---
# System prepended metadata

title: Roll & Angular Momentum
tags: [mechanics, "classic\_"]

---

# Roll & Angular Momentum
## Cylinder rolling
![image](https://hackmd.io/_uploads/HJhM3exEel.png)

* center
$$v_{com} = 𝜔r$$
* vertex
$$v_{top} = 2𝜔r = 2v_{com}$$
* energy
$$\frac{1}{2}mv_{com}^2+\frac{1}{2}I_{com}^2$$
## Down hill rolling
![image](https://hackmd.io/_uploads/H1C7nll4xg.png)

$$f_s = \mu mg\cdot cos\theta$$

$$F_{net} = -f_s+mg\cdot sin\theta$$

$$F_{net} = -\mu mg\cdot cos\theta-mg\cdot sin\theta$$
:::success
$$a = g(sin\theta-\mu cos\theta)$$
:::
* rolling moment
$$𝜏 = Fr = f_sr = r\mu mg\cdot cos\theta = I𝛼$$

$$f_s = \frac{Ia}{R^2}$$

$$a = (mg\cdot sin\theta-\frac{Ia}{R^2})/m$$
:::success
$$a = g\cdot sin\theta-\frac{Ia}{mR^2}$$
:::
## YO-YO
<div style="text-align: center;">
<img src="https://hackmd.io/_uploads/Hk643gxNlx.png" alt="image"width="300">
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$$𝜏 = Fr = f_sr = r\mu mg\cdot cos\theta = I𝛼$$

$$f_s = \frac{Ia}{R^2}$$

$$F_{net} = mg-\frac{Ia}{R^2}$$
:::success
$$a = F_{net}/m = \frac{Ia}{mR^2}+g$$
:::
## Angular Momentum
$$\vec L = \vec r\times\vec p = m(\vec r\times\vec v)$$

$$L = rmvsin\theta$$
* if $\theta = \frac{\pi}{2}$
$$L = rp = rmv = I𝜔$$

$$𝜏 = \frac{dL}{dt}$$

$$F = \frac{dL}{rdt}$$


## Conservation of angular momentum
$$L_i = L_f$$

$$I_i𝜔_i = I_f𝜔_f$$

## Precession & Nutation
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<img src="https://hackmd.io/_uploads/rkdT2lx4ex.png" alt="image"width="500">
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angular velocity of precession : $𝜔_p$

angular velocity of spin : $𝜔_s$

The distance from the center of mass to the axis of rotation : $r$

* Angular velocity of precession

$$𝜔_p = \frac{d\phi}{dt}$$

$$𝜏 = L_s𝜔_p$$

$$𝜏 = mgr\cdot sin\theta$$

$$mgr\cdot sin\theta = I𝜔_s𝜔_p$$
:::success
$$𝜔_p = \frac{mgr\cdot sin\theta}{I𝜔_s}$$
$$T_p = 2\pi/𝜔_p = \frac{2\pi I𝜔_s}{mgr\cdot sin\theta} = 
\frac{4\pi^2 I}{T_s𝜏}$$
:::
* Angular velocity of spin

if $𝜔_p >>$ 0 and $\theta = 90^o$
$$𝜏+L_p𝜔_p = L_s𝜔_p$$

$$𝜏 = L_s𝜔_p-L_p𝜔_p$$

$$mgr= I_s𝜔_s𝜔_p-I_p𝜔_p^2$$

$$𝜔_p = \frac{-I_s𝜔_s\pm\sqrt{(I_s𝜔_s)^2-4I_pmgr}}{2I_p}$$

If precession is stable

$$(I_s𝜔_s)^2-4I_pmgr>0$$
:::success
$$𝜔_s>\frac{2\sqrt{I_pmgr}}{I_s}$$
:::
* Angular momentum

$$L_{orb} = mv_{com}r = m𝜔_pr^2$$

$$L_{spin} = I_s𝜔_s+I_p𝜔_p$$

moment of inertia in spin : $I_s$

moment of inertia in Precession : $I_p$

$$L = I_s𝜔_s+I_p𝜔_p+m𝜔_pr^2$$

* Energy
$$E = U+K$$

$$U = mgr\cdot cos\theta$$

$$K = \frac{1}{2}I_s𝜔_s^2+\frac{1}{2}I_p𝜔_p^2+\frac{1}{2}mr^2𝜔_p^2$$

* Nutation
![image](https://hackmd.io/_uploads/SyzHpelEeg.png)
