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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 * 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: | 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 | 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) | + | |
- | 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 = Iω |
- | 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 | ||
- | 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 | 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) | + | |
------- | ------- | ||
Back to [[Physics]] page or [[00_start|Start]] page. | Back to [[Physics]] page or [[00_start|Start]] page. |