WM1AC formula sheet

Maths

Want	Formula	Guide
Product rule	$(uv{)}^{\prime }=u^{\prime }v+u{v}^{\prime }$	Calculus for kinematics
Chain rule	$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$	Calculus for kinematics
By parts	$\int udv=uv-\int vdu$	Calculus for kinematics
Dot product	$\mathbf{a}\cdot \mathbf{b}=a_x{b}_x+a_y{b}_y+a_z{b}_z=|\mathbf{a}||\mathbf{b}|\cos \theta$	Vectors
Cross product	$\mathbf{a}\times \mathbf{b}$ = determinant with $\mathbf{i},\mathbf{j},\mathbf{k}$ on top	Vectors
Moment	$\mathbf{M}=\mathbf{r}\times \mathbf{F}$	Vectors
Implicit	$\frac{d}{dt}f(x)=f^{\prime }(x)\dot{x}$	Implicit differentiation

Particle kinematics

Want	Formula	Guide
Definitions	$v=\dot{s}$, $a=\dot{v}=\ddot{s}$, $ads=vdv$	Rectilinear motion
Constant $a$	$v=v_0+at$ · $s=s_0+v_{0}t+\frac{1}{2}a{t}^2$ · $v^2=v_0^2+2a(s-s_0)$	Rectilinear motion
x–y components	$\mathbf{v}=\dot{x}\mathbf{i}+\dot{y}\mathbf{j}$, $\mathbf{a}=\ddot{x}\mathbf{i}+\ddot{y}\mathbf{j}$	Curvilinear and circular motion
Circular motion	$v=r\omega$ · $a_t=r\alpha$ · $a_n=v^2/r={\omega }^{2}r$ · $a=\sqrt{a_t^2+a_n^2}$	Curvilinear and circular motion

Rigid bodies and mechanisms

Want	Formula	Guide
Angular definitions	$\omega =\dot{\theta }$, $\alpha =\dot{\omega }$, $\omega d\omega =\alpha d\theta$	Rigid body rotation
Constant α	$\omega ={\omega }_0+\alpha t$ · $\theta ={\theta }_0+{\omega }_{0}t+\frac{1}{2}\alpha t^2$ · ${\omega }^2={\omega }_0^2+2\alpha (\theta -{\theta }_0)$	Rigid body rotation
Gears, belts (no slip)	$r_A{\omega }_A=r_B{\omega }_B$ · $r_A{\alpha }_A=r_B{\alpha }_B$	Gear trains and belt drives
Relative velocity	${\mathbf{v}}_B={\mathbf{v}}_A+\mathbf{\omega }\times {\mathbf{r}}_{B/A}$	Relative motion analysis
Relative acceleration	${\mathbf{a}}_B={\mathbf{a}}_A+\mathbf{\alpha }\times {\mathbf{r}}_{B/A}-{\omega }^2{\mathbf{r}}_{B/A}$	Relative motion analysis
Cosine rule	$c^2=a^2+b^2-2ab\cos C$	Absolute motion analysis

Statics and kinetics

Want	Formula	Guide
Moment	$M_O=Fd$ (d perpendicular) · $M_O=F_{y}x-F_{x}y$	Forces and moments
Couple	$M=Fd$, the same about every point	Forces and moments
Resultant	$F_{Rx}=\sum F_x$ · $F_{Ry}=\sum F_y$ · $(M_R{)}_O=\sum M+\sum Fd$	Forces and moments
Equilibrium	$\sum F_x=0$ · $\sum F_y=0$ · $\sum M=0$	Static equilibrium and free body diagrams
Friction	static $F\le {\mu }_{s}N$ · kinetic $F={\mu }_{k}N$	Friction and Newton’s second law
Newton’s 2nd law	$\sum F_x=m{a}_x$ · $\sum F_y=m{a}_y$ · $\sum M=I\alpha$, $I=m{r}^2$ (particle)	Friction and Newton’s second law
Centre of mass	$\bar{x}=\sum A_i{x}_i/\sum A_i$ · $\bar{x}=\int xydx/A$ · $\bar{y}=\int \frac{1}{2}{y}^{2}dx/A$	Centre of mass

Conversions

• 1 rev = $2\pi$ rad · rpm × $2\pi /60$ = rad/s
• Angles in radians for $s=r\theta$ and $v=r\omega$