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--- urp:physrot [2021-11-08]
nerf_herder
+++ urp:physrot [2021-11-08]
nerf_herder
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 =====Rotation=====
+Tension on rope being swung = force, centripetal force
-tension on rope being swung = force, centripetal force
 F = m * v^2/r
-kg, 2 meter rope, v = 5 m/s
+Example:
-F = 4 * 25/2 = 50
+kg, 2 meter rope, v = 5 m/s
----> now need to subtract gravity, for instance at the top of the swing
+  F = 4 * 25/2 = 50
-- 4*9.8
+  ---> now need to subtract gravity, for instance at the top of the swing
+- 4*9.8
 https://www.wikihow.com/Calculate-Tension-in-Physics
--------
+====Pendulum====
 simple pendulum:
-T = 2pi * (L/g)^0.5 (T = time, L = length, g = gravity)
+T = 2π * √(L/g) (T = time, L = length, g = gravity)
-(T/2pi)^2 = L/g
+(T/2π)² = L/g
-potential energy: U = 1/2 kx^2 (spring), or P = mgh (at mass at some height)
+potential energy: U = 1/2 kx² (spring), or P = mgh (at mass at some height)
-kinetic energy:   K = 1/2 mv^2
+kinetic energy:   K = 1/2 mv²
 Change in potential energy is given by
 U=mgh
-Joule = kg * m^2/s^2
+Joule = kg * m²/s²
--------
+====Pivot====
-force on a pivot = moment
+  * force on a pivot = moment
-moment = f * d (distance)
+  * moment = f * d (distance)
-w (greek letter omega) = angular velocity,
-    measured in rpm, or rads (2pi rads in a circle)
+  ω (greek letter omega) = angular velocity, measured in rpm, or rads (2π rads in a circle)
-w = 2pi/T = 2pi*f  T = time for full rotation, f = frequency
+  ω = 2π/T = 2π*f  T = time for full rotation, f = frequency
-  = delta theta / delta t
+    = delta theta / delta t
-v = rw (v is distance, not radians)
+  v = rω (v is distance, not radians)
+====Moment of Inertia====
+I = moment of inertia (rotational inertia), resistance to angular acceleration, units = kg * m²
+Depends on arrangement of mass about the point of rotation, distance from point of rotation = the radius R (sometime L, or d for distance)
-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
+   point mass         I = MR²
-   solid cylinder     I = 1/2 * MR^2
+   solid cylinder     I = 1/2 * MR²
-   solid sphere       I = 2/5 * MR^2
+   solid sphere       I = 2/5 * MR²
-   thin shell sphere  I = 2/3 * MR^2
+   thin shell sphere  I = 2/3 * MR²
-   hoop (around axis) I = MR^2
+   hoop (around axis) I = MR²
-   hoop (on end?)     I = 1/2 * MR^2
+   hoop (on end?)     I = 1/2 * MR²
-   rod (rotating from one end) I = 1/3 * MR^2
+   rod (rotating from one end) I = 1/3 * MR²
-   rod (centered on axis)      I = 1/12 * MR^2
+   rod (centered on axis)      I = 1/12 * MR²
-t = fr (torque), (technically cross product r x f, or r x (m*alpha x r)
+τ = f*r (torque), (technically cross product r x f, or r x (m*α x r)
-t = I*alpha
+τ = I*α
 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)
+τ (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)
+τ = Iα is rotational equivalent to f = ma (many parallels to linear forces, etc)
-Angular Momentum L = Iw
+Angular Momentum L = Iω
-  If L1 is angular momentum of ice skater with arms out:
+  If L₁ is angular momentum of ice skater with arms out:
-     the velocity (w) is low, but I is big
+     the velocity (ω) is low, but I is big
-  If L2 is with skater with arms in:
+  If L₂ is with skater with arms in:
-     velocity is higher, I is smaller.  L1 = L2 for conservation of energy
+     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
+  L = ->r x ->p (cross product of vectors r and p
-   = r * p * sin(theta)
+    = r * p * sin(θ)
-   r = hypotenuse, p is
+    r = hypotenuse, p is
-cross product of vectors: ->A x ->B = ||->A|| ||->B|| sin(theta)
+cross product of vectors: ->A x ->B = ||->A|| ||->B|| sin(θ)
-dot product of vectors:   ->A . ->B = ||->A|| ||->B|| cos(theta)
+dot product of vectors:   ->A . ->B = ||->A|| ||->B|| cos(θ)
   ||->A|| = magnitude (norm) of vector A, sometimes written with single bars
-rotational kinetic energy:
+**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
- total kinetic energy of a rolling marble is the linear kinetic energy of it moving
+  E = 1/2*I*ω²  (similar to E = 1/2 mv² for linear kinetic energy)
-  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
+**Tangential acceleration** = acceleration * radius
-a = delta w/delta t
+a = Δω/Δt
  (a = angular acceleration) in rad/s^2
-a rolling object picks up angular inertia as it accelerates, so an object rolling
+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
- 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
+**Translational motion:** movement of the center of mass for a rolling object
 Two ways of looking at it:
 ) rolling object has combination of rotational and translational motion
-)  "" object rotates around the contact point with the ground,
+)  "" object rotates around the contact point with the ground, but this point continuously changes ... not as easy concept to grasp
-     but this point continuously changes ... not as easy concept to grasp
-distance s = r*theta  (theta in radians)
+  distance s = r*θ  (θ in radians)
-  v = delta theta/delta time,  (velocity of center of mass)
+  v = Δ θ/Δ 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 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
+  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 = 2Vcm. Conversely, when in contact with the ground, Vrel = -Vcm,
- so vPt = 0.
+  so vPt = 0.
-----------
+====Optics====
-Optics:
+Refraction on going into a different medium
- refraction on going into a different medium
- Snell's law:
+**Snell's law**
-   sin(theta1) / sin(theta2) = v1/v2 = n2/n1  (note that the n values are reversed)
+   sin(θ₁) / sin(θ₂) = 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)
+   Critical angle : smallest angle that results in total reflection, no refraction
+   θc = arcsin(n₂/n₁)
 -------
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