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 &#8211; x_i)<br />&#10;

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 &#8211; \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


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  &#8211;  \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 &#8211; 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 &#8211; 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}


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 &#8211; \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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