Mass and Weight

Weight and mass - the difference

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.

The 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. Transforming Newton's Second Law related to the weight as a force due to gravity can be expressed as

W = m g (2)

where

W = weight (N)

m = mass (kg)

g = acceleration of gravity (m/s²)

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

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

1 kg = 2.20462 lbf
1000 kg = 1 tonne (metric) = 0.9842 tons (imperial)
1 ton (US) = 2000 lb = 907.185 kg

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:

w = (1 kg) (9.807 m/s²)

= 9.807 (N) (2b)

where

9.807 m/s² = 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/s²
a body with mass of 1 kg weighs 9.807 N

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

The 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 = (1 slug)(1 ft/s²)

In other words, 1 lb (pound) force acting on 1 slug mass will give the mass an acceleration of 1 ft/s².

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

w (lb) = m (slugs) g (ft/s²)

With a standard gravity - g = 32.17405 ft/s² - 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/s².

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

w = m 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 lbf = (1 lb_m)(32.174 ft/s²) / g_c

or

g_c = (1 lb_m)(32.174 ft/s²)/(1 lb_f)

Since 1 lb_f gives a mass of 1 lb_m an acceleration of 32.17405 ft/s² and a mass of 1 slug an acceleration of 1 ft/s², then

1 slug = 32.17405 lb_m

Example - Weight versus Mass

A car's mass is 1,644 kg. The weight can be calculated:

w = (1,644 kg)(9.807 m/s²)

= 16122.7 N

= 16.1 kN

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

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