Table of Symbols
The symbols used in this document are summarized in the Table of Symbols along with their units. Throughout the document, symbols in bold will represent vectors, and scalars otherwise. The symbols are listed in alphabetical order. For vector quantities, the units shown are for each component of the vector.
| Symbol | Description | Units |
|---|---|---|
| \(a\text{(}t\text{)}\) | Linear acceleration | \(\frac{\text{m}}{\text{s}^{2}}\) |
| \(\boldsymbol{a}\text{(}t\text{)}\) | Acceleration | \(\frac{\text{m}}{\text{s}^{2}}\) |
| \({\boldsymbol{a}\text{(}t\text{)}_{j}}\) | J-Th Body’s Acceleration | \(\frac{\text{m}}{\text{s}^{2}}\) |
| \({C_{\text{R}}}\) | Coefficient of restitution | – |
| \({d_{j}}\) | Distance Between the J-Th Particle and the Axis of Rotation | \({\text{m}}\) |
| \(\boldsymbol{d}\) | Distance between the center of mass of the rigid bodies | \({\text{m}}\) |
| \(\boldsymbol{\hat{d}}\) | Unit vector directed from the center of the large mass to the center of the smaller mass | \({\text{m}}\) |
| \(|\boldsymbol{d}|\) | Euclidean norm of the distance between the center of mass of two bodies | \({\text{m}}\) |
| \({|\boldsymbol{d}|^{2}}\) | Squared distance | \({\text{m}^{2}}\) |
| \(\boldsymbol{F}\) | Force | \({\text{N}}\) |
| \({\boldsymbol{F}_{1}}\) | Force exerted by the first body (on another body) | \({\text{N}}\) |
| \({\boldsymbol{F}_{2}}\) | Force exerted by the second body (on another body) | \({\text{N}}\) |
| \({\boldsymbol{F}_{\boldsymbol{g}}}\) | Force of gravity | \({\text{N}}\) |
| \({\boldsymbol{F}_{j}}\) | Force Applied to the J-Th Body at Time T | \({\text{N}}\) |
| \(G\) | Gravitational constant | \(\frac{\text{m}^{3}}{\text{kg}\text{s}^{2}}\) |
| \(\boldsymbol{g}\) | Gravitational acceleration | \(\frac{\text{m}}{\text{s}^{2}}\) |
| \(h\) | Height | \({\text{m}}\) |
| \(\boldsymbol{I}\) | Moment of inertia | \(\text{kg}\text{m}^{2}\) |
| \({\boldsymbol{I}_{\text{A}}}\) | Moment of Inertia of Rigid Body A | \(\text{kg}\text{m}^{2}\) |
| \({\boldsymbol{I}_{\text{B}}}\) | Moment of Inertia of Rigid Body B | \(\text{kg}\text{m}^{2}\) |
| \(\boldsymbol{J}\) | Impulse (vector) | \(\text{N}\text{s}\) |
| \(j\) | Impulse (scalar) | \(\text{N}\text{s}\) |
| \(KE\) | Kinetic energy | \({\text{J}}\) |
| \(L\) | Length | \({\text{m}}\) |
| \(M\) | Mass of the Larger Rigid Body | \({\text{kg}}\) |
| \(m\) | Mass | \({\text{kg}}\) |
| \({m_{1}}\) | Mass of the first body | \({\text{kg}}\) |
| \({m_{2}}\) | Mass of the second body | \({\text{kg}}\) |
| \({m_{\text{A}}}\) | Mass of Rigid Body A | \({\text{kg}}\) |
| \({m_{\text{B}}}\) | Mass of Rigid Body B | \({\text{kg}}\) |
| \({m_{j}}\) | Mass of the J-Th Particle | \({\text{kg}}\) |
| \({m_{T}}\) | Total Mass of the Rigid Body | \({\text{kg}}\) |
| \(\boldsymbol{n}\) | Collision normal vector | \({\text{m}}\) |
| \(|\boldsymbol{n}|\) | Length of the normal vector | \({\text{m}}\) |
| \(PE\) | Potential energy | \({\text{J}}\) |
| \(\boldsymbol{p}\text{(}t\text{)}\) | Position | \({\text{m}}\) |
| \({\boldsymbol{p}\text{(}t\text{)}_{\text{CM}}}\) | Center of Mass | \({\text{m}}\) |
| \({\boldsymbol{p}\text{(}t\text{)}_{j}}\) | Position Vector of the J-Th Particle | \({\text{m}}\) |
| \(\boldsymbol{r}\) | Position vector | \({\text{m}}\) |
| \(t\) | Time | \({\text{s}}\) |
| \({t_{\text{c}}}\) | Denotes the time at collision | \({\text{s}}\) |
| \(u\text{(}t\text{)}\) | Linear displacement | \({\text{m}}\) |
| \(\boldsymbol{u}\) | Displacement | \({\text{m}}\) |
| \(|{\boldsymbol{u}_{\text{A}\text{P}}}\text{*}\boldsymbol{n}|\) | Length of the Perpendicular Vector to the Contact Displacement Vector of Rigid Body A | \({\text{m}}\) |
| \(|{\boldsymbol{u}_{\text{B}\text{P}}}\text{*}\boldsymbol{n}|\) | Length of the Perpendicular Vector to the Contact Displacement Vector of Rigid Body B | \({\text{m}}\) |
| \({\boldsymbol{u}_{\text{O}\text{B}}}\) | Displacement vector between the origin and point B | \({\text{m}}\) |
| \(v\text{(}t\text{)}\) | Linear velocity | \(\frac{\text{m}}{\text{s}}\) |
| \(\boldsymbol{v}\text{(}t\text{)}\) | Velocity | \(\frac{\text{m}}{\text{s}}\) |
| \(Δ\boldsymbol{v}\) | Change in velocity | \(\frac{\text{m}}{\text{s}}\) |
| \({\boldsymbol{v}\text{(}t\text{)}^{\text{A}\text{P}}}\) | Velocity of the Point of Collision P in Body A | \(\frac{\text{m}}{\text{s}}\) |
| \({\boldsymbol{v}\text{(}t\text{)}^{\text{B}\text{P}}}\) | Velocity of the Point of Collision P in Body B | \(\frac{\text{m}}{\text{s}}\) |
| \({\boldsymbol{v}\text{(}t\text{)}_{1}}\) | Velocity of the First Body | \(\frac{\text{m}}{\text{s}}\) |
| \({\boldsymbol{v}\text{(}t\text{)}_{2}}\) | Velocity of the Second Body | \(\frac{\text{m}}{\text{s}}\) |
| \({\boldsymbol{v}\text{(}t\text{)}_{\text{A}}}\) | Velocity at Point A | \(\frac{\text{m}}{\text{s}}\) |
| \({\boldsymbol{v}\text{(}t\text{)}_{\text{B}}}\) | Velocity at Point B | \(\frac{\text{m}}{\text{s}}\) |
| \({{\boldsymbol{v}\text{(}t\text{)}_{\text{f}}}^{\text{A}\text{B}}}\) | Final Relative Velocity Between Rigid Bodies of A and B | \(\frac{\text{m}}{\text{s}}\) |
| \({{\boldsymbol{v}\text{(}t\text{)}_{\text{i}}}^{\text{A}\text{B}}}\) | Initial Relative Velocity Between Rigid Bodies of A and B | \(\frac{\text{m}}{\text{s}}\) |
| \({\boldsymbol{v}\text{(}t\text{)}_{j}}\) | Velocity of the J-Th Body | \(\frac{\text{m}}{\text{s}}\) |
| \({\boldsymbol{v}\text{(}t\text{)}_{\text{O}}}\) | Velocity at Point Origin | \(\frac{\text{m}}{\text{s}}\) |
| \(α\) | Angular acceleration | \(\frac{\text{rad}}{\text{s}^{2}}\) |
| \({α_{j}}\) | J-Th Body’s Angular Acceleration | \(\frac{\text{rad}}{\text{s}^{2}}\) |
| \(θ\) | Angular displacement | \({\text{rad}}\) |
| \(\boldsymbol{τ}\) | Torque | \(\text{N}\text{m}\) |
| \({\boldsymbol{τ}_{j}}\) | Torque applied to the j-th body | \(\text{N}\text{m}\) |
| \(ω\) | Angular velocity | \(\frac{\text{rad}}{\text{s}}\) |
| \(ϕ\) | Orientation | \({\text{rad}}\) |
Table of Symbols