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urp:physrot [2021-11-08]
nerf_herder
urp:physrot [2021-11-08] (current)
nerf_herder
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 =====Rotation===== =====Rotation=====
  
- +Tension ​on rope being swung = force, centripetal force
-tension ​on rope being swung = force, centripetal force+
 F = m * v^2/r F = m * v^2/r
-4 kg, 2 meter rope, v = 5 m/s +Example: 
-F = 4 * 25/2 = 50 +  ​4 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 
-     ​50 - 4*9.8 +  ---> 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 https://​www.wikihow.com/​Calculate-Tension-in-Physics
  
--------+====Pendulum====
  
 simple 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 Change in potential energy is given by
 U=mgh U=mgh
-Joule = kg * m^2/s^2+Joule = kg * /
  
-------- +====Pivot==== 
-force on a pivot = moment +  ​* ​force on a pivot = moment 
-moment = f * d (distance) +  ​* ​moment = f * d (distance) 
-(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) 
-2pi/T = 2pi*f  T = time for full rotation, f = frequency +  ​ω ​/T = *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: 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
  
-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)  
-= I*alpha+ 
 +τ = I*α
  
 If an object is a composite object, simply sum the inertial masses together If an object is a composite object, simply sum the inertial masses together
  
-(torque, Greek tau) = Ia  (a = acceleration),​ units are Nm (Newton-meters) +τ (torque, Greek tau) = Ia  (a = acceleration),​ units are Nm (Newton-meters) 
-Ia is rotational equivalent to f = ma (many parallels to linear forces, etc) + 
-Angular Momentum L = Iw +τ Iα is rotational equivalent to f = ma (many parallels to linear forces, etc) 
-  If L1 is angular momentum of ice skater with arms out: + 
-     the velocity (w) is low, but I is big +Angular Momentum L =  
-  If L2 is with skater with arms in: +  If L₁ is angular momentum of ice skater with arms out: 
-     ​velocity is higher, I is smaller.  ​L1 L2 for conservation of energy+     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 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   ||->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:  A number of similar articles on this on one page: 
 https://​sciencing.com/​rotational-kinetic-energy-definition-formula-units-w-examples-13720802.html 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 = angular acceleration) in rad/s^2
-rolling object picks up angular inertia as it accelerates,​ so an object rolling +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  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:  Two ways of looking at it: 
   1) rolling object has combination of rotational and translational motion   1) rolling object has combination of rotational and translational motion
-  2)  ""​ object rotates around the contact point with the ground, ​ +  2)  ""​ 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   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:   velocity of center of mass:
   Vpt = Vrel + Vcm   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: +
- ​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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urp/physrot.1636342431.txt.gz · Last modified: 2021-11-08 by nerf_herder