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## Physics ### 8.01 Classical Mechanics ### Massachusetts Institute of Technology * Physics Department ## Equation Sheet ### Kinematics $$\overrightarrow{v}=\frac{d \overrightarrow{r}}{d t}$$ $$\overrightarrow{a}=\frac{d \overrightarrow{v}}{d t}$$ $$v_{x}(t)=v_{x, 0}+\int_{t^{\prime}=0}^{t^...

## Physics ### 8.01 Classical Mechanics ### Massachusetts Institute of Technology * Physics Department ## Equation Sheet ### Kinematics $$\overrightarrow{v}=\frac{d \overrightarrow{r}}{d t}$$ $$\overrightarrow{a}=\frac{d \overrightarrow{v}}{d t}$$ $$v_{x}(t)=v_{x, 0}+\int_{t^{\prime}=0}^{t^{\prime}=t} a_{x}\left(t^{\prime}\right) d t^{\prime}$$ $$x(t)=x_{0}+\int_{t^{\prime}=0}^{t^{\prime}=t} v_{x}\left(t^{\prime}\right) d t^{\prime}$$ $$x(t)=x_{0}+v_{x, 0} t+\frac{1}{2} a_{x} t^{2}$$ $$v_{x}^{2}(t)=v_{x, 0}^{2}+2 a_{x}\left(x(t)-x_{0}\right)$$ ### Polar Coordinates $$\overrightarrow{r}=r \hat{r}$$ $$\overrightarrow{v}=\frac{d \overrightarrow{r}}{d t}=\dot{r} \hat{r}+r \dot{\theta} \hat{\theta}$$ $$\overrightarrow{a}=\frac{d \overrightarrow{v}}{d t}=\left(\ddot{r}-r \dot{\theta}^{2}\right) \hat{r}+(r \ddot{\theta}+2 \dot{r} \dot{\theta}) \hat{\theta}$$ ### Dynamics $$\overrightarrow{F}_{\text {net }}=m \overrightarrow{a}$$ $$\overrightarrow{\mathrm{F}}_{1,2}=-\overrightarrow{\mathrm{F}}_{2,1} \quad \text { Newton's Third Law }$$ ### Types of Forces $$\overrightarrow{\mathrm{F}}_{\text {grav }}=-\mathrm{mg} \hat{\mathrm{j}}$$ $$F_{\text {spring }}=-k \Delta x$$ $$F_{\text {rope }}=T$$ $$f_{\text {static }} \leq \mu_{\mathrm{s}} N$$ $$f_{\text {kinetic }}=\mu_{\mathrm{k}} N$$ $$F_{\text {quad }}=\mathrm{bv}^{2}$$ $$F_{\text {lin }}=\mathrm{bv}$$ ### Momentum & Impulse $$\overrightarrow{\mathrm{p}}=m \overrightarrow{\mathrm{v}}$$ $$\overrightarrow{\mathrm{F}}_{\text {net }}=\frac{d \overrightarrow{\mathrm{p}}}{d t}$$ $$\overrightarrow{\mathrm{J}}=\int_{t_{0}}^{t_{f}} \overrightarrow{\mathrm{F}} d t=\Delta \overrightarrow{\mathrm{p}}$$ ### Work & Energy $$W=\int_{\overrightarrow{r_{i}}}^{\overrightarrow{r_{f}}} \overrightarrow{\mathrm{F}} \cdot d \overrightarrow{\mathrm{r}}$$ $$K=\frac{1}{2} m v^{2}=\frac{p^{2}}{2 m}$$ $$\Delta K=W_{\text {net }}$$ ### Potential Energy $$\Delta U=-W_{\text {conservative }}$$ $$E=K+U$$ $$\Delta E=\Delta K+\Delta U=W_{\text {non-conservative }}$$ $$\mathrm{F}_{\mathrm{X}}=-\frac{d U}{d x}$$ ### Constrained Motion $$N-m g \cos (\theta)=m \frac{v^{2}}{R}$$ $$m g \sin (\theta)=\mathrm{m} \frac{d v}{d t}$$ ### Center of Mass $$\overrightarrow{\mathrm{r}}_{\mathrm{cm}}=\frac{\sum_{i=1}^{N} m_{i} \overrightarrow{\mathrm{r}}_{\mathrm{i}}}{\sum_{i=1}^{N} m_{i}}$$ $$\overrightarrow{\mathrm{v}}_{\mathrm{cm}}=\frac{\sum_{i=1}^{N} m_{i} \overrightarrow{\mathrm{v}}_{\mathrm{i}}}{\sum_{i=1}^{N} m_{i}}$$ $$\overrightarrow{\mathrm{p}}_{\mathrm{cm}}=M \overrightarrow{\mathrm{v}}_{\mathrm{cm}}$$ $$\overrightarrow{\mathrm{F}}_{\text {net ext }}=M \overrightarrow{\mathrm{a}}_{\mathrm{cm}}$$