Question

Verify the correctness of F=6πnrv

Answer 1

) Angle of depression =∅     cos∅= mgl²/4bd³Y

cos∅  is dimensionless as generally it is ratio of sides i.e, L/L =L⁰ which is dimensionless.

⇒⇒mgl²/4bd³Y must be dimensionless.

Y represents "young's modulous"   = Stress/strain =[dimensions of pressure]

Y=[Force/area]=[MLT⁻²/L²]=[ML⁻¹T⁻²]

Dimensions of mgl²/4bd³Y = MLT⁻²*L²/L*L³*ML⁻¹T⁻² =ML³T⁻²/ML³T⁻²

                        

                                                                                  =M⁰L⁰T⁰

So, mgl²/4bd³Y is dimensionless.

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2)  v=(√Elasticity/density)

Dimensions of velocity =LT⁻²  

Dimensions of elasticity = [Dimensions of pressure] 

As, it is defined as the ratio of tangential stress to the shearing strain and strain is dimensionless and stress has the dimensions of pressure.

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