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Centrifugal force (rotating reference frame)

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For the real outward-acting force that can be found in circular motion see Reactive centrifugal force

In physics, the fictitious centrifugal force (from Latin centrum "center" and fugere "to flee") is a force which is apparent in rotating reference frames and applies to every object under consideration. This force is oriented away from the axis of rotation of the reference frame.

In some cases, it is convenient to use a rotating reference frame, rather than an inertial reference frame. When this is desirable, coordinate transformations from the inertial reference frame can be applied.

However, to do this correctly, in the rotating reference frame, a centrifugal force must be applied in conjunction with a Coriolis force for the correct equation of motion to be calculated. The centrifugal force depends only on the position and mass of the object it applies to (and does not depend on its velocity), whereas the Coriolis force depends on the velocity and mass of the object but is independent of its position.

The term 'centrifugal force' is often misused in everyday discussions; the centrifugal forces discussed in this article only appear in rotating reference frames.

Rotating reference frames

Rotating reference frames are sometimes used in physics, mechanics or meteorology where they are the most convenient frame to use. For example the surface of the Earth is only stationary in a reference frame that rotates once per day. For many purposes the rotation causes negligible effects, but for some phenomena such as weather systems this rotation cannot be ignored.

In the classical approach, the inertial frame remains the true reference for the laws of mechanics and analysis. When using a rotating reference frame, the laws of physics are coordinate mapped from the most convenient inertial frame to a rotating frame. Assuming a constant rotation speed, this is achieved by adding to every object two coordinate accelerations which correct for the constant rotation of the coordinate axes.

a r o t {\displaystyle \mathbf {a} _{\mathrm {rot} }\,} = a − 2 ω × v − ω × ( ω × r ) {\displaystyle =\mathbf {a} -2\mathbf {\omega \times v} -\mathbf {\omega \times (\omega \times r)} \,}
= a + a C o r i o l i s + a c e n t r i f u g a l {\displaystyle =\mathbf {a+a_{\mathrm {Coriolis} }+a_{\mathrm {centrifugal} }} \,}

where a r o t {\displaystyle \mathbf {a} _{\mathrm {rot} }\,} is the frame acceleration relative to the rotating frame, a {\displaystyle \mathbf {a} \,} is the acceleration relative to the inertial frame, ω {\displaystyle \mathbf {\omega } \,} is the angular velocity vector describing the rotation of the reference frame, v {\displaystyle \mathbf {v} \,} is the velocity of the body relative to the rotating frame, and r {\displaystyle \mathbf {r} \,} is the position vector of a point on the body. A derivation can be found in the article fictitious force.

The last term is the centrifugal acceleration, so we have:

a centrifugal = − ω × ( ω × r ) = ω 2 r ⊥ {\displaystyle \mathbf {a} _{\textrm {centrifugal}}=-\mathbf {\omega \times (\omega \times r)} =\omega ^{2}\mathbf {r} _{\perp }}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \mathbf{r_\perp}} is the component of r {\displaystyle \mathbf {r} \,} perpendicular to the axis of rotation.

Fictitious forces

Main article: Fictitious force

An alternative way of dealing with a rotating frame of reference is to make Newton's laws of motion artificially valid by adding pseudo forces to be the cause of the above acceleration terms. In particular, the centrifugal acceleration is added to the motion of every object, and attributed to a centrifugal force, given by:

F c e n t r i f u g a l {\displaystyle \mathbf {F} _{\mathrm {centrifugal} }\,} = m a c e n t r i f u g a l {\displaystyle =m\mathbf {a} _{\mathrm {centrifugal} }\,}
= m ω 2 r ⊥ {\displaystyle =m\omega ^{2}\mathbf {r} _{\perp }\,}

where m {\displaystyle m\,} is the mass of the object.

This pseudo or fictitious centrifugal force is a sufficient correction to Newton's second law only if the body is stationary in the rotating frame. For bodies that move with respect to the rotating frame it must be supplemented with a second pseudo force, the "Coriolis force":

F c o r i o l i s = − 2 m ω → × v → {\displaystyle \mathbf {F} _{\mathrm {coriolis} }=-2\,m\,{\vec {\omega }}\times {\vec {v}}}

For example, a body that is stationary relative to the non-rotating frame, will be rotating when viewed from the rotating frame. The centripetal force of − m ω 2 r ⊥ {\displaystyle -m\omega ^{2}\mathbf {r} _{\perp }} required to account for this apparent rotation is the sum of the centrifugal pseudo force ( m ω 2 r ⊥ {\displaystyle m\omega ^{2}\mathbf {r} _{\perp }} ) and the Coriolis force ( − 2 m ω × v = − 2 m ω 2 r ⊥ {\displaystyle -2m\mathbf {\omega \times v} =-2m\omega ^{2}\mathbf {r} _{\perp }} ). Since this centripetal force includes contributions from only pseudo forces, it has no reactive counterpart.

Potential energy

The interface of two immiscible liquids rotating around a vertical axis is an upward-opening circular paraboloid.

The fictitious centrifugal force is conservative and has a potential energy of the form

E p = − 1 2 m ω 2 r ⊥ 2 {\displaystyle E_{p}=-{\frac {1}{2}}m\omega ^{2}r_{\perp }^{2}}

This is useful, for example, in calculating the form of the water surface h ( r ) {\displaystyle h(r)\,} in a rotating bucket: requiring the potential energy per unit mass on the surface g h ( r ) − 1 2 ω 2 r 2 {\displaystyle gh(r)-{\frac {1}{2}}\omega ^{2}r^{2}\,} to be constant, we obtain the parabolic form h ( r ) = ω 2 2 g r 2 + C {\displaystyle h(r)={\frac {\omega ^{2}}{2g}}r^{2}+C} (where C {\displaystyle C} is a constant).

Similarly, the potential energy of the centrifugal force is often used in the calculation of the height of the tides on the Earth (where the centrifugal force is included to account for the rotation of the Earth around the Earth-Moon center of mass).

The principle of operation of the centrifuge also can be simply understood in terms of this expression for the potential energy, which shows that it is favorable energetically when the volume far from the axis of rotation is occupied by the heavier substance.

The coriolis force has no equivalent potential, as it acts perpendicular to the velocity vector and hence rotates the direction of motion, but does not change the energy of a body.


Confusion and misconceptions

One can often avoid dealing with pseudo forces entirely by analyzing systems using inertial frames of reference for the physics; and when convenient, one simply maps to a rotating frame without forgetting about the frame rotation, as shown above. Such is standard practice in mechanics textbooks.

Although centrifugal force is often described as 'fictitious', it isn't imaginary in the sense that Sherlock Holmes is fiction with no relation to reality. Centrifugal force gives physically very real effects, ultimately the centrifugal and Coriolis effects are behaviours which are due to inertia. Some writers, notably Richard Feynman preferred to use the term 'pseudo force' which has connotations of pretense rather than outright fiction.

Because rotating frames are not vital for understanding mechanics, they are often not discussed in science education. Therefore teachers who need to impress on their students that centrifugal forces have no place in their calculations often do not have occasion to give a matching emphasis to the fact that a centrifugal force does occur in a rotating frame. As a result, even students who master the physics curriculum may leave school with the false impression that it is never scientifically valid to speak about centrifugal forces. Nevertheless, many popular discussions of forces do use the term "centrifugal", without pointing out that it is fictitious, and assume the reader is knowledgeable of the true inertial character of the force, leading to misconceptions and bad use of the term.

Applications

  • A centrifugal governor regulates the speed of an engine by using spinning masses that respond to centrifugal force generated by the engine. If the engine increases in speed, the masses move and trigger a cut in the throttle.
  • A centrifugal clutch is used in small engine powered devices such as chain saws, go-karts and model helicopters. It allows the engine to start and idle without driving the device but automatically and smoothly engages the drive as the engine speed rises.
  • Centrifugal forces can be used to generate artificial gravity. Proposals have been made to have gravity generated in space stations designed to rotate. The Mars Gravity Biosatellite will study the effects of Mars level gravity on mice with simulated gravity from centrifugal force.
  • Centrifuges are used in science and industry to separate substances by their relative masses.
  • Some amusement park rides make use of centrifugal forces. For instance, a Gravitron’s spin forces riders against a wall and allows riders to be elevated above the machine’s floor in defiance of Earth’s gravity.
  • Spin casting and centrifugal casting are production methods that uses centrifugal force to disperse liquid metal or plastic throughout the negative space of a mold.

See also

References

External links

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