# Mass from Weight

Scales measure , but to calculate body we need . 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.

can be determined from a because weight is just the 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 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. Understanding why the constant

*g*is called the

*acceleration due to gravity*requires introducing acceleration, which we will do in a later unit, so for now we recognize it as a constant value that relates mass and 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 always down toward Earth’s center. We can summarize the previous equation in symbol form:

(2)

### Reinforcement Exercise

In 2016 Helen Maroulis became the first American woman to win an Olympic gold medal in wrestling. She competes in the 53 **kg **class, which most people call her weight class. However, 53 **kg** is not actually a , it’s a . Use the formula provided above to calculate Helen’s weight in Newtons.

Find a between Newtons and pounds and convert Helen’s competition weight to **lbs**.

# Calculating Body Density

We now know how to measure by and how to determine mass from a weight measurement so we should be able to determine body . 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

A person weighs 902 **N** (203 **lbs**) What is the persons mass? (Assume they are on Earth’s surface)

The same person displaces 0.089 **m ^{3} **of water volume when fully submerged. What is the body density of the person?

# 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 doesn’t change just because you go to the moon (there isn’t suddenly less matter inside you), but your 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.

# Universal Law of Gravitation*

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

The 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

Look up the mass and radius of the Earth and enter these into the along with the value for provided earlier. Use the mass of the Earth as so is the only thing left unknown in the equation. Multiply and divide everything other than as indicated by the equation to get multiplied by a single number. What number did you find should be multiplied by an object’s mass to find the force of gravity? How does the resulting equation compare to the equation for the force of gravity near the surface of Earth that we stated earlier?

- "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 number that relates two different units of measure for the same quantity and allows conversion between the two units

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