Lecture 13

Centripetal and Centrifugal Forces

Spanish version at the end available

As indicated by the expression “centri”, while the ‘centripetal’ is directed towards the center, the ‘centrifugal is directed outwards, in the opposite direction of the center. Already with this definition there is something that does not sound complete. First, this suggests that this is the description of a circular motion. The mass in question has to be subjected to a movement around a center, such as when holding a stone attached to a string spinning around our head, when a visitor to the amusement park is entertained sitting in a swivel chair or on a rotating platform, or simply by experiencing a ‘strange’ force that tends to pull us out of the chair outward when we give a quick curve in the car we drive. Second, this force suggests that it is perceived by an observer in a non-inertial frame of reference, in which, of course, the observer is part of that system, since an observer outside the non-inertial system does not see or experience this force. Typically these forces are considered ‘fictitious’ as they disappear when there is no angular velocity, and do not follow the laws of Newton (laws of mechanics). If in a resting system S, a force F is leading to an acceleration, then according to Newton, the force can be given in the form F=ma. Bold lettering is indicating the vector nature of the physical quantity. In a rotating system S’, the acceleration a’ experienced by a body is not equal to the acceleration of the system at rest, (a a’), since the S’ system is not an inertial system. For the observer sitting on a rotating platform with angular velocity w, a body that slides (or rolls) out of the center of rotation with a velocity v’ follows a curved trajectory, created by a force perpendicular to the velocity v’ (the velocity is perpendicular to this force). The acceleration seen by the observer in S’, is given by -2 (w x v’ ) . Multiplying by the mass of the body that slides (or rolls), we obtain a force, which is not Newtonian, but a fictional force that is called the force of Coriolis:

Fc = -2m(w x v’). The observer in the rotating S’ system senses a force throwing it outwards; this force is the so-called Centrifugal Force given by the expression Fz = -mw x (w x r). Note that these two forces disappear when the platform stops rotating, or in other words when w = 0. Both forces are perpendicular to w. These forces, although considered fictional in physics, are real for an observer in the rotation system. In astrophysics, considering that the planets revolve around their axis (as well as other celestial bodies), we speak of centripetal and centrifugal forces, because, although we are making observations from an inertial system  (the Earth is an inertial system if considered fixed with respect to stars), we can consider ourselves momentarily observant in the rotational system of another planet. That allows us to talk about centrifugal forces, in case of tidal forces, for example. Once the centrifugal force disappears, the body will follow a straight-line tangent to the curve trajectory, according to the first axiom of Newton.

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Spanish version

Dos Fuerzas: La Centrípeta y la Centrífuga