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### Centripetal Acceleration

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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
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?
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```
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
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Physics/Chemistry

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