Science—Physics 1 Equations

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1-D Motion

$LaTeX: \left. \begin{array}{ll} \Delta x=\frac{1}{2}(v_i+v_f)t \\ v_f=v_i+at \\ \Delta x=v_it+\frac{1}{2}at^{2} \\ v_f^{2}=v_i^{2}+2a\Delta x \end{array} \right \}x-dir\\ \left. \begin{array}{ll} \Delta y=\frac{1}{2}(v_i+v_f)t \\ v_f=v_i+at \\ \Delta y=v_it+\frac{1}{2}at^{2} \\ v_f^{2}=v_i^{2}+2a\Delta y \end{array} \right \}y-dir\\ a=-9.8\frac{m}{s^{2}} \ (free\ fall)$ $\left. \begin{array}{ll} \Delta x=\frac{1}{2}(v_i+v_f)t \\ v_f=v_i+at \\ \Delta x=v_it+\frac{1}{2}at^{2} \\ v_f^{2}=v_i^{2}+2a\Delta x \end{array} \right \}x-dir\\ \left. \begin{array}{ll} \Delta y=\frac{1}{2}(v_i+v_f)t \\ v_f=v_i+at \\ \Delta y=v_it+\frac{1}{2}at^{2} \\ v_f^{2}=v_i^{2}+2a\Delta y \end{array} \right \}y-dir\\ a=-9.8\frac{m}{s^{2}} \ (free\ fall)$

Vectors

$LaTeX: R_{adj}=Rcos\theta\\ R_{opp}=Rsin\theta\\ R=\sqrt{R_{adj}^{2}+R_{opp}^{2} } \\ \theta=tan^{-1}\frac{R_{opp}}{R_{adj}}$ $R_{adj}=Rcos\theta\\ R_{opp}=Rsin\theta\\ R=\sqrt{R_{adj}^{2}+R_{opp}^{2} } \\ \theta=tan^{-1}\frac{R_{opp}}{R_{adj}}$

Projectile Motion

$LaTeX: \left. \begin{array}{ll} v_{xi}=v_icos\theta \\ v_{xi}=v_x \\ \Delta x=v_ysin\theta \\ \end{array} \right \}x-dir\\ \left. \begin{array}{ll} v_{yi}=v_ysin\theta \\ \Delta y=v_{iy}t+\frac{1}{2}at^{2} \\ v_{fy}=v_{iy}+at\\ v_{fy}^{2}=v_{iy}^{2}+2a\Delta y \end{array} \right \}y-dir\\ \theta=tan^{-1}\frac{v_{fy}}{v_{fx}}\ (impact\ angle) \\ \left. \begin{array}{ll} R=\frac{v_i^{2}sin(2\theta)}{g} \\ H=\frac{(v_isin\theta)^{2}}{2g} \\ t=\frac{2v_isin\theta}{g}\\ \end{array} \right \}Special\ Case:\ h_i=h_f\\$ $\left. \begin{array}{ll} v_{xi}=v_icos\theta \\ v_{xi}=v_x \\ \Delta x=v_ysin\theta \\ \end{array} \right \}x-dir\\ \left. \begin{array}{ll} v_{yi}=v_ysin\theta \\ \Delta y=v_{iy}t+\frac{1}{2}at^{2} \\ v_{fy}=v_{iy}+at\\ v_{fy}^{2}=v_{iy}^{2}+2a\Delta y \end{array} \right \}y-dir\\ \theta=tan^{-1}\frac{v_{fy}}{v_{fx}}\ (impact\ angle) \\ \left. \begin{array}{ll} R=\frac{v_i^{2}sin(2\theta)}{g} \\ H=\frac{(v_isin\theta)^{2}}{2g} \\ t=\frac{2v_isin\theta}{g}\\ \end{array} \right \}Special\ Case:\ h_i=h_f\\$

Forces and Torque

$LaTeX: \left. \begin{array}{ll} \sum\overset{\rightharpoonup}{F}=0 \\ \sum{F_x} = 0 \\ \sum {F_y} = 0 \\ \sum {\tau} = 0 \end{array} \right \}static\\ \left. \begin{array}{ll} \sum\overset{\rightharpoonup}{F}=m\overset{\rightharpoonup}{a} \\ \sum{F_x} = ma_x \\ \sum {F_y} = ma_y \\ \sum {\tau} = I\alpha \end{array} \right \}dynamic\\ F_g = mg\ (force\ due\ to\ gravity)\\ \left. \begin{array}{ll} f_s\le\mu_sF_n \\ f_s=\mu_kF_n \\ \end{array} \right \}frictional\ forces\\ \left. \begin{array}{ll} \overset{\rightharpoonup}{R}_{liquid}=-b\overset{\rightharpoonup}{v} \\ \overset{\rightharpoonup}{R}_{air}=\frac{1}{2}D\rho Av^{2} \\ v_{term} =\sqrt[]{\frac{2mg}{D\rho A} } \\ \end{array} \right \}restrictive\ forces\\ F=\frac{Gm_1m_2}{r^{2}} \ (Gravitational\ force\ between\ two\ objects)\\ \tau=Frsin\theta$ $\left. \begin{array}{ll} \sum\overset{\rightharpoonup}{F}=0 \\ \sum{F_x} = 0 \\ \sum {F_y} = 0 \\ \sum {\tau} = 0 \end{array} \right \}static\\ \left. \begin{array}{ll} \sum\overset{\rightharpoonup}{F}=m\overset{\rightharpoonup}{a} \\ \sum{F_x} = ma_x \\ \sum {F_y} = ma_y \\ \sum {\tau} = I\alpha \end{array} \right \}dynamic\\ F_g = mg\ (force\ due\ to\ gravity)\\ \left. \begin{array}{ll} f_s\le\mu_sF_n \\ f_s=\mu_kF_n \\ \end{array} \right \}frictional\ forces\\ \left. \begin{array}{ll} \overset{\rightharpoonup}{R}_{liquid}=-b\overset{\rightharpoonup}{v} \\ \overset{\rightharpoonup}{R}_{air}=\frac{1}{2}D\rho Av^{2} \\ v_{term} =\sqrt[]{\frac{2mg}{D\rho A} } \\ \end{array} \right \}restrictive\ forces\\ F=\frac{Gm_1m_2}{r^{2}} \ (Gravitational\ force\ between\ two\ objects)\\ \tau=Frsin\theta$

Work and Energy

$LaTeX: W=F\Delta x cos\theta\\ \Delta K = \frac{1}{2} m\Delta v^{2}\ (Kinetic\ Energy)\\ \Delta U_g = mg\Delta h\ (Gravitational\ P.E.)\\ \Delta U_s=\frac{1}{2} k\Delta x^{2} \ (Elastic\ P.E.)\\ \left. \begin{array}{ll} W_{net} = \Delta K \\ W_{nc} +W_c=\Delta K \end{array} \right \}Work\ Energy\ Theory\\ W_c=-\Delta U\ (Conservative)\\ W_{nc} =-f_kd\ (Non- Conservative)$ $W=F\Delta x cos\theta\\ \Delta K = \frac{1}{2} m\Delta v^{2}\ (Kinetic\ Energy)\\ \Delta U_g = mg\Delta h\ (Gravitational\ P.E.)\\ \Delta U_s=\frac{1}{2} k\Delta x^{2} \ (Elastic\ P.E.)\\ \left. \begin{array}{ll} W_{net} = \Delta K \\ W_{nc} +W_c=\Delta K \end{array} \right \}Work\ Energy\ Theory\\ W_c=-\Delta U\ (Conservative)\\ W_{nc} =-f_kd\ (Non- Conservative)$

Impulse and Momentum

$LaTeX: \overset{\rightharpoonup}{p}=m\overset{\rightharpoonup}{v}\\ \overset{\rightharpoonup}{I}=\overset{\rightharpoonup}{F}\Delta t\\ \overset{\rightharpoonup}{p_i}=\overset{\rightharpoonup}{p_f}\\ \overset{\rightharpoonup}{p}_{i} =\overset{\rightharpoonup}{p}_{f}\\ m_1v_{2i} +m_2v_{2i} =m_1v_{2f} +m_2v_{2f}\ (inelastic\ and\ elastic)\\ m_1v_{2i} +m_2v_{2i} =(m_1+m_2)v_{f} \ (perfectly\ inelastic)\\ v_{1i} -v_{2i} = -(v_{1f} -v_{2f} )\ (elastic)$ $\overset{\rightharpoonup}{p}=m\overset{\rightharpoonup}{v}\\ \overset{\rightharpoonup}{I}=\overset{\rightharpoonup}{F}\Delta t\\ \overset{\rightharpoonup}{p_i}=\overset{\rightharpoonup}{p_f}\\ \overset{\rightharpoonup}{p}_{i} =\overset{\rightharpoonup}{p}_{f}\\ m_1v_{2i} +m_2v_{2i} =m_1v_{2f} +m_2v_{2f}\ (inelastic\ and\ elastic)\\ m_1v_{2i} +m_2v_{2i} =(m_1+m_2)v_{f} \ (perfectly\ inelastic)\\ v_{1i} -v_{2i} = -(v_{1f} -v_{2f} )\ (elastic)$

Angular Momentum

$LaTeX: \overset{\rightharpoonup}{L}=I\omega\\ \overset{\rightharpoonup}{L}=\overset{\rightharpoonup}{r}\times \overset{\rightharpoonup}{p}\\ \overset{\rightharpoonup}{L}_i=\overset{\rightharpoonup}{L}_f$ $\overset{\rightharpoonup}{L}=I\omega\\ \overset{\rightharpoonup}{L}=\overset{\rightharpoonup}{r}\times \overset{\rightharpoonup}{p}\\ \overset{\rightharpoonup}{L}_i=\overset{\rightharpoonup}{L}_f$

Angular Motion

$LaTeX: \left. \begin{array}{ll} \omega_{av}=\frac{1}{2}(\omega_i+\omega_f) \\ \omega_f=\omega_i+\alpha t \\ \Delta\theta=\omega_it+\frac{1}{2}\alpha t \\ \omega_f^{2}=\omega_i^{2}+2\alpha\Delta\theta \end{array} \right \}Angular\ Equations\ of\ Motion\\ \overset{\rightharpoonup}{r}_{cm} =\frac{\displaystyle\sum\limits_{i=1}^nm_i\overset{\rightharpoonup}{r}_{i} }{m_1+m_2+m_3+...+m_n}\ (Center\ of\ Mass)\\ \left. \begin{array}{ll} v_t=r\omega\\ a_t=r\alpha \\ a_c=\frac{v^{2}}{r} \\ a_c=r\omega^{2} \\ F_c=ma_c \end{array} \right \}Circular\ Motion\\ \left. \begin{array}{ll} I=\displaystyle\sum\limits_{i}^nm_i{r}_i^{2} \\ I_p=I_{cm}+mr^{2} \\ I_z=I_x+I_y \\ \end{array} \right \}Moment\ of\ Inertia$ $\left. \begin{array}{ll} \omega_{av}=\frac{1}{2}(\omega_i+\omega_f) \\ \omega_f=\omega_i+\alpha t \\ \Delta\theta=\omega_it+\frac{1}{2}\alpha t \\ \omega_f^{2}=\omega_i^{2}+2\alpha\Delta\theta \end{array} \right \}Angular\ Equations\ of\ Motion\\ \overset{\rightharpoonup}{r}_{cm} =\frac{\displaystyle\sum\limits_{i=1}^nm_i\overset{\rightharpoonup}{r}_{i} }{m_1+m_2+m_3+...+m_n}\ (Center\ of\ Mass)\\ \left. \begin{array}{ll} v_t=r\omega\\ a_t=r\alpha \\ a_c=\frac{v^{2}}{r} \\ a_c=r\omega^{2} \\ F_c=ma_c \end{array} \right \}Circular\ Motion\\ \left. \begin{array}{ll} I=\displaystyle\sum\limits_{i}^nm_i{r}_i^{2} \\ I_p=I_{cm}+mr^{2} \\ I_z=I_x+I_y \\ \end{array} \right \}Moment\ of\ Inertia$

Moments of Inertia of Homogeneous Rigid Objects

$LaTeX: I_{CM} =MR^2\ (Hoop\ or\ thin\ cylindrical\ shell)\\ I_{CM}=\frac{1}{2} M(R_1^2+R_2^2) \ (Hollow\ cylinder)\\ I_{CM}=\frac{1}{2} MR^2 \ (Solid\ cylinder\ or\ disk)\\ I_{CM}=\frac{1}{12} M(a^2+b^2) \ (Rectangular\ plate)\\ I_{CM}=\frac{1}{12} ML^2\ (Long,\ thin\ rod\ with\ rotation\ axis\ through\ center)\\ I_{CM}=\frac{1}{3} ML^2 \ (Long,\ thin\ rod\ with\ rotation\ axis\ through\ end)\\ I_{CM}=\frac{2}{5} MR^2 \ (Solid\ sphere)\\ I_{CM}=\frac{2}{3} MR^2 \ (Thin\ spherical\ shell)\\$ $I_{CM} =MR^2\ (Hoop\ or\ thin\ cylindrical\ shell)\\ I_{CM}=\frac{1}{2} M(R_1^2+R_2^2) \ (Hollow\ cylinder)\\ I_{CM}=\frac{1}{2} MR^2 \ (Solid\ cylinder\ or\ disk)\\ I_{CM}=\frac{1}{12} M(a^2+b^2) \ (Rectangular\ plate)\\ I_{CM}=\frac{1}{12} ML^2\ (Long,\ thin\ rod\ with\ rotation\ axis\ through\ center)\\ I_{CM}=\frac{1}{3} ML^2 \ (Long,\ thin\ rod\ with\ rotation\ axis\ through\ end)\\ I_{CM}=\frac{2}{5} MR^2 \ (Solid\ sphere)\\ I_{CM}=\frac{2}{3} MR^2 \ (Thin\ spherical\ shell)\\$

Rolling Without Slipping

$LaTeX: \Delta s=r\Delta\theta\\ v_{cm}=r\omega\\ a_{cm}=r\alpha\\ \omega=\frac{v_{cm}}{r}\\ K_r=\frac{1}{2} I\omega^{2}\ (Rolling\ Kinetic\ Energy)\\ K=\frac{1}{2} mv^{2}+\frac{1}{2} I\omega^{2}\ (Total\ Kinetic\ Energy)$ $\Delta s=r\Delta\theta\\ v_{cm}=r\omega\\ a_{cm}=r\alpha\\ \omega=\frac{v_{cm}}{r}\\ K_r=\frac{1}{2} I\omega^{2}\ (Rolling\ Kinetic\ Energy)\\ K=\frac{1}{2} mv^{2}+\frac{1}{2} I\omega^{2}\ (Total\ Kinetic\ Energy)$

Simple Harmonic Motion (SHM)

$LaTeX: F=-kx\ (Hooke's\ Law)\\ a=-\frac{k}{m} x\ (SHM\ acceleration)\\ v=\pm \sqrt{\frac{k}{m} (A^{2}-x^{2})} \ (SHM\ velocity)\\ T=2\pi\sqrt{\frac{m}{k} } \ (Period\ of\ a\ Spring)\\ \omega=\sqrt{\frac{k}{m} } \ (Angular\ Frequency)\\ \left. \begin{array}{ll} x=Acos2\pi ft \\ v=-A\omega sin2\pi ft \\ a= -A\omega^{2} cos2\pi ft \\ \end{array} \right \}SHM\ Equations\ of\ Motion$ $F=-kx\ (Hooke's\ Law)\\ a=-\frac{k}{m} x\ (SHM\ acceleration)\\ v=\pm \sqrt{\frac{k}{m} (A^{2}-x^{2})} \ (SHM\ velocity)\\ T=2\pi\sqrt{\frac{m}{k} } \ (Period\ of\ a\ Spring)\\ \omega=\sqrt{\frac{k}{m} } \ (Angular\ Frequency)\\ \left. \begin{array}{ll} x=Acos2\pi ft \\ v=-A\omega sin2\pi ft \\ a= -A\omega^{2} cos2\pi ft \\ \end{array} \right \}SHM\ Equations\ of\ Motion$