Dynamics Particle and Rigid Body Motion Cheat Sheet

Key Facts

• Newton’s second law for a particle is $\sum \vec{F} = m\vec{a}$, where the net external force determines the particle acceleration.
• Linear impulse and momentum are related by $\int_{t_1}^{t_2} \sum \vec{F}\,dt = m\vec{v}_2 - m\vec{v}_1$.
• The work-energy equation for a particle is $T_1 + U_{1 \to 2} = T_2$, with translational kinetic energy $T = \frac{1}{2}mv^2$.
• For a rigid body in plane motion, the velocity of point $B$ relative to point $A$ is $\vec{v}_B = \vec{v}_A + \vec{\omega} \times \vec{r}_{B/A}$.
• For a rigid body in plane motion, the acceleration relation is $\vec{a}_B = \vec{a}_A + \vec{\alpha} \times \vec{r}_{B/A} + \vec{\omega} \times (\vec{\omega} \times \vec{r}_{B/A})$.
• The rotational equation of motion about a fixed axis through the mass center is $\sum M_G = I_G\alpha$.
• Rigid-body kinetic energy in plane motion is $T = \frac{1}{2}mv_G^2 + \frac{1}{2}I_G\omega^2$.
• Lagrange’s equation for generalized coordinate $q_i$ is $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = Q_i$, where $L = T - V$.

Vocabulary

Particle

A body whose size and rotation are neglected so its motion is described only by the position, velocity, and acceleration of its mass center.

Rigid body

An idealized body whose particles keep fixed distances from one another during motion.

Mass moment of inertia

The quantity $I = \int r^2\,dm$ that measures how strongly a body resists angular acceleration about an axis.

Instantaneous center of zero velocity

The point in a plane-moving rigid body or its extension that has zero velocity at one instant.

Generalized coordinate

An independent variable $q_i$ chosen to describe the configuration of a system using the fewest convenient coordinates.

Lagrangian

The scalar function $L = T - V$ used in analytical dynamics to derive equations of motion.