Chapter 8, Problem 15RQ

Consider two tubes of equal cross-sectional area. One is a 76-cm-tall tube of mercury and the other a 10.3-cm-tall tube of water. Which weighs more? Which exerts more pressure at the bottom of the tube?

To determine

The tube that weighs more and exert more pressure.

Answer to Problem 15RQ

Both the tubes weighs same and exerts equal amount of pressure at bottom.

Explanation of Solution

Given Info: The length of the tube of mercury is 76 cm and the length of the tube of water is 10.3 cm .

Write the expression for the pressure exerted by the mercury.

$P_m=h{\rho }_{m}g$

Here,

h is the height of the tube

${\rho }_m$ is the density of the mercury

g is the acceleration due to gravity

Substitute, 76 cm for h , 13600 kg/m³ for ${\rho }_m$ , and 9.8 m/s² for g in the above expression to get P_m .

$\begin{array}{c}{P}_m=\left(76\text{cm}\times \frac{1\text{m}}{100\text{cm}}\right)\times \left(13600\text{kg}/{\text{m}}^{\text{3}}\right)\times 10\text{m}/{\text{s}}^2\\ P_m=1.03\times {10}^5\text{N}/{\text{m}}^{\text{2}}\\ P_m\approx 1.03\times {10}^5\text{N}/{\text{m}}^{\text{2}}\end{array}$

Write the expression for the pressure exerted by the water.

$P_w=h{\rho }_{w}g$

Here,

h is the height of the tube

${\rho }_m$ is the density of the mercury

g is the acceleration due to gravity

Substitute, 10.3 m for h ,1000 kg/m³ for ${\rho }_w$ , and 9.8 m/s² for g in the above expression to get P_m .

$\begin{array}{c}{P}_w=\left(10.3\text{m}\right)\times \left(1000\text{kg}/{\text{m}}^{\text{3}}\right)\times 10\text{m}/{\text{s}}^2\\ P_w=1.03\times {10}^5\text{N}/{\text{m}}^{\text{2}}\\ P_w\approx {10}^5\text{N}/{\text{m}}^{\text{2}}\end{array}$

Thus, both the tubes exert equal amount of pressure at bottom.

Write the formula for the force exerted by the mercury.

$F_m=P_m{A}_m$ ……(i)

Here,

P_m is the pressure exerted by the mercury

A_m is the area of cross-section of tube.

Write the formula for the force exerted by the water.

$F_w=P_w{A}_w$ ……(ii)

Here,

P_w is the pressure exerted by the water

P_w is the area of cross-section of tube.

As the cross-sectional area of both the tubes are same, and the pressure exerted by both the liquids is also equal, then the force exerted by mercury and water will be same.

By using equation (i) and (ii),

$F_w=F_m$

The force exerted by the liquid at the bottom of the tube is the weight of the liquid. Thus, both the liquid weighs same.

Conclusion:

Therefore, both the tubes weighs same and exerts equal amount of pressure at bottom.