. The London, Edinburgh and Dublin philosophical magazine and journal of science. ormer expression is of the same dimensions 244 Mr. W. Williams on the Relation of Dimensions irm 0 as r^r in the latter. Thus —~— is of the dimensionsMXYT-2, To determine from this the dimensions of 7r we have [7mr-40]=MXYT-2,[6] =XY~1Z-1, [n] =MZ(XY)1T2; /. [9r]MZ(XY)-lT-2. Y4. (XY^Z-^MXYT-2, .-. [>]=XY^\ Thus it in this case is of the dimensions of a plane anglein the plane XY. 28. Viscosity.—The viscous resistance between planes moving with relative velocity w = — is given by F=J( NwuoL^r- I (Olausius). Tak
. The London, Edinburgh and Dublin philosophical magazine and journal of science. ormer expression is of the same dimensions 244 Mr. W. Williams on the Relation of Dimensions irm 0 as r^r in the latter. Thus —~— is of the dimensionsMXYT-2, To determine from this the dimensions of 7r we have [7mr-40]=MXYT-2,[6] =XY~1Z-1, [n] =MZ(XY)1T2; /. [9r]MZ(XY)-lT-2. Y4. (XY^Z-^MXYT-2, .-. [>]=XY^\ Thus it in this case is of the dimensions of a plane anglein the plane XY. 28. Viscosity.—The viscous resistance between planes moving with relative velocity w = — is given by F=J( NwuoL^r- I (Olausius). Taking1 the components of co and L along Z(normal to direction of motion) this becomes dimensionallv a tangential force in the plane of motion (XY). The co-efficient of viscosity is F irr2v ^ = xz-1T~1 -MZ(YZ) T dz = Tangential force per unit area -f- shear per unit time. 29. Surface Tension.—Tangential force per unit length normal to itself. Let Y be the direction of force in the plane YZ, and K the surface tension. Then MY2 T-2[K] = MYZ1T-2= yZ = Energy per unit Let AOB be a normal section of a cylindrical liquid film. of Physical Quantities to Directions in Space. 245 axis Z, radius X. The normal pressure due to the curvatureis KI - V or dimensional! y MYZ-lT-2(XY-2)==MX(YZ)~lT-2 . as should be the case. The above examples are sufficient to illustrate the methodof expressing dimensional formulae in terms of X, Y, Z. Inall the above cases isolated quantities only are dealt with, andtherefore the relation of the directions of directed quantitiesto the dimensional axes is a matter of indifference. This isnot so in the case of equations between quantities. Here,however, we have only to transform the equations to Carte-sian co-ordinates and then express the dimensions of eachterm by means of the above. Thus, let A, B, C, . . bequantities connected by the equation A + B + C + D+ ... =0. Expressed in Cartesian co-ordinates this becomes (AI+Ay+AJ+(BI+B^+B,) + (C.+Cy +
Size: 2710px × 922px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1800, bookdecade1840, booksubjectscience, bookyear1840
