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
Frictional forces tend to be proportional to a fixed power of velocity
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.
Linear Air Resistance
- Horizontal motion
Ifis unity then the velocity is exponentially approaching zero.
- : negative sign indicates a retarding force and is a proportionality constant
The displacement is given by
Example: falling object with air friction
Consider a ball falling under the influence of gravity and a frictional force that is proportion to its velocity squared
Find the fall distance
Here is a trick to convert the integral over time to one over distance so you don't need to integrate twice as inthe previous example
The integral becomes
Another block on incline example
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.
- Lorentz Force
- the work done by a magnetic field is zero if the particle's kinetic energy (mass and velocity) don't change.
No work is done on a charged particle forced to move in a fixed circular orbit by a magnetic field (cyclotron)
Apply Newton's 2nd Law
- Motion in the z-direction has no acceleration and therefor constant (zero) velocity.
- Motion in the x-y plane is circular
- = fundamental cyclotron frequency
Then we have two coupled equations
determine the velocity as a function of time
- = complex variable used to change variables
the complex variable solution may be written in terms ofand
The above expression indicates thatand oscillate at the same frequency but are 90 degrees out of phase. This is characteristic of circular motion with a magnitude of such that
Determine the position as a function of time
To determine the position as a function of time we need to integrate the solution above for the velocity as a function of time
Using the same trick used to determine the velocity, define a position function using complex variable such that
Using the definitions of velocity
The position is also composed of two oscillating components that are out of phase by 90 degrees
The radius of the circular orbit is given by
The momentum is proportional to the charge, magnetic field, and radius