Mechanics

Energy

Potential (stored) energy, due to gravity or an energy store Kinetic energy, due to a linear or angular velocity

Energy in a closed system must be conserved

Some energy forms are

magnetic (flux and magnetic intensity)
mechanical - kinetic
thermal - heat
light (EM waves)
gravitational potential
chemical - chemical bonds
sound - pressure and velocity
electrical (voltage and current)
elastic potential
nuclear (mass to energy exchange)

Mechanisms to dissipate energy

Heat
Sound
Light
Chemical reaction
Any others?

efforts (force)

force
torque
Shear
pressure
stress

flows (velocity)

position, velocity, acceleration
angular speed, angular acceleration
strain, deflection

Centre of mass, momentum and moment of inertia (vector forms)

Please familiarise yourself with the concept of centre of mass (see youtube links below)

Center of mass: Matt Anderson

Center of mass: Khan Academy

Centre of mass: Doodle Science - GCSE

Newton's equation

Applies to forces acting on a point coincident with the centre of mass of an object

\[ \sum\vec{f}=m\vec{a} \]

Question does the object respond to the force or the acceleration?

Newton's equation can also be considered as the force causing a change in momentum, that is

\[f=\frac{d\vec{p}}{dt}\]

where $\vec{p}=mv$ is the momentum (mass $\times$ velocity) of the object.

Mass $m$ is a (scalar) property of an object in space and dictates the acceleration of that object when a force $\vec{f}$ is applied.
Momentum ($\vec{p}$) can be expressed as a vector and is a conserved property leading to Noether's idea of translational invariance.
Weight is a property of an object in a gravitational field. If that object is not moving then the Weight relates to the force that the gravitational field applies to the mass.

Question What is the weight of a 1Kg mass when it is on the moon where $g=\SI{1.625}{\meter\per\second\squared}$?

Newton-Euler's equation

N-E applies when considering an object responding to a torque by spinning.

\begin{equation} \sum{\vec\tau}=J\dot{\vec\omega}+\vec\omega \times J\vec{\omega}\label{eq:NE} \end{equation}

where $\tau$ is a torque and $\omega$ is an angular velocity.

Also

Moment of inertia $J$ (or sometimes $I$) is a (matrix) property of an object that relates to spinning and dictates the angular acceleration $\dot{\vec\omega}$ of that object when a torque $\vec\tau$ is applied.
Angular momentum $L$(or moment of momentum), is $J\vec\omega$.

A special case applies when an object is spinning around its centre of mass. In this case $J$ is a diagonal matrix, so the term $\vec\omega \times J\vec{\omega}$ vanishes (angle between vectors is 0 hence cross product is 0).

Newton-Eulers equation parallels Newtons equation for linear momentum so $\sum\vec\tau=\frac{d L}{dt}$

Gyroscopic precessions: Veritassium

Bizarre spinning toys: Physics girl

Space-DRUMS handle (Dynamically Responding Ultrasonic Matrix System)

Conserved quantities (Noether's first theorem)

Noether's theorem - named after Emmy Noether - relates conserved quantities to invariance[banados16:_noeth]. See (https://www.discovermagazine.com/the-sciences/how-mathematician-emmy-noethers-theorem-changed-physics)

continuous symmetry	continuity
time (experiments that give the same result today as yesterday)	Conservation of energy/mass
space (things happen the same in different places)	Conservation of momentum
orientation (ditto orientation)	Conservation of angular momentum
wave function/Gauge invariance (e.g. light is the same everywhere)	Conservation of electric charge

	Common symbol	relationships
Energy	$E$	$\frac12 m v^2$ (KE), $\frac12Kx^2$ (PE spring), $mgh$ (PE mass
Linear momentum	$p$	$f=\frac{d p}{dt}$
Angular momentum	$L$	$\vec{L}=J\vec\omega=\vec{r}\times \vec{p}$ $\tau=\frac{dL}{dt}$

Example of conservation of energy and angular momentum

Spheres and cylinders rolling down an incline: Flipping physics

Angular Momentum in gymnastics: OpenStax

W.S. Harwin 3/10/2022