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Since the sum of all forces is the centripetal force, drawing centripetal force into a free body diagram is not necessary and usually not recommended. Using F net = F c {\displaystyle F_{\text{net}}=F_{c}} , we can draw free body diagrams to list all the forces acting on an object and then set it equal to F c {\displaystyle F_{c}} .
The rope example is an example involving a 'pull' force. The centripetal force can also be supplied as a 'push' force, such as in the case where the normal reaction of a wall supplies the centripetal force for a wall of death or a Rotor rider. Newton's idea of a centripetal force corresponds to what is nowadays referred to as a central force.
Vectors are defined in cylindrical coordinates by (ρ, φ, z), where ρ is the length of the vector projected onto the xy-plane, φ is the angle between the projection of the vector onto the xy-plane (i.e. ρ) and the positive x-axis (0 ≤ φ < 2π), z is the regular z-coordinate. (ρ, φ, z) is given in Cartesian coordinates by:
Upper panel: Ball on a banked circular track moving with constant speed ; Lower panel: Forces on the ball.The resultant or net force on the ball found by vector addition of the normal force exerted by the road and vertical force due to gravity must equal the required force for centripetal acceleration dictated by the need to travel a circular path.
The only requirement is that the central force exactly equals the centripetal force, which determines the required angular velocity for a given circular radius. Non-central forces (i.e., those that depend on the angular variables as well as the radius) are ignored here, since they do not produce circular orbits in general.
For illustration, the first term on the left-hand side is zero for circular orbits, and the applied inwards force equals the centripetal force requirement, as expected. If L is not zero the definition of angular momentum allows a change of independent variable from t {\displaystyle t} to θ {\displaystyle \theta }
It is only in very special circumstances that the vector of the centripetal force and the centrifugal term drop away against each other at every distance from the center of rotation. This is the case if and only if the centripetal force is a harmonic force. In this case, only the Coriolis term remains in the equation of motion.
The actual equation given in Rankine is that of a cubic curve, which is a polynomial curve of degree 3, at the time also known as a cubic parabola. In the UK, only from 1845, when legislation and land costs began to constrain the laying out of rail routes and tighter curves were necessary, were the principles beginning to be applied in practice.