Question

A 1.0-m-diameter vat of liquid is 2.0 m deep. The pressure at the bottom of the vat is 1.3 atm. What is the mass of the liquid in the vat?

Short answer

The mass of liquid is 2435 kg

Step-by-step solution

Given information

Diameter of vat = 1 m

so radius = 0.5 m

depth of liquid = 2.0 m.

The pressure at the bottom of the vat = 1.3 atm.

Explanation

Total pressure is calculated by

P_Total = P_atm + hρg

P_Total = pressure at the bottom = 1.3 atm

P_atm = atmospheric pressure = atm

h = depth of liquid = 2 m

ρ= density of liquid

g = gravitational acceleration

As given

$$\begin{gathered} P_{Total}=P_{atm}+h\rho g=1.3\mathrm{atm} \\ \Rightarrow \mathrm{h\rho g}=0.3\mathrm{atm} \end{gathered}$$

Substitute the value and find density

$$\begin{gathered} \rho =\frac{P}{hg} \\ =\frac{30397.5}{2\times 9.8}=1551\mathrm{kg}·{\mathrm{m}}^{-3} \end{gathered}$$

From basic mensuration find the volume of liquid

$$\begin{gathered} V=\pi r^{2}h \\ =3.14\times (0.5{)}^2\times 2=1.57{\mathrm{m}}^3 \end{gathered}$$

Now find the mass using Mass = Volume x Density

$$\begin{gathered} M=V\times \rho \\ =1.57\times 1551=2435\mathrm{kg} \end{gathered}$$

So the mass of liquid is 2435 kg.