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--- urp:physrot [2021-11-05]
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+++ urp:physrot [2021-11-08] (current)
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-=====Rotation and Oscillation=====
+=====Rotation=====
+Tension on rope being swung = force, centripetal force
+F = m * v^2/r
+Example:
+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
+- 4*9.8
-====Spring and Lever====
+https://www.wikihow.com/Calculate-Tension-in-Physics
-**Hooke's law** for springs: F=-kx, k=spring constant, x = displacement
-**Fulcrum**: t = r * f  (torque = radius * force)
+====Pendulum====
-just add the torques for multiple objects on one side of a fulcrum
-====SHM - Simple Harmonic Motion====
+simple pendulum:
-F = ma => ma = -kx
+T = 2π * √(L/g) (T = time, L = length, g = gravity)
+(T/2π)² = L/g
-angular frequency ω = √(k/m)
+potential energy: U = 1/2 kx² (spring), or P = mgh (at mass at some height)
+kinetic energy:   K = 1/2 mv²
-period of oscillation T = 2π √(m/k)  (horizontal or vertical springs)
+Change in potential energy is given by
+U=mgh
+Joule = kg * m²/s²
-In a vertical spring, the weight Mg of the body produces an initial elongation to equilibrium, such that Mg − kyₒ = 0.
+====Pivot====
+  * force on a pivot = moment
+  * moment = f * d (distance)
-If y is the displacement from this equilibrium position the total restoring force will be Mg − k(yₒ + y) = −ky
+  ω (greek letter omega) = angular velocity, measured in rpm, or rads (2π rads in a circle)
+  ω = 2π/T = 2π*f  T = time for full rotation, f = frequency
+    = delta theta / delta t
+  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 for:
+   point mass         I = MR²
+   solid cylinder     I = 1/2 * MR²
+   solid sphere       I = 2/5 * MR²
+   thin shell sphere  I = 2/3 * MR²
+   hoop (around axis) I = MR²
+   hoop (on end?)     I = 1/2 * MR²
+   rod (rotating from one end) I = 1/3 * MR²
+   rod (centered on axis)      I = 1/12 * MR²
+τ = f*r (torque), (technically cross product r x f, or r x (m*α x r)
+τ = I*α
+If an object is a composite object, simply sum the inertial masses together
+τ (torque, Greek tau) = Ia  (a = acceleration), units are Nm (Newton-meters)
+τ = Iα is rotational equivalent to f = ma (many parallels to linear forces, etc)
+Angular Momentum L = Iω
+  If L₁ is angular momentum of ice skater with arms out:
+     the velocity (ω) 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(θ)
+    r = hypotenuse, p is
+cross product of vectors: ->A x ->B = ||->A|| ||->B|| sin(θ)
+dot product of vectors:   ->A . ->B = ||->A|| ||->B|| cos(θ)
+  ||->A|| = magnitude (norm) of vector A, sometimes written with single bars
+**Rotational kinetic energy:**
+Total kinetic energy of a rolling marble is the linear kinetic energy of it moving plus the rotational energy
+  E = 1/2*I*ω²  (similar to E = 1/2 mv² for linear kinetic 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 = Δω/Δt
+ (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:
+) rolling object has combination of rotational and translational motion
+)  "" object rotates around the contact point with the ground, but this point continuously changes ... not as easy concept to grasp
+  distance s = r*θ  (θ in radians)
+  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 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.
+-------
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