33 Hydrostatic Weighing

The method of allows us to determine the average (\rho) of an object without the need for a  measurement. Instead, we measuring only the objects weight (W_0) and  apparent weight (F_A) when submerged and enter them into the equation below to calculate the density. To see how we arrive at this useful result, follow the steps in the at the end of this chapter.

(1)   \begin{equation*} \rho = \frac{W_O}{W_O-F_A}\rho_W \end{equation*}

Reinforcement Exercises

The previous equation is very similar to the equation used to determine body density from , but you will notice a slight difference. To ignore air and other gasses trapped inside the body, known as the residual volume (RV), the previous equation is modified to approximate the body density (\rho_B)::

(2)   \begin{equation*} \rho_B = \frac{W_O}{\frac{W_O-F_A}{\rho_W}-RV + 0.1} \end{equation*}

The residual volume needed to determine body density is approximated from equations based on empirical observations:

For women:

RV = [0.009\, \times\, Age(\bold{yrs})]+[0.032\,\times\,Ht(\bold{cm})]-3.90

For men:

RV = [0.0115\, \times\, Age(\bold{yrs})]+[0.019\,\times\,Ht(\bold{cm})]-2.24

Finally, the body fat percentage (\%BF) can be calculated using equations based on empirical measurements. Two of the most common are the Siri Equation and the Schutte Equation:

Siri Equation:

(3)   \begin{equation*} \%BF = \frac{495}{\rho_B}-450 \end{equation*}

Schutte Equation:

(4)   \begin{equation*} \%BF = \frac{437}{\rho_B}-393 \end{equation*}

Keep in mind that if you look up these equations from other sources you might see different symbols used, but the equations are actually the same. For example, the image below shows how the body density, residual volume, and body fat equations are related, but the symbols used are: body density = D_b, water density = D_{H2O}, body weight = BW, and apparent weight = UWW (for under-water weight).

Equations for residual volume are given for men and women. For men: 0.0115 x age (years) + 0.019 x height (cm) -2.24. For women: 0.009 x age (years) + 0.032 x height (cm) -3.90. An arrow shows where these values are used in an equation calculating body density: Db = BW/[(BW-UWW)/Dh2o –(RV +0.1)]. Arrows indicate where the body density is used in computing body fat percentage by two methods. Siri: BF% = 495/Db -450. Shutte: BF% = 437/Db -393
Formulas used in calculating residual lung volume, body density, and body fat percentage. Image Credit: Adapted from  Measure Body Fat Via Under Water Weighing by  via Instructables

[1]

Specific Gravity

The ratio of the of a substance to that of water is known as the . Specific gravity can be determined by . If we simply divide both sides of our density equation by the density of water we will have a formula for the specific gravity with weight and apparent weight as input:

(5)   \begin{equation*} SG = \frac{\rho}{\rho_W} = \frac{W_O}{W_O-F_A} \end{equation*}

Reinforcement Exercises

Hydrostatic Weighing Equation Derivation

We arrived at equation (1) by starting with the definition of an object’s as object divided by object :

    \begin{equation*} \rho = \frac{m_O}{V_O} \end{equation*}

We can find the mass of an object if we divide its weight by g:

    \begin{equation*} m_O = \frac{W_O}{g} \end{equation*}

Inserting that result for mass into the density equation we have:

    \begin{equation*} \rho = \frac{W_O}{gV_O} \end{equation*}

For a completely submerged object the volume of water is equal to the volume of the object, so we can replace V_O with V_D.

    \begin{equation*} \rho = \frac{W_O}{gV_D} \end{equation*}

Using the definition of density again, we can replace V_D with the displaced water mass (m_D) divided by water density (\rho_W) and then simplify a bit:

    \begin{equation*} \rho = \frac{W_O}{g(m_D/\rho_W)} = \frac{W_O}{g m_D}\rho_W \end{equation*}

We can look up the density of water, but it depends on the water temperature, which is why its important to measure the water temperature when . Notice that we happen to have the mass of displaced water multiplied by g in the previous equation. That is exactly how we calculate the weight of the displaced water (W_D), so we can make that substitution:

    \begin{equation*} \rho = \frac{W_O}{W_D}\rho_W \end{equation*}

which tells us that the   pushing upward on objects in a fluid is equal to the weight displaced fluid. Therefore we can replace W_D with F_B.

    \begin{equation*} \rho = \frac{W_O}{F_B}\rho_W \end{equation*}

For an object in (holding still), the forces must all cancel out. Therefore, when the buoyant force helps to lift the submerged object, a smaller force will be required to hold it still and its apparent weight will be less than the actual weight by an amount equal to the buoyant force. We know the bouyant force (F_B) must then be equal in size to the difference between the weight (W_O) and the apparent weight (F_A):

    \begin{equation*} F_B = W_O - F_A \end{equation*}

Making that replacement in our density equation we have:

    \begin{equation*} \rho = \frac{W_O}{W_O-F_A}\rho_W \end{equation*}

We now have an equation that allows us to calculate the density of an object by measuring only its and , as long as we know the of the fluid we are using.


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