# The Ungerecht Family Tree

urp:physrot

## Rotation

Tension on rope being swung = force, centripetal force F = m * v^2/r Example:

```4 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
50 - 4*9.8 ```

### Pendulum

simple pendulum: T = 2π * √(L/g) (T = time, L = length, g = gravity) (T/2π)² = L/g

potential energy: U = 1/2 kx² (spring), or P = mgh (at mass at some height) kinetic energy: K = 1/2 mv²

Change in potential energy is given by U=mgh Joule = kg * m²/s²

### 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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