# 7.8: Summary of Linear and Angular Analogs

• • Dina Zhabinskaya
• UC Davis
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# Putting it all together

The chart on below shows all of the linear motion and dynamic variables along with their rotational counterparts. Keep this chart out and handy for ready reference to help you from getting “lost” in all the symbols. You should make sure that you recognize the meaning behind the symbols when you see on of these relationships.

Summary Listing Fundamental Concepts Used in Mechanics Emphasizing Translational and Rotational Counterparts
Category Concept Translation Rotation Relation
Kinematic Variables

Position

Velocity

Acceleration

$$x$$

$$v=\dfrac{dx}{dt}$$

$$a=\dfrac{dv}{dt}$$

$$\theta$$

$$\omega=\frac{d\theta}{dt}$$

$$\alpha=\frac{d\omega}{dt}$$

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

$$\omega=\frac{v}{r}$$

$$\alpha=\frac{a}{r}$$

Fundamental Dynamic Variables

Force/Torque

Mass/Inertia

Momentum

Impulse

Momentum-Impulse

$$F$$

$$m$$

$$p=mv$$

$$J=\int Fdt$$

$$J_{ext}=\Delta p$$

$$\tau$$

$$I$$

$$L=I\omega$$

$$\textrm{ang}~J=\int \tau dt$$

$$\textrm{ang}~ J_{ext}=\Delta L$$

$$\tau=rF_{\perp}$$

$$I=\sum mr^2$$

$$L=rp_{\perp}$$

Newton's Laws

First Law

Second Law

Third Law

$$\textrm{if}~ F_{net}=0, \textrm{then}~\Delta p=0$$

$$F_{net}=ma ~\textrm{or}~ F_{net}=\frac{dp}{dt}$$

$$F_{1~on~2}=-F_{2~on~1}$$
$$J_{1~on~2}=-J_{2~on~1}$$

$$\textrm{if}~ \tau_{net}=0, \textrm{then}~\Delta L=0$$

$$\tau_{net}=I\alpha ~\textrm{or}~\tau_{net}=\frac{dL}{dt}$$

$$\tau_{1~on~2}=-\tau_{2~on~1}$$
$$\textrm{ang}~J_{1~on~2}=-\textrm{ang}~J_{2~on~1}$$

Energy

Kinetic Energy

Work

$$KE=\frac{1}{2}mv^2$$

$$W=\int \limits_{x_1}^{x_2} \vec F \cdot d\vec s$$

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

$$W=\int \limits_{\theta_1}^{\theta_2} \vec\tau \cdot d\vec\theta$$

# Contributors

This page titled 7.8: Summary of Linear and Angular Analogs is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Dina Zhabinskaya.