Calculus Based Physics Formulas: Mechanics

April 23, 2007 by aaron

This is just a basic equation list, explanations can be found elsewhere. For the most part derivations are done for you, but it is beneficial to understand how an equation goes from one form to another.

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 : $I_{cm} = \frac{2} {3} mr^2$

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