Centripetal Acceleration

Date: 05/05/97 at 00:17:12
From: Steven C. WARD
Subject: Circular motion (centripetal acceleration)

I am a State Trooper. One of my duties is to conduct in-depth 
investigations of motor vehicle accidents. I have had a different 
interpretation regarding what forces are acting when an automobile is 
in critical speed sideslip (sliding on the highway while trying to 
negotiate a curve). For argument, assume the vehicle's center of mass 
is traveling in a curved path with a known radius (determined by 
measurement of tire marks).  

I think that the vehicle is following a curved path and it therefore 
accelerating. There are equations that presume a "balance" of forces 
and which calculate the forces acting upon the vehicle.  Are the 
forces really balanced in this situation? Or is the equation 
unbalanced due to the acceleration?  

For simplicity, assume the road is level. Neglect air friction and 
suspension loading (weight shift). Forces are limited to friction and 
weight. Forces in lateral frame are related to tire friction only.  
Since the vehicle is sliding, the friction (drag factor = f) is 
dynamic (mu) and constant.

What do you think?

Date: 05/05/97 at 19:14:25
From: Doctor Anthony
Subject: Re: Circular motion (centripetal acceleration)

When the vehicle is going round a curved path, there must be an 
unbalanced force, equal to mv^2/r, acting towards the center of 
curvature of the path.  This is the equation of Newton's second law, 
F = ma, where F is the RESULTANT force acting.  The acceleration is 
given by v^2/r, where v is the velocity and r is the radius of the 
curve.

The only way (in the absence of banking) that the road can produce the 
inward acting force is by friction between the tires and the road.  
This force is given by (mu)mg where (mu) is the coefficient of 
friction. We thus have:
                      (mu)mg = mv^2/r
                       (mu)g = v^2/r

For a curve of radius r the maximum velocity is given by the equation:

                      v^2 = mu*gr   
                        v = sqrt(mu*gr)

The INWARD acting force is referred to as CENTRIPETAL force.

An outward acting force, equal and opposite to the centripetal force, 
is called the CENTRIFUGAL force.  This is an imaginary force which can 
be introduced to make the problem one of static equilibrium with all 
the foces in balance and no resulting acceleration. However it is an 
unnecessary device which has led to much confusion among those meeting 
the topic for the first time. It sometimes called d'Alembert's 
principle when a dynamic problem is reduced to one of static 
equilibrium.

-Doctor Anthony,  The Math Forum
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