Chapter 10 Key Equations: Rotational Dynamics

Vocabulary flashcards mapping each rotational motion concept and theorem to its corresponding key formula from Chapter 10 of University Physics Volume 1.

Angular position

$$\theta = \frac{s}{r}$$

Angular velocity

$$\omega = \lim_{\Delta t \to 0} \frac{\Delta \theta}{\Delta t} = \frac{d\theta}{dt}$$

Tangential speed

$$v_t = r\omega$$

Angular acceleration

$$\alpha = \lim_{\Delta t \to 0} \frac{\Delta \omega}{\Delta t} = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$$

Tangential acceleration

$$a_t = r\alpha$$

Average angular velocity

$$\bar{\omega} = \frac{\omega_0 + \omega_f}{2}$$

Angular displacement

$$\theta_f = \theta_0 + \bar{\omega}t$$

Angular velocity from constant angular acceleration

$$\omega_f = \omega_0 + \alpha t$$

Angular velocity from displacement and constant angular acceleration

$$\theta_f = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$$

Change in angular velocity

$$\omega_f^2 = \omega_0^2 + 2\alpha(\Delta\theta)$$

Total acceleration

$$\vec{\mathbf{a}} = \vec{\mathbf{a}}_c + \vec{\mathbf{a}}_t$$

Rotational kinetic energy

$$K = \frac{1}{2}\left(\sum_j m_j r_j^2\right)\omega^2$$

Moment of inertia

$$I = \sum_j m_j r_j^2$$

Rotational kinetic energy in terms of the moment of inertia of a rigid body

$$K = \frac{1}{2}I\omega^2$$

Moment of inertia of a continuous object

$$I = \int r^2\,dm$$

Parallel-axis theorem

$$I_{\text{parallel-axis}} = I_{\text{center of mass}} + md^2$$

Moment of inertia of a compound object

$$I_{\text{total}} = \sum_i I_i$$

Torque vector

$$\vec{\boldsymbol{\tau}} = \vec{\mathbf{r}} \times \vec{\mathbf{F}}$$

Magnitude of torque

$$|\vec{\boldsymbol{\tau}}| = r_{\perp} F$$

Total torque

$$\vec{\boldsymbol{\tau}}_{\text{net}} = \sum_i \vec{\boldsymbol{\tau}}_i$$

Newton's second law for rotation

$$\sum_i \tau_i = I\alpha$$

Incremental work done by a torque

$$dW = \left(\sum_i \tau_i\right) d\theta$$

Work-energy theorem

$$W_{AB} = K_B - K_A$$

Rotational work done by net force

$$W_{AB} = \int_{\theta_A}^{\theta_B} \left(\sum_i \tau_i\right) d\theta$$

Rotational power

$$P = \tau\omega$$