Centripetal AccelerationDate: 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 Check out our web site! http://mathforum.org/dr.math/ |
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