Misplaced Pages

Revision as of 09:57, 10 February 2007

• A centrifugal force occurs radially outwards between any two objects that possess a mutual tangential velocity. Centrifugal force is equal in magnitude to the square of the mutual tangential velocity divided by the distance between the two objects, and it operates in tandem with the radial inverse square law force to produce elliptical, circular, parabolic or hyperbolic motion.

• Just as in the case of gravitational force, a centrifugal force can be felt when it is restrained.

• A pseudo or "fictitious" centrifugal force appears when a rotating reference frame is used for analysis.

The idea of a fictitious centrifugal force in a rotating frame of reference is based solely on the principle that the rotating frame of reference masks the tangential motion that is actually causing the force. It therefore appears that an outward force exists for no reason. However, all centrifugal forces are real and based on the principle of radial repulsion due to tangential velocity.

Somebody being thrown against the inside of the door of a car as it is swerving around a corner is experiencing real centrifugal force.

Rotating reference frames

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 mapped from the most convenient inertial frame to that rotating frame. Assuming a constant rotation speed, this is achieved by adding to every object two coordinate accelerations which correct for the rotation of the coordinate axes.

${\displaystyle \mathbf {a} _{\mathrm {rot} }\,}$	${\displaystyle =\mathbf {a} -2\mathbf {\omega \times v} -\mathbf {\omega \times (\omega \times r)} \,}$
	${\displaystyle =\mathbf {a+a_{\mathrm {Coriolis} }+a_{\mathrm {centrifugal} }} \,}$

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

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

${\displaystyle \mathbf {a} _{\textrm {centrifugal}}=-\mathbf {\omega \times (\omega \times r)} =\omega ^{2}\mathbf {r} _{\perp }}$

where ${\displaystyle \mathbf {r_{\perp }} }$ is the component of ${\displaystyle \mathbf {r} \,}$ perpendicular to the axis of rotation.

Derivation

If we have two frames, one inertial and one rotating with a constant angular velocity ${\displaystyle {\vec {\omega }}}$, a time derivative of a vector in the rotating frame, ${\displaystyle \left({\frac {d}{dt}}\right)_{r}}$, is transformed to the time derivative in the inertial frame, ${\displaystyle \left({\frac {d}{dt}}\right)_{i}}$, by the following relation:

${\displaystyle \left({\frac {d}{dt}}\right)_{i}=\left({\frac {d}{dt}}\right)_{r}+{\vec {\omega }}\times }$

This relationship is one between two operators. Now, acceleration is the second derivative of position with respect to time. So, applying the above transformation to the position vector ${\displaystyle {\vec {r}}}$ once gets you:

${\displaystyle {\dot {\vec {r}}}_{i}=\left({\frac {d{\vec {r}}}{dt}}\right)_{i}=\left({\frac {d{\vec {r}}}{dt}}\right)_{r}+\omega \times {\vec {r}}}$

Putting ${\displaystyle {\dot {\vec {r}}}_{i}}$ back into the transformation, you get:

${\displaystyle {\ddot {\vec {r}}}_{i}=\left({\frac {d{\dot {\vec {r}}}}{dt}}\right)_{i}=\left({\frac {d{\dot {\vec {r}}}}{dt}}\right)_{r}+\omega \times {\dot {\vec {r}}}}$

${\displaystyle {\ddot {\vec {r}}}_{i}=\left({\frac {d^{2}{\vec {r}}}{dt^{2}}}\right)_{i}=\left({\frac {d}{dt}}\right)_{r}\left(\left({\frac {d{\vec {r}}}{dt}}\right)_{r}+\omega \times {\vec {r}}\right)+{\vec {\omega }}\times \left(\left({\frac {d{\vec {r}}}{dt}}\right)_{r}+\omega \times {\vec {r}}\right)}$

Because ${\displaystyle {\vec {\omega }}}$ is a contant vector - that is the rotating reference frame is rotating constantly in the same direction - its time derivative is zero. So, simplifying:

${\displaystyle {\ddot {\vec {r}}}_{i}=\left({\frac {d^{2}{\vec {r}}}{dt^{2}}}\right)_{i}=\left({\frac {d^{2}{\vec {r}}}{dt^{2}}}\right)_{r}+\omega \times \left({\frac {d{\vec {r}}}{dt}}\right)_{r}+{\vec {\omega }}\times \left({\frac {d{\vec {r}}}{dt}}\right)_{r}+\omega \times \omega \times {\vec {r}}}$

${\displaystyle {\ddot {\vec {r}}}_{i}=\left({\frac {d^{2}{\vec {r}}}{dt^{2}}}\right)_{i}=\left({\frac {d^{2}{\vec {r}}}{dt^{2}}}\right)_{r}+2{\vec {\omega }}\times \left({\frac {d{\vec {r}}}{dt}}\right)_{r}+\omega \times \omega \times {\vec {r}}}$

Finally, putting in ${\displaystyle {\vec {a}}}$ for ${\displaystyle \left({\frac {d^{2}{\vec {r}}}{dt^{2}}}\right)}$ and ${\displaystyle {\vec {v}}_{r}}$ for ${\displaystyle \left({\frac {d{\vec {r}}}{dt}}\right)_{r}}$, we get the following:

${\displaystyle {\vec {a}}_{i}={\vec {a}}_{r}+2{\vec {\omega }}\times {\vec {v}}_{r}+{\vec {\omega }}\times \left({\vec {\omega }}\times {\vec {r}}\right)}$

Moving things to the other side, but reversing one cross-product in each term, you find:

${\displaystyle {\vec {a}}_{r}={\vec {a}}_{i}-2{\vec {v}}_{r}\times {\vec {\omega }}-{\vec {\omega }}\times \left({\vec {r}}\times {\vec {\omega }}\right)}$

This tells us that ${\displaystyle {\vec {a}}_{r}}$, the acceleration of some object at ${\displaystyle {\vec {r}}}$ as observed by someone at rest in the rotating frame is equal to the acceleration, ${\displaystyle {\vec {a}}_{i}}$, as observed by an observer in the inertial, non-rotating frame, plus ${\displaystyle 2{\vec {v}}_{r}\times {\vec {\omega }}}$, which is the Coriolis effect's contribution to the acceleration, and ${\displaystyle {\vec {\omega }}\times \left({\vec {r}}\times {\vec {\omega }}\right)}$, which is the centrifugal acceleration term.

Fictitious forces

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:

${\displaystyle \mathbf {F} _{\mathrm {centrifugal} }\,}$	${\displaystyle =m\mathbf {a} _{\mathrm {centrifugal} }\,}$
	${\displaystyle =m\omega ^{2}\mathbf {r} _{\perp }\,}$

where ${\displaystyle m\,}$ 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":

${\displaystyle \mathbf {F} _{\mathrm {coriolis} }=-2m\mathbf {\omega \times v} =-2m\omega ^{2}\mathbf {r} _{\perp }}$

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 ${\displaystyle -m\omega ^{2}\mathbf {r} _{\perp }}$ required to account for this apparent rotation is the sum of the centrifugal pseudo force (${\displaystyle m\omega ^{2}\mathbf {r} _{\perp }}$) and the Coriolis force (${\displaystyle -2m\mathbf {\omega \times v} =-2m\omega ^{2}\mathbf {r} _{\perp }}$). Since this centripetal force includes contributions from only pseudo forces, it has no reactive counterpart.

Potential energy

The fictitious centrifugal force can be described by a potential energy of the form

${\displaystyle E_{p}=-{\frac {1}{2}}m\omega ^{2}r_{\perp }^{2}}$

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

It is commonly taught today that real centrifugal force does not exist. This teaching is based on the argument that when we release an object that has been constrained to move in a circle, it flies off at a tangent and not radially outwards. In actual fact, the object flies off both tangentially and radially outwards.

Maxwell on Real Centrifugal Force

James Clerk-Maxwell uses real centrifugal force in his 1861 paper 'On Physical lines of Force' . He uses focused centrifugal force in a sea of molecular vortices, in order to account for ferromagnetic and electromagnetic repulsion. He also uses it to account for diamagnetic repulsion and paramagnetic attraction.

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.

See also

• Centripetal force
• Circular motion
• Coriolis force
• Fictitious force

References

• Newton's description in Principia
• Centrifugal reaction force - Columbia electronic encyclopedia
• Fictitious centrifugal force - from ScienceWorld
• A Comic strip about Centrifugal Force.
• M. Alonso and E.J. Finn, Fundamental university physics, Addison-Wesley
• Centripetal force vs. Centrifugal force - from an online Regents Exam physics tutorial by the Oswego City School District
• Centrifugal force acts inwards near a black hole

Notes

• Force
• Mechanics