123: Centrifugal Force
Title text: You spin me right round baby, right round, in a manner depriving me of an inertial reference frame. Baby.
Black Hat has strapped James Bond to a centrifuge and claims the centrifugal force will be lethal. Bond objects that there is no such thing, but just centripetal force. The notion of centrifugal force is a common one, as we experience it whenever we turn. Teachers will initially teach Newtonian mechanics in an inertial frame, and in inertial frames the centrifugal force is zero. Instead, a body that moves in a circle does so because of a centripetal force (acting towards the centre of the rotation). This is a reasonable, and correct view, but is a subtle point that many students find hard to grasp as it seems to contradict their personal experience of centrifugal forces. For the sake of exposition, teachers may claim that "There is no such thing as centrifugal force". This however is also a misconception which is addressed in the explanation below:
- Observers' point of view (Black Hat, us, etc.)
- James Bond is moving in a circle, and is therefore accelerating. The force keeping him there is an inward force of contact against the centrifuge, a centripetal force. Via Newton's third law, since the centrifuge is pushing Bond inward, Bond is pushing the centrifuge outward. The centrifuge's material is strong enough not to break under this force, however.
- James Bond's point of view
- In James Bond's frame of reference, Bond is at rest. He is kept there by two forces: the above-mentioned inward force of contact against the centrifuge, and an outward centrifugal force. He feels both forces.
As mentioned in the explanation, as the centrifuge rotates faster, the forces needed to keep him in motion get larger, so the force he feels gets larger. This will eventually kill him. The conclusion will be the same regardless of which frame of reference is chosen.
Teachers of mechanics are well aware of this, however in introductory expositions these ideas are often not taught. In theoretical mechanics, one describes the positions and velocities of the particles in a model relative to a frame of reference. This means that a time is chosen to be time 0, and positions are chosen to be (0,0,0), (1,0,0), (0,1,0) and (0,0,1). With these chosen, the position and time of any particle in the system can be described. It is an axiom of Newtonian Mechanics that there exist "Inertial Frames". In an inertial frame a particle will remain at rest or at a constant speed unless acted on by an external force, and Newton's second law takes a simple form: F=ma. The surface of the Earth approximates an inertial frame. In a non-inertial frame, such as one rotating with a giant centrifuge, or moving with an accelerating vehicle, a particle will accelerate, relative to the frame. Newton's second law, when formed in such a frame is much more complicated, as it has terms for the linear acceleration of the frame, the angular acceleration of the frame, the centrifugal force and the Coriolis force. These extra terms are sometimes called "fictitious forces" as they result from the choice of the frame of reference. The mathematics required to describe problems in a non-inertial frame is more sophisticated, and all problems may be solved using an inertial frame. Thus is reasonable that teachers at school level "lie to children" and teach the mechanics in inertial frames.
- [James Bond is strapped to a giant wheel suspended from the ceiling. Black Hat is standing next to two levers.]
- Black hat: How do you like my centrifuge, mister Bond? When I throw this lever, you will feel centrifugal force crush every bone in your body.
- [Same scene, but a closer shot.]
- Bond: You mean centripetal force. There's no such thing as centrifugal force.
- Black hat: A laughable claim, mister Bond, perpetuated by overzealous teachers of science. Simply construct Newton's laws in a rotating system and you will see a centrifugal force term appear as plain as day.
- [Closer shot, only Bond's head is visible.]
- Bond: Come now, do you really expect me to do coordinate substitution in my head while strapped to a centrifuge?
- Black hat: No, Mister Bond. I expect you to die.