Forest UCM PnCP

Air Resistance (A Damping force that depends on velocity (F(v)))

Newton's second law

Consider the impact on solving Newton's second law when there is an external Force that is velocity dependent

[math]\sum \vec {F}_{ext} = \vec{F}(v) = m \frac{dv}{dt}[/math]

[math]\Rightarrow \int_{v_i}^{v_f} \frac{dv}{F(v)} = \int_{t_i}^{t_f} \frac{dt}{m}[/math]

Frictional forces tend to be proportional to a fixed power of velocity

[math]F(v) \approx v^n[/math]

Linear air resistance (n=1) arises from the viscous drag of the medium through which the object is falling.

Quadratic air resistance (n=2) arises from the objects continual collision with the medium that causes the elements in the medium to accelerate.

Air resistance for rain drops or ball bearings in oil tends to be more linear while canon balls and people falling through the air tends to be more quadratic.

Example: A Sphere moving through air at STP

Linear

[math]F_f = bv = \beta D v = \left ( 1.6 \times 10^{-4} \frac{N \cdot s}{m^2}\right ) D v[/math]

Quadratic

[math]F_f = cv^2 = \gamma D^2 v^2 = \left ( 2.5 \times 10^{-1} \frac{N \cdot s^2}{m^4}\right ) D^2 v^2[/math]

[math]\frac{F_f(\mbox{Quadratic})}{F_f(\mbox{linear})} = \left (1.6 \times 10 ^{2} \frac{s}{m^2} \right ) D v[/math]

Thus in order for the above ratio to be near unity,[math] Dv \lt 10^{-3} \Rightarrow[/math] D is very small like a raindrop and has a small velocity < 1 m/s.

Linear Air Resistance

Forest_UCM_PnCP_LinAirRes

quadratic friction

Forest_UCM_PnCP_QuadAirRes

Another block on incline example

Forest_UCM_NLM_BlockOnIncline

Projecile Motion

Forest_UCM_PnCP_ProjMotion

Charged Particle in uniform B-Field

Forest_UCM_PnCP_QubUniBfield

Forest_Ugrad_ClassicalMechanics