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, lbf)
m = mass (kg, slugs)
a = acceleration (m/s2, ft/s2)
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
Fg = m ag (2)
where
Fg = gravitational force - or weight (N, lbf)
m = mass (kg, slugs)
ag = acceleration of gravity on earth (9.81 m/s2, 32.17405 ft/s2)
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
Fg_earth = (1 kg) (9.81 m/s2)
= 9.81 N
The weight of the same body on the moon can be calculated as
Fg_moon = (1 kg) ((9.81 m/s2) / 6)
= 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 (gF) or 0.101972 kgf (kgF, kp)
- ≈ 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:
Fg = (1 kg) (9.807 m/s2)
= 9.807 (N)
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 weights 9.807 N
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 lbf = (1 slug)(1 ft/s2)
In other words,
1 lbf (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:
Fg (lbf) = m (slugs) ag (ft/s2)
With standard gravity -
ag = 32.17405 ft/s2 - the mass of
1 slug weights
32.17405 lbf (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 (lbm)
- the unit of force is the pound (lb) alternatively pound-force (lbf).
In the EE system
1 lb of force will give a mass of
1 lbm 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 / gc (3)
where
gc = a proportionality constant
or transformed to weight
Fg = m ag / gc (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 lbm) (32.174 ft/s2) / gc
or
gc = (1 lbm) (32.174 ft/s2) / (1 lbf)
Since
1 lbf gives a mass of
1 lbm an acceleration of
32.17405 ft/s2 and a mass of
1 slug an acceleration of
1 ft/s2, then
1 slug = 32.17405 lbm
Example - Weight versus Mass
The mass of a car is
1644 kg. The weight can be calculated:
Fg = (1644 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.
- 1 kg gravitation force = 9.81 N = 2.20462 lbf
