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Dimensions

# Dimensions - previous index next Using Dimensions Michael...

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previous index next Using Dimensions Michael Fowler, UVa Some of the most interesting results of hydrodynamics, such as the sixteen-fold increase in flow down a pipe on doubling the radius, can actually be found without doing any calculations, just from dimensional considerations. We symbolize the “dimensions” mass , length and time by M , L , T . We then write the dimensions of other physical quantities in terms of these. For example, velocity has dimensions 1 LT , and acceleration 2 . LT We shall use square brackets [] to denote the dimensions of a quantity, for example, for velocity, we write [ ] 1 . vL T = Force must have the same dimensions as mass times acceleration, so [ ] 2 . FM L T = This “dimensional” notation does not depend on the units we use to measure mass, length and time. All equations in physics must have the same dimensions on both sides . We can see from the equation defining the coefficient of viscosity , η 0 / FA v d / = , (the left hand side is force per unit area, the right hand v 0 / d is the velocity gradient) that [ ] [ ] [ ] () [ ] [ ] ( ) 22 1 11 // / / FAdv M L T L L L T M L T . −− =⋅ = = How can thinking dimensionally help us find the flow rate I through a pipe? Well, the flow

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