PHYS 121 Mechanics
Contents
Kinematics
| Quantity | Unit | Definition |
|---|---|---|
| Displacement | $\mathrm{m}$ | $\Delta \vec{r} = \vec{r}_{f} - \vec{r}_{i}$ |
| Instantaneous velocity | $\mathrm{m/s}$ | $\vec{v} = \dfrac{d\vec{r}}{dt}$ |
| Instantaneous acceleration | $\mathrm{m/s^{2}}$ | $\vec{a} = \dfrac{d\vec{v}}{dt} = \dfrac{d^{2}\vec{x}}{dt^{2}}$ |
| Description | Equations |
|---|---|
| Kinematics equations at constant acceleration | $x = x_{0} + v_{0}t + \dfrac{1}{2}at^{2}$ $v = v_{0} + at$ $v_{f}^{2} = v_{i}^{2} + 2a\Delta x$ |
| Relative velocity | $\vec{v}_{P/A} = \vec{v}_{P/B} + \vec{v}_{B/A}$ |
| Centripetal acceleration | $a_{\mathrm{rad}} = \dfrac{v^{2}}{R} = \dfrac{4\pi^{2}R}{T^{2}}$ |
Dynamics
| Quantity | Unit | Definition |
|---|---|---|
| Spring force (Hooke’s law) | $\mathrm{N}$ | $F_{s} = -k\Delta x$ |
| Static friction | $\mathrm{N}$ | $f_{s} \leq (f_{s})_{\mathrm{max}} = \mu_{s} F_{N}$ $\mu_{s} = \tan(\theta)$ |
| Kinetic friction | $\mathrm{N}$ | $f_{k} = \mu_{k} F_{N}$ |
| Gravitational force near Earth surface | $\mathrm{N}$ | $\vec{F}_{g} = m\vec{g}$ $g = 9.81 \mathrm{m/s^{2}}$ |
| Description | Equations |
|---|---|
| Newton’s first law moving at constant velocity | $\sum \vec{F}_{\mathrm{ext}} = \vec{0}$ |
| Newton’s second law | $\sum \vec{F}_{\mathrm{ext}} = m\vec{a}$ |
| Newton’s third law | $\vec{F}_{AB} = - \vec{F}_{BA}$ |
| Acceleration on an inclined plane | $a = g\sin\theta$ |
Energy
| Quantity | Unit | Definition |
|---|---|---|
| Work | $\mathrm{J}$ | $W = \vec{F} \cdot \vec{x} = Fx\cos\theta$ $W = \displaystyle\int_{x_{1}}^{x_{2}} F \ dx$ |
| Kinetic energy | $\mathrm{J}$ | $K = \dfrac{1}{2}mv^{2}$ |
| Power | $\mathrm{W}$ | $P = \dfrac{dW}{dt} = \dfrac{dE}{dt}$ $P = \vec{F} \cdot \vec{v}$ |
| Gravitational potential energy | $\mathrm{J}$ | $U = mgh$ $W_{\mathrm{grav}} = - \Delta U_{\mathrm{grav}}$ |
| Elastic potential energy | $\mathrm{J}$ | $U = \dfrac{1}{2}kx^{2}$ $W_{\mathrm{el}} = - \Delta U_{\mathrm{el}}$ |
| Reduced mass | $\mathrm{kg}$ | $\mu = \dfrac{m_{1}m_{2}}{m_{1} + m_{2}}$ |
| Coefficient of restitution | - | $e = -\dfrac{v_{12, f}}{v_{12, i}}$ |
| Description | Equations |
|---|---|
| Work-energy theorem | $W_{\mathrm{total}} = \Delta K$ |
| Conservation of mechanical energy | $K_{i} + U_{i} = K_{f} + U_{f}$ |
| Energy of system with external force (non-isolated system) | $K_{i} + U_{i} + W = K_{f} + U_{f}$ |
| Conservation of energy | $\Delta K + \Delta U + \Delta U_{int} = 0$ |
| Force as a function of potential energy | $\vec{F} = -\vec{\nabla} U$ $F = -\dfrac{dU}{dx}$ |
Momentum
| Quantity | Unit | Definition |
|---|---|---|
| Momentum | $\mathrm{kg\cdot m/s}$ | $\vec{p} = m\vec{v}$ $\sum \vec{F}_{\mathrm{ext}} = \dfrac{d\vec{p}}{dt}$ |
| Impulse | $\mathrm{kg\cdot m/s}$ | $\vec{J} = \sum \vec{F} \Delta t$ $\vec{J} = \displaystyle\int_{t_{1}}^{t_{2}} \sum \vec{F} \ dt$ |
| Center of mass | $\mathrm{m}$ | $\vec{r}_{\mathrm{cm}} = \dfrac{\sum\limits_{i} m_{i}\vec{r}_{i}}{\sum\limits_{i} m_{i}}$ |
| Description | Equations |
|---|---|
| Impulse-momentum theorem | $\vec{J} = \Delta \vec{p}$ |
| Conservation of momentum (closed system) | $\vec{p}_{i} = \vec{p}_{f}$ $\sum \vec{F}_{\mathrm{ext}} = \dfrac{d\vec{p}}{dt}$ |
| Force on extended body | $\sum \vec{F}_{\mathrm{ext}} = m\vec{a}_{\mathrm{cm}}$ |
Rotational Kinematics
| Quantity | Unit | Definition |
|---|---|---|
| Angular displacement | $\mathrm{rad}$ | $\Delta \theta = \theta_{f} - \theta_{i}$ |
| Angular velocity | $\mathrm{rad/s}$ | $\omega_{z} = \dfrac{d\theta}{dt}$ |
| Angular acceleration | $\mathrm{rad/s^{2}}$ | $\alpha_{z} = \dfrac{d\omega_{z}}{dt} = \dfrac{d^{2}\theta}{dt^{2}}$ |
| Rotational Inertia of particle | $\mathrm{kg\cdot m^{2}}$ | $I = \sum\limits_{i} m_{i}r_{i}^{2}$ |
| Rotational kinetic energy | $\mathrm{J}$ | $K = \dfrac{1}{2}I\omega^{2}$ |
| Description | Equations |
|---|---|
| Rotational kinematics equation with constant angular acceleration | $\theta = \theta_{0} + \omega_{0z}t + \dfrac{1}{2}\alpha_{z} t^{2}$ $\omega_{z} = \omega_{0z} + \alpha_{z}t$ $\omega_{fz}^{2} = \omega_{iz}^{2} + 2\alpha_{z}\Delta\theta$ |
| Relationship between linear kinematics and rotational kinematics | $s = r\theta$ $v = r\omega$ $a_{\mathrm{tan}} = r\alpha$ $a_{\mathrm{rad}} = \dfrac{v^{2}}{r} = \omega^{2}r$ |
| Parallel-axis theorem | $I_{\mathrm{parallel}} = I_{\mathrm{cm}} +md^2$ |
Rotational Dynamics
| Quantity | Unit | Definition |
|---|---|---|
| Torque | $\mathrm{N\cdot m}$ | $\vec{\tau} = \vec{r} \times \vec{F} = Fr\sin\theta$ $\sum \vec{\tau} = \dfrac{d\vec{L}}{dt}$ |
| Angular momentum of a particle | $\mathrm{kg\cdot m^{2}/s}$ | $\vec{L} = \vec{r} \times \vec{p} = mvr\sin\theta$ |
| Angular momentum of rotating body | $\mathrm{kg\cdot m^{2}/s}$ | $\vec{L} = I \vec{\omega}$ |
| Description | Equations |
|---|---|
| Rotational Newton’s second law | $\sum \tau = I\alpha_{z}$ |
| Condition of mechanical equilibrium | $\sum \vec{F}_{\mathrm{ext}} = m\vec{a}$ $\sum \tau = I\alpha_{z}$ |
| Total kinetic energy of rotating and translating object | $K = \dfrac{1}{2}mv^{2}_{\mathrm{cm}} + \dfrac{1}{2} I_{\mathrm{cm}} \omega^{2}$ |
| Rolling without slipping | $v_{\mathrm{cm}} = R\omega$ |
| Slipping (only rolling) | $v_{\mathrm{cm}} < R\omega$ |
| Skidding (only translating) | $v_{\mathrm{cm}} > R\omega$ |
| Rotational Work | $W = \tau_{z}\Delta\theta$ $W = \displaystyle\int_{\theta_{1}}^{\theta_{2}} \tau_{z} \ d\theta$ $W = \Delta K_{\mathrm{rot}}$ |
| Power | $P = \dfrac{dW}{dt}$ $P = \tau_{z}\omega_{z}$ |
| Conservation of angular momentum (closed system) | $\vec{L}_{i} = \vec{L}_{f}$ $\sum \vec{\tau} = \dfrac{d\vec{L}}{dt}$ |
Universal Gravitation
| Quantity | Unit | Definition |
|---|---|---|
| Gravitational force | $\mathrm{N}$ | $F_{g} = G\dfrac{m_{1}m_{2}}{r^{2}}$ |
| Gravitational acceleration | $\mathrm{m/s^{2}}$ | $g = G\dfrac{m_{E}}{r^{2}}$ |
| Gravitational potential energy | $\mathrm{J}$ | $U = - G\dfrac{m_{E}m}{r}$ |
| Description | Equations |
|---|---|
| Escape velocity | $v_{\mathrm{escape}} = \sqrt{\dfrac{2Gm_{E}}{R}}$ |
| Velocity in circular orbit | $v_{\mathrm{circ}} = \sqrt{\dfrac{Gm_{E}}{R}} = \dfrac{2\pi R}{T}$ |
| Period in circular orbit | $T = \dfrac{2\pi R}{v} = 2\pi R\sqrt{\dfrac{R}{Gm_{E}}} = \dfrac{2\pi r^{3/2}}{\sqrt{Gm_{E}}}$ |