Difference between revisions of "Forest UCM PnCP"

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=Air Resistance (A Damping force that depends on velocity (F(v)))=
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==Newton's second law==
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Consider the impact on solving Newton's second law when there is an external Force that is velocity dependent
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:<math>\sum \vec {F}_{ext} = \vec{F}(v) = m \frac{dv}{dt}</math>
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: <math>\Rightarrow  \int_{v_i}^{v_f} \frac{dv}{F(v)} = \int_{t_i}^{t_f} \frac{dt}{m}</math>
  
=Charged Particle in uniform B-Field=
 
  
Consider a charged particle moving the x-y plane in the presence of a uniform magnetic field with field lines in the z-dierection.
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Frictional forces tend to be proportional to a fixed power of velocity
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: <math>F(v) \approx v^n</math>
  
:<math>\vec{v} = v_x \hat i + v_y \hat j</math>
 
:<math>\vec{B} = B \hat k</math>
 
  
  
;Lorentz Force
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Linear air resistance (n=1) arises from the viscous drag of the medium through which the object is falling.
  
:<math>\vec{F} = q \vec{E} + q\vec{v} \times \vec{B}</math>
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Quadratic air resistance (n=2) arises from the objects continual collision with the medium that causes the elements in the medium to accelerate.
  
;Note: the work done by a magnetic field is zero if the particle's kinetic energy (mass and velocity) don't change.
 
:<math>W = \Delta K.E.</math>
 
  
No work is done on a charged particle force to move in a fixed circular orbit by a magnetic field (cyclotron)
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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.
  
  
:<math>\vec{F} = m \vec{a} = q \vec{v} \times \vec{B} = q\left ( \begin{matrix} \hat i  & \hat j & \hat k \\ v_x  & v_y &0 \\ 0 &0 & B  \end{matrix} \right )</math>
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ExampleA Sphere moving through air at STP
:<math>\vec{F} = q \left (v_y B \hat i - v_x B \hat j \right )</math>
 
  
==Apply Newton's 2nd Law==
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;Linear:
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:<math>F_f = bv = \beta D v = \left ( 1.6 \times 10^{-4} \frac{N \cdot s}{m^2}\right ) D v</math>
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;Quadratic:
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:<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>ma_x = qv_yB</math>
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:<math>\frac{F_f(\mbox{Quadratic})}{F_f(\mbox{linear})} = \left (1.6 \times 10 ^{2} \frac{s}{m^2} \right ) D v</math>
:<math>ma_y = -qv_x B</math>
 
:<math>ma_z = 0</math>
 
  
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Thus in order for the above ratio to be near unity,<math> Dv < 10^{-3} \Rightarrow</math> D is very small like a raindrop and has a small velocity < 1 m/s.
  
;Motion in the z-direction has no acceleration and therefor constant (zero) velocity.
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==Linear Air Resistance==
  
;Motion in the x-y plane is circular
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[[Forest_UCM_PnCP_LinAirRes]]
  
Let
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==quadratic friction==
:<math>\omega=\frac{qB}{m}</math> = fundamental cyclotron frequency
 
  
Then we have two coupled equations
 
  
:<math>\dot{v}_x = \omega v_y</math>
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[[Forest_UCM_PnCP_QuadAirRes]]
:<math>\dot{v}_y = - \omega v_x</math>
 
  
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==Another block on incline example==
  
let
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[[Forest_UCM_NLM_BlockOnIncline]]
  
:<math>v^* = v_x + i v_y</math> = complex variable used to change variables
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=Projecile Motion=
  
:<math>\dot{v}^* = \dot{v}_x + i \dot{v}_y</math>
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[[Forest_UCM_PnCP_ProjMotion]]
:: <math>= \omega v_y + i (-\omega v_x)</math>
 
:: <math>= -i \omega \left ( \omega v_x +i\omega v_y \right )</math>
 
:: <math>= -i \omega v^*</math>
 
:\Rightarrow
 
::<math>v^* = Ae^{-i\omega t}</math>
 
  
http://hep.physics.wayne.edu/~harr/courses/5200/f07/lecture10.htm
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=Charged Particle in uniform B-Field=
  
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[[Forest_UCM_PnCP_QubUniBfield]]
  
http://www.physics.sfsu.edu/~lea/courses/grad/motion.PDF
 
  
http://physics.ucsd.edu/students/courses/summer2009/session1/physics2b/CH29.pdf
 
  
http://cnx.org/contents/77faa148-866e-4e96-8d6e-1858487a520f@9
 
  
 
[[Forest_Ugrad_ClassicalMechanics]]
 
[[Forest_Ugrad_ClassicalMechanics]]

Latest revision as of 17:45, 8 September 2014

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