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Mass vs. Weight

Mass vs. 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.

  • mass electron 9.1095 10 -31 kg
  • mass proton 1.67265 10 -27 kg
  • mass neutron 1.67495 10 -27 kg

Weight

Weight - force and acceleration of gravity

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 g = gravitational force - or weight (N, lb f )

m = mass (kg , slugs (lb m ))

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) ( (9.81 m/s 2 ) / 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 (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:

F g = (1 kg) (9.807 m/s 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

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 foot, second and 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 with 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 of mass will give the mass an acceleration of 1 ft/s 2 .

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

F g (lb f ) = m (slugs) a g (ft/s 2 )

With standard gravity - a g = 32.17405 ft/s 2 - the weight (force) of 1 slug mass can be calculated as

F g =  (1 slug) ( 32.17405 ft/s 2 )

32.17405 lb f

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 f 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 (force)

F g = m a 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 m an acceleration of 32.17405 ft/s 2 and a mass of 1 slug an acceleration of 1 ft/s 2 , then

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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Related Topics

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  • Fluid Mechanics

    The study of fluids - liquids and gases. Involving velocity, pressure, density and temperature as functions of space and time.
  • Mechanics

    Forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more.
  • Statics

    Loads - forces and torque, beams and columns.

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