Table of Contents

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Angular Velocity, Acceleration

Angular velocity $(\omega)$ and angular acceleration $(\alpha)$ are $$ \omega=\frac{d\theta}{dt};\;\;\;\;\alpha=\frac{d\omega}{dt}=\frac{d^2\theta}{dt^2} $$ The direction of $\omega$ is given by the right hand grip rule (thumb = direction, fingers = rotation). Direction of $\alpha$ uses the same rule, but the fingers are the rate of change of $\omega$.

Linear Speed In Angular Motion

Speed $(v)$ is: $$ v=\frac{ds}{dt}=r\frac{d\theta}{dt}=r\omega $$ Where $s$ is arc length, and $r$ is radius.

Acceleration

Tangential component: $$ a_{tangential}=\frac{dv}{dt}=\frac{d}{dt}(r\omega)=r\alpha $$ Centripetal Component $$ a_{centripetal}=\frac{v^2}{r}=\omega^2r $$

Moment Of Inertia

Moment of inertia is a measure of how hard it is to start an object rotating. $$I=\sum_im_ir_i^2$$

For Composite Bodies

For a body with multiple moments of inertia stacked symmetrically along the axis of rotation: $$ I=I_A+I_B+I_C=\sum_iI_i $$

For An Extended Object

Inertia in Z = I Inertia in Z’=I’ $$ I'=I_{cm}+md^2\;\;\;\;\;(cm=center \;of\; mass) $$

Kinetic Energy

For a point mass: $$ K_{Rotational}=\frac{1}{2}mv^2=\frac{1}{2}m\omega^2r^2 $$ For one particle in the system: $$ K_i=\frac{1}{2}m_iv_i^2=\frac{1}{2}m_ir_i^2\omega^2 $$ So in total: $$ K=\sum_i\frac{1}{2}m_ir_i^2\omega^2=\frac{1}{2}\omega^2\sum_im_ir_i^2 $$ BUT Moment of Inertia $(I)$ is $\sum_im_ir_i^2$, so $$ K=\frac{1}{2}I\omega^2 $$

Torque

Torque $(\tau)$: $$ \boldsymbol{\tau}=F_tl=Fr_\perp\sin(\phi) $$ As a vector: $$ \boldsymbol{\tau}=\mathbf{F}\times\mathbf{r}\;\;\;(cross\;product) $$ You can use the right hand grip rule for the direction of torque. ($\tau$=thumb) From $F=ma:$ $$ \boldsymbol{\tau}=\mathbf{I}\cdot\boldsymbol{\alpha} $$

Work and Power

$$ W=\int dW=\int \tau d\theta=\tau\Delta\theta\;\;\;(If \;\tau\;is\;constant) $$ $$ P=\frac{dW}{dt}=\tau\frac{d\theta}{dt}=\tau\omega $$

Rolling without Slipping

If a body is rolling without slipping: $$ v_{cp}=v_{cm}-\omega R=0 $$ Where $cp$=contact point and $cm$=centre of mass, so $$ V_{cm}=\omega R $$

Angular Momentum

Definition: $$ \mathbf{L}=\sum_i\mathbf{r_i}\times\mathbf{p_i} $$ Angular momentum is conserved in the absence of external torque. If there is a torque then: $$ \frac{d\mathbf{L}}{dt}=\boldsymbol{\tau} $$ If the rotation axis is an axis of symmetry then the angular momentum is: $$ \mathbf{L}=I\boldsymbol{\omega} $$ Precession Angular Velocity $$ \Omega=\frac{\tau_y}{L_x} $$ Gyroscope Equation: $$ \omega_p=\frac{\tau}{L}$$

Object Stability

Conditions for equilibrium: $$ \sum\mathbf{F}=0\;\;\;\&\;\;\;\sum\boldsymbol{\tau}=0 $$ Theorem: The gravitational torque acts completely through the objects centre of mass: $$ \boldsymbol{\tau}_W=M\mathbf{r}_{cm}\times\mathbf{g} $$

Real Bodies

Table of deformations (Young's modulus type stuff):

Type	Stress	Strain	Deformation
Uniaxial (Youngs Modulus)	$\frac{F_{\perp}}{A}$	$\frac{\Delta l}{l_0}$	$Y=\frac{F_{\perp}/A}{\Delta l /l_0}$
Bulk (Bulk Modulus)	$\Delta p$	$\frac{\Delta V}{V_0}$	$B=-\frac{\Delta p}{\Delta V/V_0}$
Shear (Shear Modulus)	$\frac{F_{//}}{A}$	$\frac{x}{h}$	$S=\frac{F_{//} /A}{x/h}$

Kepler's Laws

1st

Planets move in an elliptical orbit with the parent star as one of the two foci.

2nd

The areas swept out by the orbit from time interval $\Delta t$ is identical through all positions of orbit.

3rd

$T^2\propto r^3$, $T^2=\left (\frac{4\pi^2}{GM}\right )r^3$, or $T=\frac{2\pi a^{3/2}}{\sqrt{GM}}$ where $2a$ is the diameter of the widest point of orbit.

Escape Speed

You can use energy conservation. $U+K=\textrm{const}.$ $$v=\sqrt{\frac{2GM}{R}}$$ Following from this, the Schwarzschild radius is: $$R=\sqrt{\frac{2GM}{c^2}}$$

Fluid Mechanics

Pressure depends on depth, as there will be more fluid above an object the further it descends.$$p(y)=-\rho gy+p_0$$

Fluid Equilibrium

Pascal's Principle: A change in pressure applied to an enclosed fluid is transmitted undiminished to all points in the fluid. Archimedes' Principle: When a body is partially or fully immersed, the buoyancy force is equal to the weight of the fluid displaced.

Conservation Of Liquid

The volume of liquid through a point will exit the point. The flow rate is constant. $$\ddot V=A_1v_1=A_2v_2=\textrm{constant}$$

Bernoulli's Equation

Along a streamline of incompressible fluid where energy is conserved: $$p+\rho gy+\frac{1}{2}\rho v^2=\textrm{constant}$$ Awful proof, just learn this lol

Viscosity

$\eta$ …the measure of fluid resistance.

$$F=\eta\frac{vA}{L}$$ Laminar Flow $$\ddot V=(p_{in}-p_{out})\frac{\pi R^4}{8\eta l}$$