# Mass and Weight

## Mass and Weight - the Gravity Force

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**Mass** and **Weight** are two often misused and misunderstood terms in mechanics and fluid mechanics.

The fundamental relation between mass and weight is defined by Newton's Second Law. Newton's Second Law can be expressed as

F = m a(1)

where

F= force (N, lb_{f})

m= mass (kg,slugs)

a= acceleration (m/s^{2},)ft/s^{2}

### Mass

**Mass** is a measure of the amount of material in an object, being directly related to the number and type of atoms present in the object. Mass does not change with a body's position, movement or alteration of its shape, unless material is added or removed.

- an object with mass
*1 kg*on earth would have the same mass of*1 kg*on the moon

Mass is a fundamental property of an object, a numerical measure of its inertia and a fundamental measure of the amount of matter in the object.

### Weight

**Weight** is the **gravitational force** acting on a body mass. The generic expression of Newton's Second Law *(1)* can be transformed to express weight as a force by replacing the acceleration - *a* - with the acceleration of gravity - *g* - as

F_{g}= m a_{g}(2)

where

F= gravitational force - or_{g}weight(N, lb_{f})

m= mass(kg,slugs)

a_{g}= acceleration of gravity on earth(9.81 m/s^{2},32.17405 ft/s)^{2}

#### Example - The Weight of a Body on Earth vs. Moon

The acceleration of gravity on the moon is approximately *1/6* of the acceleration of gravity on the earth. The weight of a body with mass *1 kg* on the earth can be calculated as

*F _{g_}*

_{earth}= (1 kg)

*)**(9.81 m/s*^{2}* = 9.81 N*

The weight of the same body on the moon can be calculated as

*F _{g_}*

_{moon}= (1 kg) (

*) / 6)**(9.81 m/s*^{2}* = 1.64 N*

The handling of mass and weight depends on the systems of units used. The most common unit systems are

- the International System -
*SI* - the British Gravitational System -
*BG* - the English Engineering System -
*EE*

One newton is

- ≈ the weight of one hundred grams -
*101.972 gf (g*_{F}) or 0.101972 kgf (kg_{F}or kilopond - kp (pondus is latin for weight)) - ≈ halfway between one-fifth and one-fourth of a pound -
*0.224809 lb or 3.59694 oz*

### The International System - *SI*

In the SI system the mass unit is the *kg* and since the weight is a force - the weight unit is the *Newton* (*N*). Equation *(2)* for a body with *1 kg* mass can be expressed as:

= (1 kg) (9.807 m/sF_{g}^{2})

= 9.807 (N)

where

9.807 m/s^{2}= standard gravity close to earth in the SI system

As a result:

- a
*9.807 N*force acting on a body with*1 kg*mass will give the body an acceleration of*9.807 m/s*^{2} - a body with mass of
*1 kg*weights*9.807 N*

- More about the SI System - A tutorial introduction to the SI-system.

### The Imperial British Gravitational System - *BG*

The British Gravitational System (Imperial System) of units is used by engineers in the English-speaking world with the same relation to the foot - pound - second system as the meter - kilogram - force second system (SI) has to the meter - kilogram - second system. For engineers who deals with forces, instead of masses, it's convenient to use a system that has as its base units length, time, and force, instead of length, time and mass.

The three base units in the Imperial system are the *foot, the second, and the pound-force*.

In the BG system the mass unit is the *slug* and is defined from the Newton's Second Law *(1)*. The unit of mass, the *slug*, is derived from the pound-force by defining it as the mass that will accelerate at *1 foot per second per second* when a *1 pound-force* acts upon it:

1 lb_{f}= (1 slug)(1 ft/s^{2})

In other words, *1 lb _{f} (pound-force)* acting on

*1 slug*mass will give the mass an acceleration of

*1 ft/s*

^{2}.

The weight of the mass from equation *(2)* in BG units can be expressed as:

(lbF_{g}_{f}) = m (slugs) a_{g}(ft/s^{2})

With standard gravity -* a _{g} = 32.17405 ft/s^{2}* - the mass of

*1 slug*weights

*32.17405 lb*

_{f }*(pound-force)*.

### The English Engineering System - *EE*

In the English Engineering system of units the primary dimensions are are *force, mass, length, time and temperature.* The units for force and mass are defined independently

*the basic unit of mass is**pound-mass**(lb*_{m})*the unit of force is the**pound**(lb) alternatively**pound-force**(lb*_{f}).

In the EE system *1 lb of force* will give a mass of *1 lb _{m}* a standard acceleration of

*32.17405 ft/s*.

^{2}Since the EE system operates with these units of force and mass, the Newton's Second Law can be modified to

F = m a / g_{c}(3)

where

g_{c}= a proportionality constant

or transformed to weight

= m aF_{g}_{g}/ g_{c}(4)

The proportionality constant g_{c} makes it possible to define suitable units for force and mass. We can transform (4) to

1 lb_{f}= (1 lb_{m}) (32.174 ft/s^{2}) / g_{c}

or

g_{c}= (1 lb_{m}) (32.174 ft/s^{2}) / (1 lb_{f})

Since *1 lb _{f}* gives a mass of

*1 lb*an acceleration of

_{m}*32.17405 ft/s*and a mass of

^{2}*1 slug*an acceleration of

*1 ft/s*, then

^{2}

1 slug = 32.17405 lb_{m}

### Example - Weight versus Mass

The mass of a car is *1644 kg*. The weight can be calculated:

F_{g}= (1644 kg) (9.807 m/s^{2})

= 16122.7 N

= 16.1 kN

- there is a force (weight) of *16.1 kN* between the car and the earth.

*1 kg gravitation force = 9.81 N = 2.20462 lb*_{f}

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