Notes / QB

Calculus Based Physics Formulas: Mechanics

One dimensional Equations of motion (along a single vector direction)
Velocity as a function of time : $v_{xf} = v_{xi} + a_x t$
Position as a function of time: $x_f = x_i + v_{xi}t + \frac{1}{2} a_x t^2$
Velocity as a function of position: $v^2_{xf} = v^2_{xi} + 2a_x ( x_f - x_i)<br />$

Projectile Motion
Horizontal motion
Velocity along x: $v_{xi} = v_i cos(\theta)$
Position from position as a function of time: $x_f = v_i cos(\theta)t$
Max Horizontal dist: $R = v^2_i \frac{sin( 2 \theta_i)}{g}$

Vertical Motion
Velocity along y: $v_{yi} = v_i sin(\theta)$
Position: from position as a function of time $y_f = y_i + v_{yi}t - \frac{1}{2} g*t^2$
Maximum Height: $h_{max} = v^2_{i} \frac{sin(\theta_i)}{2g}$

Circular Motion
Radial Acc: $a_r = v^2_r = a cos( \theta)$
Tan.

Acc: $a_t = \frac{d \mid \vec {v}\mid}{dt} = a sin(\theta)= r \alpha$
Total Acc (magnitude) from Pythagoras: $a = \sqrt{a_r^2 + a_t^2}$

The Laws of Motion
Newtons Second Law: $\sum{F_{x,y, or z}} = ma_{x,y, or z}$
Equilibrium Conditions: $\sum {F_{x,y, or z}} = 0$
Force of Static Friction $F_{s max} = \mu_s*n$
Force from Kinetic Friction $F_{k max} = \mu_k*n$

Force/Work
Constant Force: $w_{net} = \vec{f_{net}}*\delta r = F * r cos(\theta) = \delta K$
Variable Force: $w_{net} = \int f_{net} d \vec{r}$
Hooke’s Law: $f_s = -k x$
Spring Work: $w = \frac{1}{2} k x_i^2 - \frac{1}{2} k x^2_f$
Kinetic energy: $k = \frac{1}{2} m v^2$
Work – kinetic energy theorem: $w_{net} = \delta k = k_f - k_i, k_f = k_i$
Power: $P = \frac {\Delta w}{\Delta t}$ , $p = \frac {de}{dt}$ , $P = \vec{f} \vec{v}$
gravitational potential energy: $U = mgh$
conservation of mechanical energy: $E = K_f + U_f = K_i + U_i = const. + \mid f_k \delta x \mid$
elastic collision conserved moment and KE: $v_{1f} = (\frac{m_1 - m_2}{m_1 + m_2}) v_{1i} + (\frac {2 m_2}{m_1 + m_2}) v_{2i}$
2d elastic (comp): $m_1 v_{1ix} + m_2 v_{2ix} = m_1 v_{1fx} + m_2 v_{2fx}$ , $m_1 v_{1ix} = m_1 v_{1f} cos(\theta) + m_2 v_{2fx}cos(\phi)$
KE conservation for elastic: $\frac{1}{2} m_i v_i^2 +\frac{1}{2} m_{2i} v_{2i}^2 = \frac{1}{2} m_1 i v_{1f} i^2 + \frac{1}{2} m_2 i v_{2f} i^2$
Momentum: $\vec{P} = m \vec{{v}$

Mass
Center of mass (comp): $x_{cm} = \frac{\sum_{i=1}^{n} m_i x_i }{m}$
Position vector for CM: $\vec{r_{cm}} = x_{cm} \vec {i} + y_{cm} \vec {j} + z_{cm} \vec {k}$
Continuous mas dist: $x_{cm} =\frac {1}{m} \int{\lambda dx}$
Mass of Uniform: $m= \int{\lambda dx}$
Linear Mass Dist: $\lambda = \frac{m}{l} = \frac{dm}{dl}$
Area Mass Dist: $\omega = \frac{m}{a} = \frac{dm}{da}$

Rotational Motion
angular speed: $\omega = \frac{d\theta}{dt}$
angular acceleration: $\alpha = \frac{d\omega}{dt}$ , $\frac{a_t}{r}$ , $\frac{\tau}{I}$
Moment of Inertia: $I = m_i r_i^2$ , $I = \int r^2 dm$ , $I = \int (density)r^2 dv$ , $I = I_cm + mD^2$
Rotational KE: $K_R = \frac{1}{2} I \omega^2$
Work: $\frac{1}{2} I \omega_f^2 - \frac{1}{2} I \omega_i^2$
Net torque : $\sum\tau = I \alpha$ , $\sum \tau =\frac {dL}{dt}$
Work : $W = \int_{\theta_f}^{\theta_i} t d\theta$
Power : $P = \tau \omega$
Angular Momentum : $L = I \omega$
Torque: : $\tau = rF sin(\theta)$

Moments of Inertia
Hoop : $I_{cm} =mr^2$
Cylinder (hollow) : $I_{cm} = \frac{1} {2} m(r_1^2 + r_2^2)$
Cylinder : $I_{cm} = \frac{1} {2} mr^2$
Rectangular Plate : $I_{cm} = \frac{1} {12} m(a^2 + b^2)$
Rod (center rotate): $I_{cm} =\frac{1} {12} mL^2$
Rod (end Rotate): $I_{cm} =\frac{1} {3} mL^2$
Solid Sphere : $I_{cm} = \frac{2} {5} mr^2$
Spherical Shell : $I_{cm} = \frac{2} {3} mr^2$