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?

Accepted Answer

Step 1: Understand the relationship between pressure, depth, and density. The pressure at the bottom of the vat is due to the weight of the liquid above it. The formula for pressure at a depth in a liquid is given by: P=P_{atm}+ρgh, where P is the total pressure, P_{atm} is atmospheric pressure, ρ is the density of the liquid, g is the acceleration due to gravity, and h is the depth of the liquid. Step 2: Rearrange the formula to solve for the density of the liquid. Subtract atmospheric pressure from the total pressure to isolate the term related to the liquid: ρ=(P-P_{atm})gh. Use the given values: P=1.3 atm, P_{atm}=1.0 atm, g=9.8 m/s², and h=2.0 m. Step 3: Calculate the volume of the liquid in the vat. The vat is cylindrical, so the volume can be calculated using the formula for the volume of a cylinder: V=π$$r^{2}h$$, where r is the radius of the vat and h is the depth. The diameter is given as 1.0 m, so the radius is r=0.5 m. Substitute r=0.5 m and h=2.0 m into the formula. Step 4: Use the density and volume to calculate the mass of the liquid. The formula for mass is m=ρV, where ρ is the density and V is the volume. Substitute the values of ρ and V obtained from the previous steps. Step 5: Ensure unit consistency throughout the calculations. Convert pressure from atm to Pascals (1 atm = 101325 Pa) and verify that all units are in SI (meters, kilograms, seconds). Perform the calculations step by step to find the mass of the liquid in the vat.

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?

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