The Ungerecht Family Tree

tension on rope being swung = force, centripetal force F = m * v^2/r 4 kg, 2 meter rope, v = 5 m/s F = 4 * 25/2 = 50 —> now need to subtract gravity, for instance at the top of the swing

   50 - 4*9.8 

https://www.wikihow.com/Calculate-Tension-in-Physics

simple pendulum: T = 2pi * (L/g)^0.5 (T = time, L = length, g = gravity) (T/2pi)^2 = L/g

potential energy: U = 1/2 kx^2 (spring), or P = mgh (at mass at some height) kinetic energy: K = 1/2 mv^2

Change in potential energy is given by U=mgh Joule = kg * m^2/s^2

force on a pivot = moment moment = f * d (distance) w (greek letter omega) = angular velocity,

  measured in rpm, or rads (2pi rads in a circle)

w = 2pi/T = 2pi*f T = time for full rotation, f = frequency

= delta theta / delta t

v = rw (v is distance, not radians)

I = moment of inertia (rotational inertia), resistance to angular acceleration

depends on arrangement of mass about the point of rotation
distance from point of rotation = the radius R (sometime L, or d for distance)
 units = kg * m^2

I for:

 point mass         I = MR^2
 solid cylinder     I = 1/2 * MR^2
 solid sphere       I = 2/5 * MR^2
 thin shell sphere  I = 2/3 * MR^2
 hoop (around axis) I = MR^2
 hoop (on end?)     I = 1/2 * MR^2
 rod (rotating from one end) I = 1/3 * MR^2
 rod (centered on axis)      I = 1/12 * MR^2

t = fr (torque), (technically cross product r x f, or r x (m*alpha x r) t = I*alpha

If an object is a composite object, simply sum the inertial masses together

t (torque, Greek tau) = Ia (a = acceleration), units are Nm (Newton-meters) t = Ia is rotational equivalent to f = ma (many parallels to linear forces, etc) Angular Momentum L = Iw

If L1 is angular momentum of ice skater with arms out:
   the velocity (w) is low, but I is big
If L2 is with skater with arms in:
   velocity is higher, I is smaller.  L1 = L2 for conservation of energy

Oddly, can also have angular momentum of a linearly moving object past another object L = →r x →p (cross product of vectors r and p

 = r * p * sin(theta)
 r = hypotenuse, p is 

cross product of vectors: →A x →B = ||→A|| ||→B|| sin(theta) dot product of vectors: →A . →B = ||→A|| ||→B|| cos(theta)

||->A|| = magnitude (norm) of vector A, sometimes written with single bars

rotational kinetic energy:

E = 1/2*I*w^2  (similar to E = 1/2 mv^2 for linear kinetic energy)

total kinetic energy of a rolling marble is the linear kinetic energy of it moving

plus the rotational energy

A number of similar articles on this on one page: https://sciencing.com/rotational-kinetic-energy-definition-formula-units-w-examples-13720802.html

tangential acceleration = acceleration * radius a = delta w/delta t (a = angular acceleration) in rad/s^2 a rolling object picks up angular inertia as it accelerates, so an object rolling down an incline will have a final velocity less than a frictionless object that does not roll See: https://www.asc.ohio-state.edu/gan.1/teaching/spring99/C12.pdf

translational motion: movement of the center of mass for a rolling object Two ways of looking at it:

1) rolling object has combination of rotational and translational motion
2)  "" object rotates around the contact point with the ground, 
   but this point continuously changes ... not as easy concept to grasp

distance s = r*theta (theta in radians)

v = delta theta/delta time,  (velocity of center of mass)
v = rw

velocity of a point on a disk is velocity relative to center of mass, plus

velocity of center of mass:
Vpt = Vrel + Vcm

If disk is rolling on ground, when point is at the top of the disk, Vrel = Vcm so Vpt = 2Vcm. Conversely, when in contact with the ground, Vrel = -Vcm, so vPt = 0.

Optics: refraction on going into a different medium Snell's law:

 sin(theta1) / sin(theta2) = v1/v2 = n2/n1  (note that the n values are reversed)
 v = velocity of light in that medium, n = index of refraction
 v = c/n  (c = speed of light in a vacuum)
 it bends towards the normal direction when entering denser material
 (and slows down). bend is because photons are waves.

Critical angle : smallest angle that results in total reflection, no refraction

 thetaC = arcsin(n2/n1)

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