Rotational Kinematics and Dynamics

Consider a particle moving along the circumference of a circle of radius $r$.

$\Delta s=r\Delta \theta (\theta \text{ must be in radians)}$ $\omega =\frac{d\theta }{dt}$ $\alpha =\frac{d\omega }{dt}=\frac{d^2\theta }{d{t}^2}$

Angular velocity $\omega$ is defined as a vector whose direction is perpendicular to the plane of rotation, according to the right hand rule.

$\omega =\frac{d\theta }{dt}$ Angular acceleration $\alpha$ is a vector $\alpha =\frac{d\omega }{dt}=\frac{d^2\theta }{d{t}^2}$

Tangential Velocity ${\vec{v}}_t$ ${\vec{v}}_t=\vec{R}\times \vec{\omega }\text{(vector form)}$ $v(t)=r\omega (t)$ Tangential Acceleration ${\vec{a}}_t$ ${\vec{a}}_t=\vec{R}\times \vec{\alpha }\text{(vector form)}$ $a(t)=r\alpha (t)$

We can relate angular velocity to what we learned in Circular Motion, where the radial acceleration $a_r$ is $a_r=r{\omega }^2\text{because }\frac{m{v}^2}{r}$

Rotational Kinematics Equations

For Kinematics equations, see [[notes/Kinematics Equations#Rotational Kinematics|Kinematics Equations#Rotational Kinematics]]

Rotational Kinetic Energy

$K=\frac{1}{2}m{v}^2=\frac{1}{2}m(r\omega {)}^2=\frac{1}{2}m{r}^2{\omega }^2=\frac{1}{2}{I}_O{\omega }^2$ where $I_O$ is the Moment of Inertia

Warning

Make sure NOT to double count kinetic energy. In the end, rotational and linear kinetic energy are the same thing.

Related

• Torque
• Angular Momentum
• Circular Motion