Banks, Coasters and Planets
I continue my own exploration of motion on curves.
Banks
As Newton’s second law proclaims a body in uniform rectilinear motion will continue so unless acted upon by a force. Directing frictional force allows us to negotiate curves. Banked curves provides a central force that can allow us to negotiate curves. The above free body diagram aids in determining the maximum speed that the bank will allow us to negotiate the curve. Let be the angle of the bank, represent the frictional force ( coefficient of friction) , the normal force from the surface.
The central force arises from the horizontal component:
Solving for :
Or equivalently (given ),
For this reduces to:
Roller Coasters
Loops of roller coasters pose challenges for engineers, e.g. forces imposed on those riding on the carriage and speeds required for carriages to remain constrained to the tracks. Some insights can be obtained by looking at motion around a circular loop. The following free body diagram will assist these considerations. Let represent the centripetal force, , the angular position of cart ( cart at the bottom and cart at the top) be the normal or resultant force.
There are three possible outcomes for the car on the circular loop:
 it completes the loop
 it rolls up the loop to a point, stops and then slides down
 it rolls up the loop and loses contact with the track and falls off
Completing the loop:
Rolling up, stopping and sliding down
Intermediate Scenario
Clothoid Loop
Planets
Finally, using considerations from the post Seeking the centre, the elliptical nature of orbits under the inverse square law can be derived.
and
As , the transformation provides simplification of the differential equation:
Therefore the differential equation becomes,
noting .
This is the equation for a harmonic oscillator with constant forcing function. The solution is:
Without loss 0f generality, let and substituting for yields:
Rearranging this into a more canonical form yields,
where and
This is the polar formula of a conic section.
The polar of plot of for varying values of is show below (note 0= circle>ellipse>1=parabola>hyperbola):
Comment:
The Newtonian gravitational potential energy can be derived from the force formula:
By convention, U =GMm/r where the potential energy at “infinity” is regarded as zero and hence all less than this are negative. Changes in potential difference map as above.
For distances small with respect to the radius of the earth, the usual is a good approximation.
Note: for :
For, is approximately constant.

July 10, 2011 at 5:40 amAnalytical Geometry  salesmarketingessentials