Calculus Based Physics Formulas: Mechanics

Posted on Friday the 23rd of March, 2007 at 9:52 am in Physics

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 timey_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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  1. wps posted the following on April 7, 2007 at 4:47 pm.

    Thanks, although the list would be easier to read if it as in a table or something. Also, the position vector for CM formula shows “hatted” i, j and k not vectors.

    Reply to wps
  2. Aaron posted the following on April 7, 2007 at 7:00 pm.

    Thanks I’ll fix the vectors bug.

    Reply to Aaron
  3. Syed Noman posted the following on December 24, 2007 at 1:08 pm.

    hi
    thanks alot for providin such importan informatoin in ashort way
    i would very very glad if you sen me the description if the (mechanic physics formula)

    best regards
    Syed Noman

    Reply to Syed Noman

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