Frames of Reference. Apparent Forces

Transcription

1 Fames of Refeence Newton s laws of motion ae valid in a coodinate system that is fixed in space. A coodinate system fixed in space is known as an inetial (o absolute) fame of efeence. A coodinate system that is not fixed in space, such as one defined with espect to the otating eath, is a noninetial fame of efeence. Because we ae inteested in atmospheic and oceanic motions fom an eath-based pespective, we must fomulate the laws of motion in the noninetial fame of efeence defined with espect to the otating eath. Appaent Foces In an inetial efeence fame, a body at est o in unifom motion has no net foces acting on it. A body at est o in unifom motion elative to the otating eath is not at est o in unifom motion elative to a coodinate system fixed in space. To econcile Newton s laws with the noninetial efeence fame of the otating eath, two appaent foces must be intoduced. These ae the centifugal foce and the Coiolis foce. 1

2 Centifugal Foce Conside an object that is whiled in a cicula motion at the end of sting. In an inetial fame of efeence, this object is being acceleated towad the cente of the cicle. This acceleation is called centipetal acceleation, and the foce that causes it is the pull of the sting. t=3 t=1 t= Centifugal Foce Conside a fame of efeence that is moving with the object at the end of the sting. In this noninetial fame of efeence, the object appeas stationay despite the pull of the sting towad the cente of the cicle. An additional appaent foce must be included to exactly balance the foce of the sting. This appaent foce is the centifugal foce. centipetal foce (pull of sting) centifugal foce no net foce no acceleation (in otating fame of efeence)

3 As in the pevious example, an object is attached to a sting and whiled in a cicle of adius at a constant angula velocity ω. In a vey small time incement δt the object otates though an angle δθ. dv dt dθ = V dt V = ω dv dt = ω dθ = ω dt δθ This is the centipetal acceleation. In the otating fame of efeence it is balanced by the centifugal acceleation, which has equal magnitude but opposite sign. v δv v Centifugal Acceleation on Eath In a otating fame of efeence, an object at est on the eath s suface is subject to a centifugal acceleation R R R R is the angula speed of otation of the eath (= 7.9 x 10-5 ad s -1 ). is the position vecto fom the axis of otation to the object. 3

4 Gavity Foce Fo an object at est on the eath s suface, the centifugal foce patly balances the gavitational foce, thus we define a gavity foce (a.k.a. gavity) g R R g g + R * * g g Gavitational foce is always diected towad cente of eath; gavity is not, except at poles and equato. The eath has adjusted to this balance of foces by taking the shape of a spheoid with an equatoial bulge. Coiolis Foce We can apply Newton s second law of motion in a otating coodinate system to an object at est with espect to that otating efeence fame by including the centifugal foce, which is an appaent foce. If the object is moving with espect to the otating efeence fame, an additional appaent foce is equied fo Newton s second law to be valid. This is the Coiolis foce. 4

5 On a otating tuntable, daw a line that is staight when viewed fom an inetial fame of efeence. In the otating efeence fame, the esulting line cuves to the ight, as can be seen when the tuntable is stopped. An object moving in a staight path in an inetial fame of efeence will follow a cuved path with espect to a otating fame of efeence. The appaent foce that deflects the object is the Coiolis foce. Coiolis Foce Conside an object on the eath s suface that is moving in an eastwad diection (same diection that eath otates) at a speed u. The object is now otating faste than the eath, so the centifugal foce acting on it inceases to + u R total centifugal foce ur u R R = R + + R R centifugal foce due to eath s otation deflecting foces Fo lage-scale atmospheic motions, this tem is much smalle than the tem to its left and can be neglected. The deflecting foces act outwad in a diection pependicula to the axis of otation. 5

6 The Coiolis foce that esults fom zonal (i.e., east-west) motion can be divided into components in the vetical and meidional dv dt dw dt Co Co = u sinφ = u cosφ Coiolis Foce diections. R ur R φ u cosφ u sinφ This deivation and esult applies only fo zonal motion. (See Holton section 1.5 fo meidional motion.) A moe geneal way to epesent appaent foces is equied. In the Nothen Hemisphee, an eastwad-moving object will be deflected southwad, o to the ight. 6