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Two barometers are made with water and mercury. Ifthe mercury column is 30 in. tall, how tall is the watercolumn?

User Strongriley
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1 Answer

23 votes
23 votes

We know that two barometers are measuring the same pressure when the mass in their columns are equal, that is, the two barometers will measure the same when:


m_(Hg)=m_(H2O)

To find the mass of the barometer we need to remember that the mass is related to the density of the substance and its volume by:


m=\rho V

Hence the first equation takes the form:


\rho_(Hg)V_(Hg)=\rho_(H2O)V_(H2O)

Now, let's assume the barometers are cylindrical, then their volume is given by:


V=Ah

where A is the area of the base; let's further assume the area of the base is equal for both barometers; plugging this in our equation we have:


\begin{gathered} \rho_(Hg)Ah_(Hg)=\rho_(H2O)Ah_(H2O)_{} \\ \rho_(Hg)h_(Hg)=\rho_(H2O)h_(H2O) \end{gathered}

Hence we have the following equation relating the densities of the substances in the barometer and the height of the column in it:


\rho_(Hg)h_(Hg)=\rho_(H2O)h_(H2O)

Now, before we plug the values of the densities we will convert the height of the mercury column to cm (this will make the operations easier), let's do that:


30in\cdot(2.54cm)/(1in)=76.2cm

The density of mercury is 13.6 g/cm^3, the density of water is 1 g/cm^3; plugging the values we know we have that:


\begin{gathered} (13.6)(76.2)=(1)h_(H2O) \\ h_(H2O)=1036.32 \end{gathered}

Hence, the height of the column of water is 1036.32 cm, let's convert this into inches:


1036.32cm\cdot(1in)/(2.54cm)=408

Therefore, the height of the column of water is 408 in tall.

User Malia
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