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# Mass, Weight and Gravity

by Ron Kurtus (revised 23 June 2012)

The * mass* of an object is the amount of matter it contains regardless of its volume or any forces acting on it. There are two measurements of mass: gravitational and inertial.

Gravitational mass is the mass of a body as determined by its response to the force of gravity, such as done on a balance scale. Inertial mass is the measurement of the mass of an object as measured by its resistance to acceleration. Gravitational mass and inertial mass have been shown to be equivalent.

** Weight** is defined as the force of gravity on a mass. A spring scale can be used to measure weight. Although mass is the same on the Moon as it is on Earth, the weight of an object is 1/6 as much on the Moon as it is on the Earth.

Questions you may have include:

- How is gravitational mass measured?
- How is inertial mass measured?
- What is weight?

This lesson will answer those questions. Useful tool: Units Conversion

## Gravitational mass

The measurement of the gravitational mass of an object is done by comparing the arms of a balance scale between a unit mass and the mass of interest.

### Unit of mass

The unit of mass in the metric or SI system is the kilogram (kg). In the English system, the unit of mass is the pound-mass (lb-mass) or the slug.

Originally, a kilogram was defined as the amount of matter in 1 liter (1 L) of water at the temperature of melting ice (0° C). A gram is 1 cubic centimeter (1 cc) of water at 0° C.

Presently, a metal standard is used to designate a kilogram instead of a quantity of water.

### Balance scale to measure mass

A balance scale is often used to measure mass by comparing the moment arms. A unit mass (1 kg) or some known mass is used as a basis of comparison with an unknown mass.

Balance scale used to compare mass

The pivot point of the scale is adjusted so that the objects balance. Then the mass is determined from the ratio:

m_{1}d_{1}= m_{2}d_{2}

m_{2}= m_{1}d_{1}/d_{2}

where

**m**is the unit mass_{1}**d**is the moment arm of the unit mass_{1}**m**is the test mass_{2}**d**is the moment arm of the unit mass_{2 }

If **m _{1}** = 1 kg, then:

mkg_{2}= d_{1}/d_{2}

Although the official unit of mass is 1 kg, you may use a mass with a fraction of that mass, such as 10 grams. Likewise, the moment arm may be in centimeters, instead of meters. Similar variations are true if you are using the English system for mass and length.

## Inertial mass

Inertial mass is determined by applying *Newton's Second Law*, which states that a force is required to accelerate a mass and overcome its inertia. This is expressed in the equation:

F = ma

where

**F**is a force in newtons (N) or pounds (lbs) that is necessary to overcome the inertia of the mass**m**is the mass in kilograms (kg) or pound-mass (lbs-mass)**a**is the resulting acceleration in m/s^{2}or ft/s^{2}

This mass is sometimes called *inertial mass* since the force is overcoming the inertia of the mass.

### Using a spring to apply force

The mass of an object can be determined experimentally by applying *Hooke's Law* for springs, which says that the force is proportional to the length stretched or compressed in an ideal spring:

F = kΔx

where

**F**is the force applied by an ideal spring**k**is the spring constant, which depends on the spring material**Δx**is the change in the length of the spring (delta**x**)

Compressing a spring requires a force. The compressed spring then has potential energy that, when released, will accelerate the mass at the free end of the spring.

Combining the force equations:

ma = kΔx

m = kΔx/a

Thus, knowing the spring constant and measuring the compression displacement and acceleration, you can determine the mass of the object. Of course, this is an ideal situation. The mass of the spring, friction and other factors come into play.

Released spring accelerates mass

### Using ratio

If two different masses are accelerated by the same spring configuration, the ratio of those masses can be found. This eliminates the need to know **k** and **Δx**.

m_{1}a_{1}= kΔx

m_{2}a_{2}= kΔx

m_{1}a_{1}=m_{2}a_{2}

Also, if **m _{1}** is a unit mass,

**m**can be found from the ratio of accelerations:

_{2}

m_{2}= a_{1}/akg_{2}

### Gravitational and inertial equivalent

Scientists have wondered if the effect of gravity on a mass was the same as the effect of acceleration. Experiments have shown that gravitational mass was equivalent to inertial mass. This fact was instrumental in the determining of the *General Relativity Theory of Gravitation*.

## Weight

The weight of an object is the force of gravity on the mass of the object:

F = mg

or

W = mg

where

**F**is the force caused by gravity on the mass in newtons (N) or pounds (lbs)**m**is the mass of the object in kg or pound-mass**g**is the acceleration due to gravity (9.8 m/s^{2}or 32 ft/s^{2})**W**is the weight in N or lbs

### Confusion between weight and mass

Many people (and even textbooks) mix up mass and weight. They will say that an object weighs 25 kg and another object as a mass of 2 pounds. Both expressions are scientifically incorrect.

Mass is the amount of matter. The metric or SI unit of mass is the kilogram. The English unit of mass is the slug or pound-mass.

The units of weight are units of force. The metric or SI unit of weight is the Newton (N) or sometimes the Kg-force. The English unit of weight is the pound (lb).

(See Confusion about Mass and Weight Units for more information.)

### Measuring weight

A spring scale is often used to measure an object's weight. Again, Hooke's Law is applied. The displacement a spring is stretched for a given force or weight is first calibrated. Then you can use the spring scale to measure the weight of various objects.

Spring stretches according to weight

Spring scales are calibrated with a known weight, such that knowing **Δy** will give you the weight, **W**. Within a range of weights, the weight is proportional to the displacement stretched. In other words:

W_{2}= W_{1}Δy_{2}/Δy_{1}_{}

where

**W**is the unknown weight_{2}**W**is the unit weight or calibrating weight_{1}**Δy**is the stretch of the spring for the unknown weight_{2}**Δy**is the calibrating stretch of the spring_{1}

For example, if the scale had been calibrated **W _{1} =** 1 lb stretched the spring

**1 inch, then if the spring was stretched**

Δy

Δy

_{1}=**Δy**2.5 inches,

_{2}=**W**2.5 lbs.

_{2}=### Weight on the Moon

The mass of an object—or the amount of matter it contains—is the same on the Moon as it is on the Earth. However, the weight of the object is a function of the acceleration due to gravity. Since gravity on the Moon is about 1/6 of that on Earth, an object will weigh 1/6 as much on the Moon.

W_{M}= mg_{M}

where

**W**is the weight on the Moon_{M}**m**is the mass of the object**g**is the acceleration due to the Moon's gravity (1.6 m/s_{M}^{2}or 5.3 ft/s^{2})

Since **g _{M} = g/6**, then:

W_{M}= W/6

If you weigh 60 kg (132 pounds) on the Earth, you would weigh only 10 kg (22 lbs) on the Moon.

## Summary

Gravitational mass of an object is determined by using a balance scale to compare its mass with a unit mass. Inertial mass is the measurement of the mass of an object measured by its resistance to acceleration. Gravitational mass and inertial mass have been shown to be equivalent.

Weight is defined as the force of gravity on a mass. A spring scale can be used to measure weight.

The mass of an object is the same on the Moon as it is on Earth, but its weight is 1/6 as much on the Moon as on the Earth.

Gravity is a heavy subject

## Resources and references

### Websites

**Difference Between Pound-Force and Pound-Mass** - Engineerography Blog

### Books

**Top-rated books on Simple Gravity Science**

**Top-rated books on Advanced Gravity Physics**

## Questions and comments

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## Mass, Weight and Gravity