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urp:physrot [2021-11-05]
nerf_herder
urp:physrot [2021-11-08] (current)
nerf_herder
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-====Rotation ​and Oscillation====+=====Rotation=====
  
-====Spring and Lever==== +Tension on rope being swung force, centripetal force 
-**Hooke'​s law** for springs: F=-kxk=spring constant, x = displacement+F = m v^2/r 
 +Example: 
 +  4 kg, 2 meter rope, v = 5 m/s 
 +  ​F = 4 * 25/2 = 50 
 +  ​---> now need to subtract gravityfor instance at the top of the swing 
 +    50 - 4*9.8 
  
-**Fulcrum**t = r * f  (torque = radius * force) +https://​www.wikihow.com/​Calculate-Tension-in-Physics
-just add the torques for multiple objects on one side of a fulcrum+
  
-====SHM - Simple Harmonic Motion==== +====Pendulum====
-ma = -kx+
  
-angular frequency ω = √(k/m)+simple pendulum: 
 +2π * √(L/g(T = time, L = length, g = gravity) 
 +(T/2π)² = L/g
  
-period of oscillation T = 2π √(m/k ​(horizontal ​or vertical springs)+potential energy: U 1/kx² (spring)or P = mgh (at mass at some height) 
 +kinetic energy: ​  K = 1/2 mv²
  
-In a vertical spring, the weight Mg of the body produces an initial elongation to equilibrium,​ such that Mg − kyₒ 0.+Change in potential energy is given by 
 +U=mgh 
 +Joule = kg * m²/s²
  
-If y is the displacement from this equilibrium position the total restoring ​force will be Mg − k(yₒ + y= −ky+====Pivot==== 
 +  * force on a pivot = moment 
 +  * moment = f * d (distance)
  
 +  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: 
 +  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*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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urp/physrot.1636076726.txt.gz · Last modified: 2021-11-05 by nerf_herder