example of a rotating reference frame is the surface of the Earth. All non-inertial reference frames exhibit fictitious forces; rotating reference frames A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth. All… the centrifugal force, the Coriolis force, and, for non-uniformly rotating reference frames, A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth. All non-inertial reference frames exhibit fictitious forces; rotating reference frames are characterized by three: the centrifugal force, the Coriolis force, and, for non-uniformly rotating reference frames, the Euler force. Scientists in a rotating box can measure the rotation speed and axis of rotation by measuring these fictitious forces. For example, Léon Foucault was able to show the Coriolis force that results from Earth's rotation using the Foucault pendulum. If Earth were to rotate many times faster, these fictitious forces could be felt by humans, as they are when on a spinning carousel. In classical mechanics, centrifugal force is an outward force associated with rotation. Centrifugal force is one of several so-called pseudo-forces (also known as inertial forces), so named because, unlike real forces, they do not originate in interactions with other bodies situated in the environment of the particle upon which they act. Instead, centrifugal force originates in the rotation of the frame of reference within which observations are made. The mathematical expression for the Coriolis force appeared in an 1835 paper by a French scientist Gaspard-Gustave Coriolis in connection with hydrodynamics, and also in the tidal equations of Pierre-Simon Laplace in 1778. Early in the 20th century, the term Coriolis force began to be used in connection with meteorology. Perhaps the most commonly encountered rotating reference frame is the Earth. Moving objects on the surface of the Earth experience a Coriolis force, and appear to veer to the right in the Northern Hemisphere, and to the left in the Southern Hemisphere. Movements of air in the atmosphere and water in the ocean are notable examples of this behavior: rather than flowing directly from areas of high pressure to low pressure, as they would on a non-rotating planet, winds and currents tend to flow to the right of this direction north of the equator, and to the left of this direction south of the equator. This effect is responsible for the rotation of large cyclones (see Coriolis effects in meteorology). In classical mechanics, the Euler acceleration (named for Leonhard Euler), also known as azimuthal acceleration or transverse acceleration is an acceleration that appears when a non-uniformly rotating reference frame is used for analysis of motion and there is variation in the angular velocity of the reference frame's axis. This article is restricted to a frame of reference that rotates about a fixed axis. The Euler force is a fictitious force on a body that is related to the Euler acceleration by F = ma, where a is the Euler acceleration and m is the mass of the body. where subscript i {\displaystyle \mathrm {i} } means the inertial frame of reference, and r {\displaystyle \mathrm {r} } means the rotating frame of reference. where a r = d e f ( d 2 r d t 2 ) r {\displaystyle \mathbf {a} _{\mathrm {r} }\ {\stackrel {\mathrm {def} }{=}}\ \left({\tfrac {\mathrm {d} ^{2}\mathbf {r} }{\mathrm {d} t^{2}}}\right)_{\mathrm {r} }} is the apparent acceleration in the rotating reference frame, the term − Ω × ( Ω × r ) {\displaystyle -{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times \mathbf {r} )} represents centrifugal acceleration, and the term − 2 Ω × v r {\displaystyle -2{\boldsymbol {\Omega }}\times \mathbf {v} _{\mathrm {r} }} is the Coriolis acceleration. The last term, − d Ω d t × r {\displaystyle -{\tfrac {\mathrm {d} {\boldsymbol {\Omega }}}{\mathrm {d} t}}\times \mathbf {r} } , is the Euler acceleration and is zero in uniformly rotating frames. where m {\displaystyle m} is the mass of the object being acted upon by these fictitious forces. Notice that all three forces vanish when the frame is not rotating, that is, when Ω = 0 . {\displaystyle {\boldsymbol {\Omega }}=0\ .} For completeness, the inertial acceleration a i {\displaystyle \mathbf {a} _{\mathrm {i} }} due to impressed external forces F i m p {\displaystyle \mathbf {F} _{\mathrm {imp} }} can be determined from the total physical force in the inertial (non-rotating) frame (for example, force from Treat the fictitious forces like real forces, and pretend you are in an inertial frame. Obviously, a rotating frame of reference is a case of a non-inertial frame. Thus the particle in addition to the real force is acted upon by a fictitious force...The particle will move according to Newton's second law of motion if the total force acting on it is taken as the sum of the real and fictitious forces. This equation has exactly the form of Newton's second law, except that in addition to F, the sum of all forces identified in the inertial frame, there is an extra term on the right...This means we can continue to use Newton's second law in the noninertial frame provided we agree that in the noninertial frame we must add an extra force-like term, often called the inertial force.