plane motion: $T = \tfrac12 m v_G^2 + \tfrac12 I_G\omega^2$ , $I_G = m k_G^2$	used in 1 question, e.g. Dynamics L17 Worked example 1
Work of a couple $U = M\Delta\theta$ ; work of a spring $U = -\tfrac12 k(s_2^2 - s_1^2)$	used in 1 question, e.g. Dynamics L17 Worked example 1
Rolling without slipping: $v_G = \omega r$ , $\Delta s_G = r\Delta\theta$	used in 1 question, e.g. Dynamics L17 Worked example 1
Fixed-axis rotation: $T = \tfrac12 I_A\omega^2$ , $I_A = m k_A^2$	used in 1 question, e.g. Dynamics L17 Worked example 2
Work of the weight: $U = mg\,\Delta y$ for a drop $\Delta y$ of G	used in 1 question, e.g. Dynamics L17 Worked example 2
$\int x \sin(ax)\,dx = \dfrac{\sin(ax)}{a^2} - \dfrac{x\cos(ax)}{a}$	used in 1 question, e.g. Statics Progress Test 2024 Q1
Static equilibrium; pin gives two forces, roller one	used in 1 question, e.g. Statics new problem 2 (session 14)
Axial stress $\sigma = F/A \le \sigma_{all}$ ; buckling safety factor $P_{cr} \ge FS \cdot F$	used in 1 question, e.g. ENGR273 2026 Q2
Rotation about a fixed axis: $\mathbf{v} = \boldsymbol\omega\times\mathbf{r}$ , $\mathbf{a} = \boldsymbol\alpha\times\mathbf{r} - \omega^2\mathbf{r}$	used in 1 question, e.g. ENGR273 2026 Q3
Equations of motion $\sum\mathbf{F} = m\mathbf{a}_G$ , $\sum M_G = I_G\alpha$ (or $\sum M_O = I_O\alpha$ about a fixed pin)	used in 1 question, e.g. ENGR273 2026 Q3
Slender rod $I_G = \dfrac{1}{12} m L^2$ , $I_O = \dfrac{1}{3} m L^2$ about an end	used in 1 question, e.g. ENGR273 2026 Q3
Equations of motion $\sum\mathbf{F} = m\mathbf{a}_G$ , $\sum M_G = I_G\alpha$ , or about any point P: $\sum M_P = I_G\alpha + (\mathbf{r}_{G/P}\times m\mathbf{a}_G)\cdot\mathbf{k}$	used in 1 question, e.g. ENGR273 2026 Q4
Transverse shear stress $\tau = \dfrac{VQ}{It}$ with $Q = \bar y\,A'$ (first moment about the neutral axis of the area on one side of the point)	used in 1 question, e.g. ENGR216 2025 Q A1
Rectangle $I = \dfrac{bh^3}{12}$ (on the formula sheet); parallel-axis theorem $I = I_c + A d^2$ and centroid of a composite area $\bar y = \dfrac{\sum A_i \bar y_i}{\sum A_i}$	used in 1 question, e.g. ENGR216 2025 Q A1
Elastic curve $\dfrac{d^2y}{dx^2} = \dfrac{M(x)}{EI}$ , slope $\theta = dy/dx$	used in 1 question, e.g. ENGR216 2025 Q A2
Slope is extreme where $d\theta/dx = M/EI = 0$ ; deflection is extreme where $\theta = 0$	used in 1 question, e.g. ENGR216 2025 Q A2
Relative acceleration (rigid body): $\mathbf{a}_G = \mathbf{a}_O + \boldsymbol\alpha\times\mathbf{r}_{G/O} - \omega^2\mathbf{r}_{G/O}$	used in 1 question, e.g. ENGR216 2025 Q B1
Equations of motion: $\sum\mathbf{F} = m\mathbf{a}_G$ , $\sum M_G = I_G\alpha$ ; for a translating body $\sum\mathbf{F} = m\mathbf{a}$	used in 1 question, e.g. ENGR216 2025 Q B1
Slender rod about its centre: $I_G = \dfrac{1}{12} m L^2$	used in 1 question, e.g. ENGR216 2025 Q B1
Moments about another point P (check): $\sum M_P = I_G\alpha + (\mathbf{r}_{G/P}\times m\mathbf{a}_G)_z$	used in 1 question, e.g. ENGR216 2025 Q B1
Newton's third law at a pin: equal and opposite pin forces on the two bodies	used in 1 question, e.g. ENGR216 2025 Q B1
Friction: static $F \le \mu_s N$ , kinetic $F = \mu_k N$ , opposing the (impending) relative sliding	used in 1 question, e.g. ENGR216 2025 Q B1
Newton's second law for a particle: $\sum\mathbf{F} = m\mathbf{a}$ , in components	used in 1 question, e.g. ENGR216 2025 Q B2
Kinetic friction $F = \mu_k N$ , opposing the relative sliding; static $F \le \mu_s N$ only when there is no sliding	used in 1 question, e.g. ENGR216 2025 Q B2
Rectangle $I = \dfrac{bh^3}{12}$ (on the formula sheet); I-section by subtraction or by three rectangles with the parallel-axis theorem $I = I_c + Ad^2$	used in 1 question, e.g. ENGR216 2024 Statics Q A1
Degrees to radians, $1^\circ = \pi/180\ \mathrm{rad}$	used in 1 question, e.g. ENGR216 2024 Statics Q A2
Linear spring force $F_s = k_{sp}(L - L_0)$ , directed along the spring	used in 1 question, e.g. ENGR216 2024 Dynamics Q A1
Rotation about a fixed axis: $\mathbf{v} = \boldsymbol\omega\times\mathbf{r}$ , $\mathbf{a} = \boldsymbol\alpha\times\mathbf{r} - \omega^2\mathbf{r}$ , $a_t = \alpha r$ , $a_n = \omega^2 r$	used in 1 question, e.g. ENGR216 2024 Dynamics Q A2
Point sliding on a rotating body (rotating axes): $\mathbf{v}_C = \mathbf{v}_{C'} + \mathbf{v}_{rel}$ , $\mathbf{a}_C = \mathbf{a}_{C'} + \mathbf{a}_{rel} + 2\boldsymbol\omega\times\mathbf{v}_{rel}$	used in 1 question, e.g. ENGR216 2024 Dynamics Q A2
Small rotation of a straight member: transverse displacement $= L\,\theta$	used in 1 question, e.g. ENGR216 2023 Statics Q A1
Mean normal stress: $\sigma_{mean} = N/A$	used in 1 question, e.g. ENGR216 2023 Statics Q A1
Constant acceleration: $\dot r = \dot r_0 + \ddot r\,t$ , $r = r_0 + \dot r_0 t + \tfrac12\ddot r\,t^2$ (and the same for $\theta$ )	used in 1 question, e.g. ENGR216 2023 Dynamics Q B1
Normal-tangential components: $a_t = \dfrac{\mathbf{a}\cdot\mathbf{v}}{v}$ , $a_n = \sqrt{a^2 - a_t^2}$ , $\rho = \dfrac{v^2}{a_n}$	used in 1 question, e.g. ENGR216 2023 Dynamics Q B1
Instantaneous centre of zero velocity (as a check)	used in 1 question, e.g. ENGR216 2023 Dynamics Q B2
Parallel-axis theorem: $I_D = I_G + m d^2$	used in 1 question, e.g. ENGR216 2023 Dynamics Q B2
Planar kinetics: $\sum\mathbf{F} = m\mathbf{a}_G$ , $\sum M_G = I_G\alpha$ ; about a fixed pin $\sum M_D = I_D\alpha$	used in 1 question, e.g. ENGR216 2023 Dynamics Q B2
Elastic curve $EI\,\dfrac{d^2y}{dx^2} = M(x)$ , so $\theta_C - \theta_B = \dfrac{1}{EI}\displaystyle\int_B^C M\,dx$	used in 1 question, e.g. ENGR216 2022 Statics Q A1
Elastic curve $EI\,\dfrac{d^2y}{dx^2} = M(x)$ , so $M = EI\,y''$ ; clamped end: $y = 0$ , $y' = 0$	used in 1 question, e.g. ENGR216 2022 Statics Q A2
Statically indeterminate beams by compatibility/superposition; cantilever with end load $P$ : $\delta = \dfrac{PL^3}{3EI}$	used in 1 question, e.g. ENGR216 2022 Statics Q A2
Fixed-axis rotation, vector form: $\mathbf{v}_B = \boldsymbol\omega\times\mathbf{r}_{B/A}$ , $\mathbf{a}_B = \boldsymbol\alpha\times\mathbf{r}_{B/A} - \omega^2\mathbf{r}_{B/A}$	used in 1 question, e.g. ENGR216 2022 Dynamics Q A1
Polar coordinates: $\mathbf{v} = \dot r\,\mathbf{u}_r + r\dot\theta\,\mathbf{u}_\theta$ , $\mathbf{a} = (\ddot r - r\dot\theta^2)\,\mathbf{u}_r + (r\ddot\theta + 2\dot r\dot\theta)\,\mathbf{u}_\theta$ ; Coriolis term $2\dot r\dot\theta$	used in 1 question, e.g. ENGR216 2022 Dynamics Q A1
Normal-tangential: $\mathbf{v} = v\,\mathbf{u}_t$ , $a_t = \dot v$ , $a_n = v^2/\rho$ toward the centre of curvature	used in 1 question, e.g. ENGR216 2022 Dynamics Q A1
Constraint differentiation: $x = y\tan\theta$ , $\dfrac{d}{d\theta}\tan\theta = \sec^2\theta$ , $\dfrac{d}{d\theta}\sec^2\theta = 2\sec^2\theta\tan\theta$ , chain rule $\dfrac{d}{dt} = \dot\theta\dfrac{d}{d\theta}$	used in 1 question, e.g. ENGR216 2022 Dynamics Q A1
Rotation about a fixed axis through G: $\sum M_G = I_G\,\alpha$	used in 1 question, e.g. ENGR216 2022 Dynamics Q A2
Angle-dependent integration: $\alpha = \omega\,\dfrac{d\omega}{d\theta}$ , so $I\,\omega\,\dfrac{d\omega}{d\theta} = T(\omega)$	used in 1 question, e.g. ENGR216 2022 Dynamics Q A2
$\displaystyle\int \frac{\omega\,d\omega}{a - b\omega^2} = -\frac{1}{2b}\ln(a - b\omega^2)$ ; 1 rev $= 2\pi$ rad, rpm $= \omega \cdot 60/(2\pi)$	used in 1 question, e.g. ENGR216 2022 Dynamics Q A2
Rope over a pulley without slip: $a = \alpha R$ ; pulley with inertia: $(T_1 - T_2)R = I_G\alpha$ , so $m_{eq} = m + I_G/R^2$	used in 1 question, e.g. ENGR216 2022 Dynamics Q A2
Work-energy: $T_1 + \sum U_{1\to2} = T_2$ , spring work $-\tfrac{1}{2}k(s_2^2 - s_1^2)$ , gravity work $mg\,\Delta s$	used in 1 question, e.g. ENGR216 2022 Dynamics Q A2
Transverse shear stress: $\tau = \dfrac{VQ}{It}$ , $\tau_{max} = \dfrac{3V}{2A}$ for a rectangle	used in 1 question, e.g. ENGR216 2021 Statics Q A1
Safety factor and strength limit: $P_{allow} = P_{cr}/FS$ , $P \le \sigma_{all} A$	used in 1 question, e.g. ENGR216 2021 Statics Q A1
Fixed-end conditions $y = 0$ , $y' = 0$ ; roller $y = 0$ , $M = 0$	used in 1 question, e.g. ENGR216 2021 Statics Q A2
Fixed-axis rotation: $\omega = \omega_0 + \int_0^t \alpha\,dt$ , $\theta = \theta_0 + \int_0^t \omega\,dt$	used in 1 question, e.g. ENGR216 2021 Dynamics Q A1
Meshing gears (no slip at the pitch point): $r_S\,\omega_S = r_D\,\omega_D$ , $r_S\,\theta_S = r_D\,\theta_D$ , $r_S\,\alpha_S = r_D\,\alpha_D$	used in 1 question, e.g. ENGR216 2021 Dynamics Q A1
Point on a rotating body: $v = \omega r$ , $a_t = \alpha r$	used in 1 question, e.g. ENGR216 2021 Dynamics Q A1
Normal-tangential kinetics: $\sum F_t = m a_t$ , $\sum F_n = m a_n$ , $a_t = \dot v$ , $a_n = \dfrac{v^2}{\rho}$	used in 1 question, e.g. ENGR216 2021 Dynamics Q A2
Radius of curvature of $y(x)$ : $\rho = \dfrac{\left[1 + (dy/dx)^2\right]^{3/2}}{\|d^2y/dx^2\|}$	used in 1 question, e.g. ENGR216 2021 Dynamics Q A2
Slope angle: $\tan\theta = dy/dx$ ; magnitude $a = \sqrt{a_t^2 + a_n^2}$	used in 1 question, e.g. ENGR216 2021 Dynamics Q A2
Instantaneous centre of zero velocity (IC) method	used in 1 question, e.g. ENGR216 2021 Dynamics Q A3
Maximum in-plane shear stress $\tau_{max} = \sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2 + \tau_{xy}^2} = \frac{\sigma_{max}-\sigma_{min}}{2}$ (radius of Mohr's circle)	used in 1 question, e.g. ENGR216 2019 Q A1
Normal strain $\epsilon = \delta / L$	used in 1 question, e.g. ENGR216 2019 Q A1
Plane stress: $\sigma_z = \tau_{xz} = \tau_{yz} = 0$	used in 1 question, e.g. ENGR216 2019 Q A1
Normal stress $\sigma = P/A$ and allowable load with a safety factor $P_{all} = P_{cr}/FS$	used in 1 question, e.g. ENGR216 2019 Q A3
Newton's second law in polar components: $\sum F_r = m a_r$ , $\sum F_\theta = m a_\theta$	used in 1 question, e.g. ENGR216 2019 Q B1
Weight resolved along $\mathbf{u}_r$ , $\mathbf{u}_\theta$ : $W_r = -mg\sin\theta$ , $W_\theta = -mg\cos\theta$	used in 1 question, e.g. ENGR216 2019 Q B1
Parallel-axis theorem: $I_C = I_G + m d^2$	used in 1 question, e.g. ENGR216 2019 Q B3
Fixed-axis rotation of G: $a_n = \omega^2 r_{G/C}$ (towards C), $a_t = \alpha r_{G/C}$	used in 1 question, e.g. ENGR216 2019 Q B3
Safety factor: $P_{cr} = FS \cdot P$ ; allowable stress $\|\sigma\| \le \sigma_{all}$	used in 1 question, e.g. ENGR216 2018 Q A1
Uniaxial stress: maximum shear $\tau_{max} = \|\sigma\|/2$ on planes at $45^\circ$	used in 1 question, e.g. ENGR216 2018 Q A1
Principal axes $\tan 2\theta_p = \dfrac{2\tau_{xy}}{\sigma_x - \sigma_y}$ , max shear axes $\tan 2\theta_s = -\dfrac{\sigma_x - \sigma_y}{2\tau_{xy}}$	used in 1 question, e.g. ENGR216 2018 Q A1
Maximum in-plane shear $\tau_{max} = \sqrt{\left(\dfrac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$ ; Mohr's circle centre $\left(\dfrac{\sigma_x + \sigma_y}{2}, 0\right)$ and radius $\tau_{max}$	used in 1 question, e.g. ENGR216 2018 Q A2
Transverse shear stress $\tau = \dfrac{VQ}{It}$ , $\tau_{max} = \dfrac{3V}{2A}$ for a rectangle	used in 1 question, e.g. ENGR216 2018 Q A3
Axial deformation of a uniform segment $\delta = \dfrac{NL}{AE}$ , summed over segments $\delta = \sum_i \dfrac{N_i L_i}{A_i E_i}$	used in 1 question, e.g. ENGR216 2017 Q A1
Normal stress $\sigma = N/A$ and Hooke's law $\sigma = E\epsilon$	used in 1 question, e.g. ENGR216 2017 Q A1
Equilibrium along the bar $\sum F_x = 0$	used in 1 question, e.g. ENGR216 2017 Q A1
Boundary conditions at simple supports: $y = 0$ ; slope $\theta = dy/dx$ (rotation of the section, small angles)	used in 1 question, e.g. ENGR216 2017 Q A2
Strength requirement $\sigma = F/A \le \sigma_{all}$ ; buckling with a safety factor $P_{cr} \ge FS \times F$	used in 1 question, e.g. ENGR216 2017 Q A3
Equilibrium of a pin joint, $\sum F_x = 0$ , $\sum F_y = 0$ , with two-force members carrying force along their axes	used in 1 question, e.g. ENGR216 2017 Q A3
Rigid-body translation: $\alpha = 0$ , so $\sum M_G = 0$ , or about any point P, $\sum M_P = (m a_G)\,d$ where $d$ is the moment arm of $m\mathbf{a}_G$ about P	used in 1 question, e.g. ENGR216 2017 Q B1
Kinetic (sliding) friction $F = \mu N$ , opposing the sliding motion	used in 1 question, e.g. ENGR216 2017 Q B1
Tipping condition: a roller is about to lift when its normal reaction falls to zero	used in 1 question, e.g. ENGR216 2017 Q B1
Angular speed conversion $\omega = 2\pi N/60$ (rev/min to rad/s)	used in 1 question, e.g. ENGR216 2017 Q B2
Fixed-axis rotation at constant speed: $\mathbf{a}_A = -\omega^2 \mathbf{r}_{A/O}$	used in 1 question, e.g. ENGR216 2017 Q B2
No-slip cable on a drum: $a = r\alpha$ (tangential acceleration of the drum surface equals the cable acceleration)	used in 1 question, e.g. ENGR216 2017 Q B3
Axial deformation: $\delta = \dfrac{PL}{AE}$	used in 1 question, e.g. Tutorial 1 Problem 1
Equilibrium of the end plate: $P_1 + P_2 = P$	used in 1 question, e.g. Tutorial 1 Problem 1
Compatibility (rigid plate): $\delta_1 = \delta_2$	used in 1 question, e.g. Tutorial 1 Problem 1
Normal stress under centric axial load: $\sigma_z = P_z / A$	used in 1 question, e.g. Tutorial 1 Problem 2
Maximum shear stress under axial load, on planes at $45^\circ$ : $\tau_{max} = \dfrac{P}{2A}$	used in 1 question, e.g. Tutorial 1 Problem 2
Bulk modulus: $k = \dfrac{E}{3(1-2\nu)}$	used in 1 question, e.g. Tutorial 1 Problem 2
Dilatation: $e = \epsilon_x + \epsilon_y + \epsilon_z = \dfrac{1-2\nu}{E}(\sigma_x + \sigma_y + \sigma_z)$	used in 1 question, e.g. Tutorial 1 Problem 2
Transverse shear stress: $\tau = \dfrac{VQ}{It}$ , $Q = \bar y' A'$	used in 1 question, e.g. Tutorial 2 Problem 1
Elastic curve: $EI\dfrac{d^2y}{dx^2} = M(x)$ , maximum deflection where $\dfrac{dy}{dx} = 0$	used in 1 question, e.g. Tutorial 2 Problem 3
Principal stresses and maximum in-plane shear: $\sigma_{1,2} = \dfrac{\sigma_x}{2} \pm \sqrt{\left(\dfrac{\sigma_x}{2}\right)^2 + \tau^2}$ , $\tau_{max} = \sqrt{\left(\dfrac{\sigma_x}{2}\right)^2 + \tau^2}$	used in 1 question, e.g. Tutorial 3 Problem 2
Maximum in-plane shear for a uniaxial surface element: $\tau_{max} = \dfrac{\|\sigma\|}{2}$ ; principal axes along the stress directions, maximum-shear axes at $45^\circ$	used in 1 question, e.g. Tutorial 4 Problem 2
Parallel-axis theorem $I = I_c + Ad^2$ (alternative route for $I_x$ )	used in 1 question, e.g. Tutorial 4 Problem 3
Axial stress $\sigma = F/A$	used in 1 question, e.g. Tutorial 4 Problem 3
Equilibrium of the bar: $\sum F = 0$	used in 1 question, e.g. Statics Session 2 Problem 1
Axial deformation of a uniform segment $\delta_i = \dfrac{P_i L_i}{A_i E_i}$ , summed over segments $\delta = \sum_i \dfrac{P_i L_i}{A_i E_i}$	used in 1 question, e.g. Statics Session 2 Problem 1
Normal stress $\sigma = P/A$	used in 1 question, e.g. Statics Session 2 Problem 1
Axial deformation of a stepped bar $\delta = \sum_i \dfrac{P_i l_i}{A_i E}$	used in 1 question, e.g. Statics Session 2 Problem 2
Superposition: release the redundant support, find the deformation due to the loads $\delta_L$ and due to the redundant reaction $\delta_R$ , impose $\delta_L + \delta_R = 0$	used in 1 question, e.g. Statics Session 2 Problem 2
Equilibrium $\sum F_y = 0$	used in 1 question, e.g. Statics Session 2 Problem 2
Axial deformation of a stepped bar $\delta = \sum_i \dfrac{P_i L_i}{A_i E_i}$	used in 1 question, e.g. Statics Session 2 Practice Problem A
Internal force from a section FBD (take the free part, tension positive)	used in 1 question, e.g. Statics Session 2 Practice Problem A
Equilibrium of a rigid body: $\sum M_B = 0$ , $\sum M_D = 0$	used in 1 question, e.g. Statics Session 2 Practice Problem B
Rigid bar: displaced points stay on a straight line (similar triangles)	used in 1 question, e.g. Statics Session 2 Practice Problem B
Equilibrium of a rigid body: $\sum M_B = 0$ , $\sum F_y = 0$	used in 1 question, e.g. Statics Session 2 Problem (rods CE and DF)
Axial deformation $\delta = \dfrac{FL}{AE}$ , circle $A = \pi d^2/4$	used in 1 question, e.g. Statics Session 2 Problem (rods CE and DF)
Rigid bar rotating about B: displacements proportional to distance from B (similar triangles)	used in 1 question, e.g. Statics Session 2 Problem (rods CE and DF)
Normal stress $\sigma_x = P/A$ , circle $A = \pi d^2/4$	used in 1 question, e.g. Statics Session 3 Example 1
Normal strain $\epsilon = \delta/L$ , Hooke's law $\sigma = E\epsilon$	used in 1 question, e.g. Statics Session 3 Example 1
Poisson's ratio $\nu = -\dfrac{\epsilon_y}{\epsilon_x}$ (uniaxial load along $x$ )	used in 1 question, e.g. Statics Session 3 Example 1
Bulk modulus: $k = \dfrac{E}{3(1-2\nu)}$ ; dilatation $e = \epsilon_x + \epsilon_y + \epsilon_z = -\dfrac{p}{k}$ under hydrostatic pressure	used in 1 question, e.g. Statics Session 3 Example 2
Volume change $\Delta V = eV$	used in 1 question, e.g. Statics Session 3 Example 2
Average shearing strain $\gamma_{xy} \approx \tan\gamma_{xy} = \dfrac{\text{displacement}}{\text{height}}$	used in 1 question, e.g. Statics Session 4 Example
Hooke's law in shear $\tau_{xy} = G\gamma_{xy}$ , average shear stress $\tau = P/A$	used in 1 question, e.g. Statics Session 4 Example
Equilibrium $\sum F_y = 0$ , $\sum M = 0$ for the reactions; method of sections	used in 1 question, e.g. Old spec Statics Session 6 Review Problem
Superposition of load cases for a linear structure	used in 1 question, e.g. Old spec Statics Session 6 Review Problem
Shear flow (horizontal force per unit length) $q = \dfrac{VQ}{I}$ ; nail force $F = qz$$ is on the formula sheet)	used in 1 question, e.g. Old spec Statics Session 8 Example
Centroid of a composite area $\bar y = \dfrac{\sum A_i \bar y_i}{\sum A_i}$	used in 1 question, e.g. Old spec Statics Session 9 Problem 1
Simply supported beam with central load: $V_{max} = P/2$ , $M_{max} = PL/4$	used in 1 question, e.g. Old spec Statics Session 9 Problem 2
Fixed end: $y = 0$ and $\theta = 0$	used in 1 question, e.g. Old spec Statics Session 11 Example 1
Unloaded segment: $M = 0$ , so it stays straight (rigid rotation), small angles $y_B \approx y_C + L\theta_C$	used in 1 question, e.g. Old spec Statics Session 11 Example 2
Fixed end $y = 0$ , $\theta = 0$ ; roller $y = 0$	used in 1 question, e.g. Old spec Statics Session 12 Example 2
Stress transformation $\sigma_{x'}$ , $\tau_{x'y'}$ (to decide which angle gives which stress)	used in 1 question, e.g. Old spec Statics Session 16 Example
Torsion $\tau_{max} = \dfrac{Tr}{J}$ , $J = \dfrac{\pi r^4}{2}$	used in 1 question, e.g. Old spec Statics Session 16 Problem
Principal stresses $\sigma_{max,min} = \dfrac{\sigma_x+\sigma_y}{2} \pm \sqrt{\left(\dfrac{\sigma_x-\sigma_y}{2}\right)^2 + \tau_{xy}^2}$ , $\tan2\theta_p = \dfrac{2\tau_{xy}}{\sigma_x-\sigma_y}$ , $\tau_{max} = \sqrt{\left(\dfrac{\sigma_x-\sigma_y}{2}\right)^2 + \tau_{xy}^2}$	used in 1 question, e.g. Old spec Statics Session 16 Problem
Design: $P_{cr} \ge kP$ and $\sigma = P/A \le \sigma_{all}$ ; the larger radius governs	used in 1 question, e.g. Old spec Statics Session 18 Problem
Allowable load $P_{max} = \min(\sigma_{cr}/k,\ \sigma_{all})\,A$	used in 1 question, e.g. Old spec Statics Session 19 Example 1