Moment of inertia

Basic info (click to expand)

• $s = r\theta$ , $\omega = \frac{d\theta}{dt}$ , $\alpha = \frac{d\omega}{dt}$
• Anticlockwise rotation: $\Delta\theta > 0$
• Direction of $\vec\omega$ is defined by the right-hand thumb rule

$$\begin{align*} v_{tan} &= r\omega \\ a_{tan} &= r\alpha \\ a_{rad} &= \frac{v^2}{r} = \omega^2 r \end{align*}$$

SI unit: kg$\cdot$m$^2$

$$\begin{align*} I &= \sum\limits_i m_i r_i^2 = \int r^2dm = \int r^2\rho dr \\ K_{rot} &= \frac{1}{2}I\omega^2 \end{align*}$$

• $\rho = \frac{M}{L}$ (density of the rod)

Moments of inertia of various bodies

Parallel axis theorem

• $I_p$: $I$ about an axis parallel to the axis through its center of mass
• $I_{cm}$: $I$ about an axis through its center of mass
• $M$: total mass of the rigid body
• $d$: perpendicular distance between the two axises

$$\begin{align*} I_p &= I_{cm} + Md^2 \end{align*}$$

Torque

SI unit: N$\cdot$m

$$\begin{align*} \vec\tau &= \frac{d\vec L}{dt} = \vec r \times\vec F \\ \tau &= rF\sin\theta \\ &= I\alpha \\ W &= \int \tau d\theta \\ P &= \tau\omega \end{align*}$$

• $\theta$: angle (anticlockwise) from the radial arm to the line of action
• Direction of $\vec\tau$ is defined by the right-hand thumb rule

Rolling up/down without slipping

• $\vec v_{cm} = r\omega$
• Contact point must be at rest: $\vec v = 0$
• There must be static friction which is always up the slope, but it does no work