# Hydrostatic Weighing

## Materials:

- lab sheet
- writing utensil
- calculator
- video of lab for distance learners

OR

- graduated cylinder (required for displacement method, but not required for hydrostatic weighing method)
- plastic bin to contain spills
- object to submerge (potato)
- scale with at least 0.1 gram precision
- force sensor + computer with control and analysis software OR spring scale with 0.1 gram precision (required for hydrostatic weighing method, but not for required for displacement method or sphere approximation method)

## Observation

Density is defined as mass per volume. Therefore, the average density of an object can be calculated using measurements of its mass and its volume. However, it can be difficult to determine the volume of oddly-shaped and non-uniform objects, such as people.

## Question

Is there a way to determine the density of an object without actually measuring it’s volume?

## Search Existing Knowledge

1) Can you find any information on a method to determine the density of an object without measuring its volume? What measurements do you need to make for the method you found? Cite your sources.

## Hypothesis Generation:

Generate a hypothesis. Testing your hypothesis should provide information that helps to answer your question.

2) If the method for determining density that does not rely on a volume measurement is reliable, then that method will provide a result that is less than ___% different from the density calculated from mass and volume directly. (Choose a percent difference that will make testing your hypothesis helpful in answering your question.)

## Experimental Hypothesis Testing

We will determine the density of a potato using mass and volume (found two separate ways) and then compare those results to density determined using a hydrostatic weighing, which does not require a volume measurement. The following video provides you with data from a particular potato so that you can complete the lab from home without any equipment.

### Sphere Volume Method

**[Outcome 4-1]**

3) Record the potato mass here (in units of grams):__________**g**

4) Record the potato circumference here (in units of **cm**): __________**cm**

5) Look up how the circumference and radius of a sphere are related and then assume the potato is a sphere to calculate its radius. Cite your sources. Show your work.

6) Look up how the volume of a sphere depends on the radius and then calculate volume of the potato. Show your work and cite your sources.

7) Use the mass and volume to calculate the potato density. You may need to look up how density is defined by mass and volume. Cite your sources and show your work.

### Displacement Method

**[Outcome 4-1]**

9) Record the volume of water displaced by the potato when fully submerged:___________ **cm ^{3}**. Assuming there are no trapped air bubbles, this is same as the volume of the potato.

10) How do the values for volume found in the previous two methods compare? Which do you trust more?

**[Outcome 4-4]**

11) Use your recorded mass and displaced volume to calculate the density of your potato. Show your work.

12) How do the results from the previous two methods of determining density compare? Which do you trust more?

### Hydrostatic Weighing Method

13) Measure the weight of your potato in air and record here (use units of Newtons) :___________**N**

14) Fully submerge the potato and record the reading on the scale. This is known as the the apparent weight because the object appears to weigh less when it is submerged. The apparent weight is sometimes called the under-water weight. Record the apparent weight here:___________**N**

15) Based on your measured weight and apparent weight, what must be the size of the buoyant force that pushes up on the submerged potato? [Hint: The object is not moving, so it must be in static equilibrium].

16) You now know the buoyant force, so according to Archimedes principle you also know the weight of the water displaced by the object. Explain using Archimedes’ Principle

17) You now have the weight of the displaced water. Use that to calculate the mass of the water displaced. You may need to look up the equation relating mass and weight for an object near the surface of the Earth. Show your work.

18) Look up the density of water in units of **kg/m ^{3}** and record here:________

19) Now use the density of water and the mass of the displaced water to find the volume of the displaced water in **m ^{3}**. You may need to rearrange the density equation. Show your work.

20) How does this volume of displaced water compare to the volume of the potato when it is fully submerged?

21) Notice that with the hydrostatic method we found the volume of the potato from measurements of weight and apparent weight and the known water density, but we didn’t attempt to measure the volume. Convert this volume to units of **cm ^{3}**.

22) Now you know the volume of the potato, so you just need to know its mass to calculate it’s density. Use the weight that you found at the start of the hydrostatic method to calculate the mass in **kg**. You may need to look up the equation that relates mass and weight for an object near the surface of the Earth. Cite your source and show your work.

23) Convert the potato mass to units of grams.

24) Now use the potato mass (in **g**) and volume in (in **cm ^{3}**) to calculate the potato density. Show your work.

## Data Analysis

25) You have now determined density of the potato from the mass and volume measurements and also from weight measurements only (hydrostatic method). Calculate a percent difference between the density obtained from the hydrostatic method and from mass and volume (use the value that relied on the volume found using displacement, not treating the potato as a sphere). Here is how you calculate percent difference:

- Find the difference between the two values
- Find the average of the two values
- Divide the difference by the average and multiply by 100%

26) Do the results support your hypothesis? (Be sure to look back at what your hypothesis was).

## Conclusions

27) Our original question was: *Is there a way to determine the density of an object without actually measuring it’s volume?* Based on the results of your data analysis, how would you answer this question now?

## Modeling

The body fat percentage of a humans can be estimated using a rather complex that requires body density, , as an input. Hydrostatic weighing allows us to determine the density of a person to use in the model. The model looks like this (using under water weight to mean the same thing as apparent weight).

The work you did in the hydrostatic weighing part of this lab was a stepwise recreation of the equation for calculating body density that is shown in the diagram above, except for the part about residual volume of body gasses () because our potato didn’t have lungs or intestines. For people, residual volume can be estimated from empirical equations that depend on age, height, and gender, as seen in the diagram.

28) For fun we will use the potato density we found using the hydrostatic method, as an input to calculate the “body fat percentage” of our potato, just as we would for a person. Follow through with that calculation of body fat percentage using the Siri Equation seen in the above diagram. Show your work.

29) Does the Siri equation produce a body fat percentage that between 0% and 100% for the potato?

Be careful when applying any model to a system it wasn’t meant to describe! Just because it will give you a result, that doesn’t mean the result is meaningful. On the flip side, even if you are using a model as it was designed, you have to provide good input data if you want to get meaningful outputs. Models aren’t magic and they won’t make up for bad data.

mathematical explanation of the relation between measured values that is used for making predictions