# 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

If is 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

# 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
Note
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

Let

= fundamental cyclotron frequency

Then we have two coupled equations

## determine the velocity as a function of time

let

= complex variable used to change variables

the complex variable solution may be written in terms of and

The above expression indicates that and 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