To all, sorry I haven't had time to respond before now.
I've done a little reading and have to admit there's some confusion about centripetal vs centrifugal forces, even among those who claim to know. Whether the "centrifugal" force is actually a force or not, seems to be more of a philosophical point rather than a scientifically observational or mathematical one. However I'm convinced that a purely centripetal force in NOT an acceleration force (though some claim it to be), as it does NOT alter the momentum of a rotating object, but only changes its direction. To blend these forces is a mistake.
From all of the posted comments, it seems that there are 2 basic questions that need to be addressed, namely that acceleration and centripetal forces are separate forces, and that velocity (or speed) is relative.
First the relative nature of speed, and for this I’ll go back to the van and roller-skate example. Let’s suppose that the van accelerates to 15mph (relative to the fixed spot over the road that you first occupied) before the back doors meet up with you. Suppose the latch on the back doors is fairly weak, and when the doors contact you, you are accelerated to only 5mph (relative to the road surface) before the doors open and you fall out landing on your feet. At the moment you exit, the van is traveling 15mph relative to the road surface, 10 mph relative to you, and you are traveling 5mph relative to the road surface. All are familiar with the idea that IAS, true airspeed, and true ground speed are speeds relative to an airframe and different external reference points.
Momentum is a product of mass and velocity (M = m * v), where M = momentum, m = mass, and v = velocity (or speed). Just as speed is relative to other objects, momentum is relative to other objects. Momentum changes when speed changes, which happens when acceleration forces are applied.
The centripetal force is a product of the mass and velocity squared over the radius, expressed as F = m * v2 (v squared) / R, where F = centripetal force, m = mass, v = velocity, and R = radius. The velocity here is relative to the center of rotation. Centrifugal force (and I’m convinced it’s a real force) opposes the centripetal force with the same (though opposite) magnitude.
Normal momentum is linear, but with a rotating object it’s angular. Newton’s first law states that once an object is in motion, it will remain in motion unless an external force is applied to it. Since speed and momentum are relative to other objects, for an object’s momentum to change relative to another object, its speed must change relative to that object (assuming we are not changing masses). For a rotating object, speed must change relative to the center of rotation.
For the next part of this discussion we need a pair of examples. For the first example, we’ll use a 5lb steel ball attached to a rope that can be swung around be an electric motor. The second example is not perfect, but should help to make my point that the centripetal force is NOT normally an acceleration force (exceptions like gravity will be covered). At this point we also have to make distinctions between centripetal forces that are exerted mechanically (like our first example) and those that are not, such as gravity and the electrical force of orbiting electrons in atoms. So the following example will work for mechanical means of exerting centripetal force, and will also work for gravity (with a strict limitation to purely circular orbits).
In the second example suppose you have a board that’s a 2x4 about 8 feet high, with one end embedded in concrete in the ground, and the other end sticking straight up. Let’s assume the wood is of high quality with the following properties: no matter how much the 2x4 is bent it will not break, it’s perfectly resilient so it won’t bend or crack and springs back straight, and it’s quite stiff. In this example we’ll use a rope attached to the top of the 2x4 to bend it.
Now lets start the motor and get the ball rotating at 120 rpm. Remember the formula for the centripetal force is F = m * v2 (v squared) / R, and momentum is M = mv. At 120 rpm we know the mass is 5lbs, and if we knew the length of the rope (to the center of the ball) we would know the radius, thus we could calculate the velocity from the radius and the rpm. We could then derive the centripetal force being exerted by the rope on the steel ball. This would also be the value of the centrifugal force being exerted by the steel ball on the rope. I think you can see that the centrifugal force is real as the rope is being pulled on both ends, or else the ball and rope would not be in the air spinning around.
The momentum (now angular momentum) of the steel ball is its mass times its velocity. According to Newton’s first law, all objects in motion want to travel in a straight line until acted on by an external force and this property is established by its momentum. The stiffness of the 2x4 represents the momentum of an object (the steel ball) and its tendency to want to travel in a straight line. Pulling on the rope to bend the 2x4 is equivalent to pulling the motion of an object into an arc around a center, which represents the centripetal force. Notice the 2x4 exerts no force at all on the rope, unless the rope is used to bend the 2x4, then the stiffness of the 2x4 starts exerting a force opposite the pull of the rope. This is how the centrifugal force works in that the force does not exist unless a centripetal force is being exerted.
Now lets change the rotation rate to 150rpm but reduce the length of the rope so that the steel ball continues to travel at the same velocity (relative to the center of rotation) in feet/sec or whatever our unit of measure is. By using the same velocity we maintain the same momentum (M = m * v) as in the 120rpm example. So now lets look at the formula for the centripetal force again, which is F = m * v2 (v squared) / R. Notice what has happened to the centripetal force. The numerator has not changed since mass and velocity have not changed but the radius is now smaller, so you can see that the centripetal force has increased.
Hmmm…the mass is still the same, the velocity is still the same, so therefore the momentum is still the same. Only the radius and rotation rate have changed. Why then has the centripetal force increased? This is due to the fact that the steel ball is now being pulled into a tighter arc, which is farther away from the straight line that it wants to travel in. This can be expressed in degrees per second change based on the steel ball traveling around the circumference of rotation. The 120rpm example will have fewer degrees per second change compared to the 150rpm example. This is equivalent to bending the 2x4 more by pulling harder on the rope, while the stiffness of the 2x4 exerts a harder counterforce to the bending. The only possible explanation for the increase in the centripetal force, is the increase in the centrifugal force, which in the 150rpm example, more strongly resists divergence from travel in a straight line.
Increasing the mass will also increase the centripetal force, but increasing the velocity produces a much stronger increase in the centripetal force, since it’s the only term in the formula being squared. Any increase in the velocity increases the degrees per second rate of change from straight-line travel (assuming the radius stays the same).
I’m going to skip the gravity example for now, as I’m tired. Also, there are no acceleration forces being used in these examples (once rpm was established), therefore the centripetal force CANNOT be an acceleration force, since there are no momentum changes in these examples. Acceleration changes momentum, the centripetal force causes divergence from straight-line motion.
I’ll stop here.
(edited for typos)
[ 23 December 2001: Message edited by: Flight Safety ]
[ 23 December 2001: Message edited by: Flight Safety ]</p>