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Question Number 94782 by I want to learn more last updated on 21/May/20
The velocity of physical quantities is given by    v  =  (√((P  +  (1/n))/x)) ,  where  P is the pressure.  Find the dimention of   n  and  x.
$$\mathrm{The}\:\mathrm{velocity}\:\mathrm{of}\:\mathrm{physical}\:\mathrm{quantities}\:\mathrm{is}\:\mathrm{given}\:\mathrm{by} \\ $$$$\:\:\mathrm{v}\:\:=\:\:\sqrt{\frac{\mathrm{P}\:\:+\:\:\frac{\mathrm{1}}{\mathrm{n}}}{\mathrm{x}}}\:,\:\:\mathrm{where}\:\:\mathrm{P}\:\mathrm{is}\:\mathrm{the}\:\mathrm{pressure}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{dimention}\:\mathrm{of}\:\:\:\mathrm{n}\:\:\mathrm{and}\:\:\mathrm{x}. \\ $$
Answered by mr W last updated on 21/May/20
v=[(m/s)]  p=[(N/m^2 )]=(1/n)  ⇒n=[(m^2 /N)]  (([(N/m^2 )])/x)=([(m/s)])^2 =[(m^2 /s^2 )]  ⇒x=[(N/m^2 )×(s^2 /m^2 )]=[((kgm)/s^2 )×(s^2 /m^4 )]=[((kg)/m^3 )]  i.e. x is the density.
$${v}=\left[\frac{{m}}{{s}}\right] \\ $$$${p}=\left[\frac{{N}}{{m}^{\mathrm{2}} }\right]=\frac{\mathrm{1}}{{n}} \\ $$$$\Rightarrow{n}=\left[\frac{{m}^{\mathrm{2}} }{{N}}\right] \\ $$$$\frac{\left[\frac{{N}}{{m}^{\mathrm{2}} }\right]}{{x}}=\left(\left[\frac{{m}}{{s}}\right]\right)^{\mathrm{2}} =\left[\frac{{m}^{\mathrm{2}} }{{s}^{\mathrm{2}} }\right] \\ $$$$\Rightarrow{x}=\left[\frac{{N}}{{m}^{\mathrm{2}} }×\frac{{s}^{\mathrm{2}} }{{m}^{\mathrm{2}} }\right]=\left[\frac{{kgm}}{{s}^{\mathrm{2}} }×\frac{{s}^{\mathrm{2}} }{{m}^{\mathrm{4}} }\right]=\left[\frac{{kg}}{{m}^{\mathrm{3}} }\right] \\ $$$${i}.{e}.\:{x}\:{is}\:{the}\:{density}. \\ $$
Commented by I want to learn more last updated on 21/May/20
I appeciate sir.
$$\mathrm{I}\:\mathrm{appeciate}\:\mathrm{sir}. \\ $$

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