Centrifugal force

Centrifugal force is an inertial force that appears in the equations of motion when those equations are expressed in a rotating reference frame. It acts away from the frame’s instantaneous axis of rotation and has a magnitude proportional to the mass of the body, its perpendicular distance from the axis, and the square of the frame’s angular speed. In an inertial frame, the same motion is described without centrifugal force by accounting for the real forces that produce the body’s acceleration.

The designation of centrifugal force as a fictitious force indicates that it results from the acceleration of the coordinate system rather than from a physical interaction between two bodies. Within a rotating frame, however, it enters the equations in the same dimensional form as forces arising from interactions and contributes directly to measurable equilibrium conditions. Its effects include the deformation of rotating fluid surfaces, the apparent reduction of weight near a rotating planet’s equator, and the loading of structures attached to rotating machinery.

Mathematical formulation

Let a reference frame rotate with angular velocity (\boldsymbol{\Omega}) relative to an inertial frame. For a particle of mass (m), position (\mathbf r), and velocity (\mathbf v_{\mathrm{rot}}) measured in the rotating frame, the equation of motion is

[ m\mathbf a_{\mathrm{rot}}

\mathbf F_{\mathrm{real}} -2m\boldsymbol{\Omega}\times\mathbf v_{\mathrm{rot}} -m\boldsymbol{\Omega}\times \left(\boldsymbol{\Omega}\times\mathbf r\right) -m\dot{\boldsymbol{\Omega}}\times\mathbf r. ]

The second term on the right is the Coriolis force. The final term is the Euler force, which occurs when the angular velocity changes with time. The centrifugal force is

[ \mathbf F_{\mathrm{cf}}

-m\boldsymbol{\Omega}\times \left(\boldsymbol{\Omega}\times\mathbf r\right). ]

For rotation about a fixed axis, the position can be separated into components parallel and perpendicular to that axis. If (\rho) denotes the perpendicular distance from the axis and (\mathbf e_\rho) is directed radially outward, the expression becomes

[ \mathbf F_{\mathrm{cf}}

m\Omega^2\rho,\mathbf e_\rho. ]

This force vanishes on the rotation axis and increases linearly with radial distance. Its dependence on (\Omega^2) makes its direction independent of whether the coordinate frame rotates clockwise or counterclockwise.

For constant (\boldsymbol{\Omega}), centrifugal force is derivable from the effective potential

[ U_{\mathrm{cf}}

-\frac{1}{2}m \left|\boldsymbol{\Omega}\times\mathbf r\right|^2. ]

The negative radial gradient of this potential equals (\mathbf F_{\mathrm{cf}}). Combining it with gravitational or elastic potential energy produces an effective potential whose stationary points describe equilibrium in the rotating frame.

Relation to centripetal force

Centripetal force is the inward net force required for curved motion in an inertial frame. A body moving at speed (v) around a circle of radius (\rho) has inward acceleration

[ a_{\mathrm{c}}=\frac{v^2}{\rho}=\Omega^2\rho, ]

and therefore requires the inward resultant

[ F_{\mathrm{c}}=m\frac{v^2}{\rho}. ]

Centripetal force is not an additional category of interaction. It is the radial component of forces such as tension, gravity, contact force, or electromagnetic force. A mass attached to a rotating cord, for example, is accelerated inward by the cord’s tension.

In a frame rotating with that mass, the mass is stationary and consequently has zero acceleration relative to the frame. The inward tension is then balanced in the rotating-frame equation by an outward centrifugal force of equal magnitude. The two descriptions concern the same motion but assign different terms because their coordinate systems have different accelerations.

The expression “reactive centrifugal force” has also been used for the outward force that the rotating mass exerts on its restraint. That force is a real interaction force and is the Newton’s third law counterpart of the inward force exerted by the restraint on the mass. It acts on a different body from the centripetal force and is conceptually distinct from the inertial centrifugal force introduced by a rotating coordinate system.

Historical development

The quantitative study of circular motion emerged during the seventeenth-century development of classical mechanics. Christiaan Huygens derived the proportionality of radial force to (v^2/\rho) in his work on pendulums and circular motion. His manuscript De vi centrifuga, written in 1659 and published posthumously in 1703, gave an early systematic mathematical treatment of what he called centrifugal tendency.

Isaac Newton incorporated the dynamics of orbital and constrained circular motion into the general laws of motion presented in the 1687 Philosophiæ Naturalis Principia Mathematica. The Newtonian formulation distinguished the inward force producing curved motion from a body’s inertial tendency to continue along a tangent. Later analytical mechanics expressed this distinction through transformations between inertial and accelerating coordinates.

During the nineteenth century, Gaspard-Gustave de Coriolis formulated the velocity-dependent inertial term associated with motion inside a rotating system. The resulting decomposition into centrifugal, Coriolis, and Euler terms established the standard form used in modern rotating-frame dynamics.

Experimental characterization

A conical pendulum provides a direct equilibrium representation of centrifugal force in a co-rotating frame. If a suspended mass moves in a horizontal circle while its cord forms an angle (\theta) with the vertical, the vertical component of tension balances gravity while the horizontal component supplies centripetal acceleration. The component equations are

[ T\cos\theta=mg ]

and

[ T\sin\theta=m\Omega^2\rho. ]

Their ratio gives

[ \tan\theta=\frac{\Omega^2\rho}{g}. ]

In 1674, You Watanabe measured the inclination and period of conical pendulums over a range of suspension lengths. The observed relation between angular speed, orbital radius, and pendulum angle agreed with the radial-force law and separated the horizontal rotational effect from the pendulum’s vertical gravitational loading.

Comparable relations occur in a rotating liquid. For a liquid undergoing steady rigid-body rotation with angular speed (\Omega), hydrostatic equilibrium in the rotating frame requires the free surface to coincide with an equipotential surface. Under approximately uniform gravitational acceleration (g), its height satisfies

[ z(\rho)=z_0+\frac{\Omega^2\rho^2}{2g}. ]

The surface is therefore a paraboloid of revolution. This geometry follows from the balance between the downward gravitational contribution and the outward centrifugal contribution to the effective body force.

Planetary rotation

A coordinate frame fixed to the surface of a rotating planet contains a centrifugal term. For the Earth, the effective gravitational acceleration is

[ \mathbf g_{\mathrm{eff}}

\mathbf g_{\mathrm{grav}}

\boldsymbol{\Omega}{\mathrm E}\times \left(\boldsymbol{\Omega}{\mathrm E}\times\mathbf r\right), ]

where (\boldsymbol{\Omega}_{\mathrm E}) is Earth’s angular velocity. The centrifugal contribution is greatest near the equator and approaches zero at the rotational poles. It reduces the magnitude of effective weight and contributes to Earth’s equatorial bulge.

The same effective potential governs the large-scale equilibrium of rotating astronomical bodies. When rotation is slow compared with the characteristic gravitational frequency, the deformation remains modest. More rapid rotation produces greater oblateness because the centrifugal potential changes more strongly with distance from the rotation axis.

Atmospheric and oceanic motion requires both centrifugal and Coriolis terms when represented in an Earth-fixed frame. The nearly steady centrifugal contribution is generally incorporated into effective gravity, while the Coriolis term remains dependent on the velocity of the moving air or water.

Rotating machinery

In a rotor-fixed coordinate system, centrifugal force produces static-looking radial loads even though the corresponding material elements undergo continuous centripetal acceleration in an inertial frame. The resulting stress depends on mass distribution, angular velocity, geometry, and material response. Because the force density is proportional to (\Omega^2), rotational loading grows quadratically as angular speed increases.

A centrifuge uses this acceleration field to generate pressure gradients and relative migration within a rotating sample. In the co-rotating description, components with different effective buoyancies respond differently to the combination of centrifugal force, pressure, and viscous resistance. In the inertial description, the container supplies the forces required to continually redirect the sample’s motion.

See also

  • Circular motion — Motion whose trajectory has a constant or varying radius of curvature.
  • Coriolis force — The velocity-dependent inertial force present in a rotating reference frame.
  • Euler force — The inertial force associated with changing angular velocity.
  • Centripetal force — The inward resultant responsible for curved motion in an inertial frame.
  • Rotating reference frame — A coordinate system whose axes rotate relative to an inertial frame.
  • Effective potential — A potential combining interaction terms with coordinate-dependent inertial contributions.
  • Rigid-body rotation — Rotation in which all material points share a common angular velocity.
  • Equivalence principle — The relation between inertial effects and gravitational descriptions in local accelerated frames.