What is the expression of centripetal force


V.2.1 Investigation of the centrifugal force
In the following we want to investigate which force keeps the mass on the circular path. The radial acceleration calculated above

is made by a coercive force evoked. This constraining force can be, for example, the thread tension or the rail guide for a train. According to Newton, this force can be written as .

In the sense of d'Alembert's principle, one feels in the system that is moved along, which is not an inertial system, an apparent force, that of the real force, the centripetal force , opposite is the same size. This force creates an apparent dynamic equilibrium in the moving system, the moving mass point does not move. However, the constraining force acts on the mass point that is moved along with it not, he only feels the apparent power.

V.2.2 Experiments on centrifugal force
The better known of the two forces introduced above is probably the apparent force: the centrifugal force. One of the reasons for this is that in circular movements, e.g. in vehicles, the constraining force does not act directly on your own body, but the centrifugal force does. An example of this is when a car is cornering, in which one feels a force out of the curve, i.e. away from the center of the circle. Let us calculate the force acting on the driver who is driving at 120 km / h into a curve with a radius r = 10 m and braking to half the speed, i.e. to 60 km / h. The angular velocity is
the radial acceleration
With the above values, the driver experiences an acceleration of 27.8 m / s2 , that corresponds to about 3g, i.e. three times the acceleration due to gravity.
In this experiment, a round sheet of paper is quickly set in a circular motion via an axis of a drill that is fixed in the middle. Thus, every single point on the paper moves at an angular velocity. The square of this angular velocity increases the acceleration experienced by the points on the paper. This acceleration results in a stabilizing force. If the drive speed is selected accordingly, this force is so great that the blade is sufficiently stabilized to saw wood.
The same principle can be used to bring a metal chain into a circular shape and stabilize it. For this purpose, a disc is driven by a drill. A metal chain is stretched over the disc. After the chain has been driven fast enough, it can be released from the pulley. It now rolls tangentially, stabilized by the radial force. The energy of the chain is sufficient to fly a few meters high and high when bumping into an obstacle.
In this test arrangement, two carriages stand opposite one another on a rail, on the center of which a rod is attached vertically, at the end of which a spring hangs over the rod. The carriages are attached to the spring via pulleys, so they experience the spring return force (Hook's force ), which accelerates it towards the center of the rail. The rail can be set in uniform rotation with adjustable angular speed via a vertical axis. The centrifugal force now also acts on the car, which accelerates the car away from the axle. At a certain angular velocity an equilibrium is established which is independent of the distance x between the carriages and the center of the rails. This angular velocity can easily be calculated: The opposing forces

must be of the same magnitude so that the resulting force is zero and the body is at rest. With the well-known formulas for the Hook force

and the centrifugal force
and the radius r of the circular path

the deflection of the spring over the pulley follows

Dr .
This equation is independent of the deflection r, which the experiment also showed. Reformulating according to provides the expression for the angular velocity at which equilibrium occurs:

The centrifugal governor consists of two heavy balls that can be set in rotation via an axis. The balls hang on rods of length l, which are fixedly mounted on the driven axle. Two more rods of length s are connected to the axle in such a way that they can be moved up and down, whereby they are stuck to the balls. Set in rotation, the balls and thus the lower rod rise to a certain height at which a conductor connected to the balls moves so far up that it loses contact with its circuit and thus cuts it. The drive of the axle is also connected to this circuit, so that the balls no longer rotate and sink. Contact is re-established and the balls rise again. The displayed rods form the angle with the axis, the balls are the distance r from the axis. In equilibrium, the rod exerts a constraining force on the ball from, the absolute value of the vertical component of the resulting force vector from weight and the centrifugal force must correspond. The geometrical consideration of the acting force components shows that the relation applies to the angle of incidence: