# 30 Density from Displacement and Weight

# Mass from Weight

Scales measure weight, but to calculate body density we need mass. Some scales read off mass, such as the electronic scale in the image below, even though they actually measure weight as discussed in the previous chapter.

Mass can be determined from a weight because weight is just the force of gravity on the body and force of gravity depends on mass in a known way. On the surface of the Earth, the force gravity on an object is related to its mass by the equation:

(1)

The acceleration due to gravity on Earth, typically abbreviated to *g*, has a value of 9.8 **m/s**** ^{2 }** and doesn’t change much over the entire surface of the Earth. Therefore we (and scales) can measure weight and then use equation (1) above to calculate mass. Unit 1 introduced this value as the accelration of an object in free-fall near the surface of Earth. Now we recognize that

*g*also serves as a constant value that relates mass to weight for objects on the surface of Earth.

Force is a vector, so we need to specify a direction for the gravitational force, which is toward Earth’s center. We can summarize the previous equation in symbol form:

(2)

# Body Weight and Mass on the Moon

The value of *g *only holds constant near the surface of the Earth, and therefore scales that use equation (1) to calculate mass from measured weight will read incorrect results. For example, your mass doesn’t change just because you go to the moon (there isn’t suddenly less of you), but your weight does change. In fact if you stood on a scale on the moon it would measure a weight about 1/6 of what it would read on Earth. The scale wouldn’t know you were on the moon instead of the Earth, so if the scale then tried to calculate your mass from weight, it would read a mass that is 1/6 the actual value. Of course you didn’t lose 5/6 of yourself on the way there, so that would not be correct.

# Calculating Body Density

We now know how to measure volume by displacement and how to determine mass from a weight measurement so we should be able to determine body density. First we measure the weight, then calculate the mass. Dividing the mass by the volume found from our displacement measurement will give us the body density. Give it a try:

### Reinforcement Exercises: Body Density

# More information on Gravity

When you do want to calculate the force of gravity and you are not near the surface of the Earth then use the Universal Law of Gravitation**. **

The Universal Law of Gravitation states that the gravitational force between two objects depends on the mass of each object ( and ) and the distance between their centers, (). To calculate the gravitational force we need to multiply the two masses together, divide by the distance between them squared, and finally multiply by the universal gravitational constant , which always has the same value of . Written in equation form the universal law of gravitation is:

(3)

### Reinforcement Exercise

- "Digi-keukenweegschaal1284" by Algont [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0/)] via wikimedia commons. ↵

the force of gravity on on object, typically in reference to the force of gravity caused by Earth or another celestial body

relation between the amount of a material and the space it takes up, calculated as mass divided by volume.

a measurement of the amount of matter in an object made by determining its resistance to changes in motion (inertial mass) or the force of gravity applied to it by another known mass from a known distance (gravitational mass). The gravitational mass and an inertial mass appear equal.

attraction between two objects due to their mass as described by Newton's Universal Law of Gravitation

the rate at which an object changes velocity when gravity is the only force acting on the object

a quantity of space, such as the volume within a box or the volume taken up by an object.

method for determining the volume of an object by measuring how much water it displaces

every particle attracts every other particle in the universe with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers