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. This is a common misconception among science teachers which is addressed in the explanation below:
|Thorough explanation of centripetal and centrifugal forces|
|Newton's famous second law of motion states that the net force acting on a body is equal to its mass times its acceleration (if its mass is constant). The law is a vector law, however, so both force and acceleration are vector quantities. (Vectors have not only a size but a direction, and are usually represented as arrows.) Acceleration is defined as the rate of change of velocity with respect to time, where velocity is a vector. Therefore, whenever the length of the velocity vector (known as the speed) changes or the direction changes, there must be an acceleration, and therefore a net force.
Now, consider a body moving in a circle at a uniform speed. Although the length of the velocity vector is uniform, it is constantly changing direction. This means that there must be an acceleration and therefore a net force on the body. It turns out that the acceleration and force are both pointing inward to the center of the circle of motion. This force is the real centripetal force.
The sun exerts an inward gravitational force on the Earth to keep it moving in its near-uniform near-circular motion. Another example of a centripetal force is as follows: if you were to hold one end of a string whose other end is attached to a ball, and then swung the ball around over your head, you would be supplying the inward force to keep the ball moving in its circle.
Now, Newton's first law states that an object at rest will stay at rest and an object in uniform linear motion will stay in its motion unless acted on by a nonzero net force. Any frame of reference for which this is the case is termed an inertial reference frame. (A reference frame is simply a "point of view," if you will; observations are made from the observer's reference frame.) Evidently, a reference frame moving with a constant speed and direction with respect to another reference frame is inertial via this law.
But what if you are rotating with respect to an inertial frame? We know from experience (having been in a car as it has rounded a turn, or any similar turning motion) that objects at rest do not stay at rest; they are flung outward as if acted on by a force. It turns out that it is as if a force equal in magnitude to the centripetal force but pointing outward is acting on all objects in the rotating frame. This is called the centrifugal force. Since this force and the resulting motion are due to the frame's being non-inertial instead of some unbalanced force, the centrifugal force is termed a fictitious force.
The term "fictitious" implies that it does not really exist, and therefore zealous science teachers can go out of their way to claim that they are not real.
But, since James Bond is rotating with respect to Black Hat and any other stationary onlookers (who are all in an essentially inertial frame; see the parenthesized paragraph below), he will experience a real centrifugal force. As the centrifuge rotates faster, the centripetal force needed increases, therefore the centrifugal force he feels increases, and eventually the force will crush and kill him.
(The observing frame is not technically inertial either, as it is on a planet which is rotating and revolving about a star which is moving non-inertially. The resulting pseudo-forces are very, very minor, however, and can only be detected by more sensitive instruments [e.g. a Foucault pendulum].)
The thorough explanation above is summarized as follows:
- 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.
- [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.