How to Calculate the Weight of Displaced Water

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The displacement method is a reliable way to determine the volume of an object that doesn't have measurable dimensions, such as a statue or a rock. You simply immerse the rock in water in a container large enough to hold it and measure the volume of water it displaces. The principle dates to the Greek mathematician Archimedes, who may have run through the streets shouting "Eureka" when he discovered it. If you want to know the weight of the displaced water, simply measure its volume and multiply by the density of water.

TL;DR (Too Long; Didn't Read)

You can determine the weight of a volume of displaced water by multiplying by the density of water. In CGS metric units, the density of water at 4 C is 1 gm/ml, so if you're using those units, the volume in milliliters and weight in grams are the same number to a high degree of accuracy.

Water Density Varies With Temperature

The density of water changes with temperature. Its maximum occurs at 4 degrees Celsius (39.2 degrees Fahrenheit). In metric units, it's 1 gm/ml in the CGS (centimeters, grams, seconds) system and 1,000 kg/m3 in the MKS (meters, kilograms, seconds) system. In the Imperial system, it's 62.42 lb/cu. ft. Water is the only compound that actually gets less dense when it freezes, and as the temperature rises, the density also decreases. At slightly less than room temperature, which is the temperature you're likely to do most experiments, the density is 0.9982 g/ml or 62.28 lb/cu.ft. That's a difference of only two thousandths of a percent, so it's important for only very accurate calculations.

Measure the Volume

When you use the displacement method, you have your choice of two ways to measure the volume of displaced water. One is to fill the container to the brim and catch the water that overflows in a graduated container. The other is to measure the change in the water level and calculate the volume using the dimensions of the container. If you're measuring the volume of a small sample, you can fill a graduated container to a certain mark, and simply read the scale when the water rises to determine the volume change. That's the customary procedure in a lab.

Determine Weight

Once you know the volume of the displaced water, you can immediately determine its weight by multiplying by the density of water at the relevant temperature. That's because the definition of density (d) is mass (m) divided by volume (v), so m = dv. In this context, mass and weight are synonymous, unless you're conducting the experiment in space.

If accuracy doesn't demand otherwise, simply stick with the density at 4 C. If you measure volume in CGS metric units, the measured volume in milliliters will then be equal to the weight (mass) in grams. In MKS units, multiply the volume in liters by 1,000 to get the weight in kilograms. If you're using imperial units, multiply the volume in cu.ft. by 62.42 to get the weight in pounds. If you measure volume in ounces, gallons or cubic yards, use these conversion factors:

  • 1 ounce = 10-3 cu. ft. 
  • 1 gallon = 0.134 cu. ft.
  • 1 cubic yard = 27 cu. ft.

References

About the Author

Chris Deziel holds a Bachelor's degree in physics and a Master's degree in Humanities, He has taught science, math and English at the university level, both in his native Canada and in Japan. He began writing online in 2010, offering information in scientific, cultural and practical topics. His writing covers science, math and home improvement and design, as well as religion and the oriental healing arts.

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