Rotations

Rotation

Centripetal Force

$$ \large F = ma = \frac{mv^2}{r^2} = mr\omega^2 $$

where

$v = \omega r$

$\omega$: angular velocity.

Centrifugal Force

Angular Momentum

$$ \vec{L} = m\vec{v}r $$

Angular Momentum Conservation

Rotational Kinetic Energy

$$ E_\text{rot} = \frac{1}{2} I \omega^2 $$

where

$I$: moment of inertia.

Moment of Inertia

Parallel Axis Theorem

$$ I = I_{\text{cm}} + md^2 $$
  • GRE Physics GR0177 Problems 025 and 089

Torque

$$ \vec{\tau} = I \vec{\alpha} = \vec{r} \times \vec{F} $$

where $\vec{\alpha}$ is the angular acceleration.

Rotation Matrix

Rotation about z-axis of a coordinate system by an angle of $\theta$.

$$ \large R_z = \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix} $$
  • GRE Physics GR0177 Problem 075

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