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

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

Object Stability

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

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 \[\ddot V=(p_{in}-p_{out})\frac{\pi R^4}{8\eta l}\]