Mass vs. Weight

Date: 01/22/99 at 18:56:47
From: Charlene Grubb
Subject: Mass vs. weight

My math students often confuse mass with weight. Can you give me a 
definition of mass and how it differs from weight? Thank you.

Date: 01/23/99 at 16:04:20
From: Doctor Rick
Subject: Re: Mass vs. weight

Hi, Charlene. Thanks for writing!

"Mass" and "weight" are often used interchangeably in everyday use - or 
rather, since mass is not exactly an everyday term, "weight" is 
often used when "mass" is meant. Let's try to clear this up.

Weight is the force exerted on an object by gravity. It is measured in
newtons in the SI, and in poundals (foot-pounds per second squared) in 
the "English" system.

Mass has two meanings, both of which can be viewed as proportionality
factors. "Inertial mass" is the factor that determines how much force 
it takes to produce a given acceleration in an object, according to 
Newton's first law of motion:

  F = ma

"Gravitational mass" is the factor that determines how much 
gravitational force is developed between two objects at a given 
distance, according to Newton's law of gravity:

  F = -GMm/r^2

where G is the gravitational constant, r is the distance between the
objects, and M and m are the masses of the two objects.

Inertial mass and gravitational mass are the same. (This is still the
subject of experiments seeking to verify it more and more precisely, 
because it is intimately connected with Einstein's general theory of 
relativity.) Because of this, the two equations above combine into

  a = GM/r^2

On the surface of the earth, M (the mass of the earth) and r (the 
radius of the earth) are fixed, and so a (the acceleration due to 
gravity) is a constant, which we call g.

  g = 9.8 meters/sec^2 = 32 feet/sec^2

Therefore the force exerted by gravity (i.e., weight) and the mass of 
an object are proportional:

  W = mg

Because of this, we have gotten used to measuring weight in pounds or
kilograms, which properly speaking are units of mass, not weight. In 
part, this is just a matter of saying "weight" when we mean mass, and 
it's harmless because, if two objects weigh the same, then they have 
the same mass. But we go further and talk, for instance, of a pressure 
of 15 pounds per square inch. Here we mean that the FORCE exerted on 
one square inch is the same as the force exerted by a 15-pound mass.

To make allowances for this habit, units called "kilograms force" 
(kgf) and "pounds force" (lbf) are sometimes used. A kilogram force, 
for instance, is the force exerted by gravity on a mass of 1 kg.

  1 kgf = (1 kg)(9.8 m/s^2) = 9.8 Newtons

  1 lbf = (1 lb)(32 ft/s^2) = 32 poundals

This allows us to be scientifically correct and say that a bag of 
sugar weighs 5 pounds force.

When we leave the earth's surface, or when we need to be very accurate 
while comparing objects at different elevations, or when an object is 
not at rest, the distinction between mass and weight becomes important. 

For example, an astronaut in orbit is WEIGHTLESS. He/she is not 
MASSLESS. And this weightlessness is not due to being so far from earth 
(100 miles or so is insignificant compared with the radius of the 
earth, about 4000 miles). Rather, it is because the astronaut is not 
stationary, but is in free fall, along with the entire spacecraft, that 
gravity exerts no force. It is only when something prevents an object 
from falling with an acceleration equal to g that gravity exerts a 
force on the object.

Likewise, an astronaut on the moon has the same mass as on earth. But 
since the moon has about 0.272 times the radius of the earth, and about 
0.0123 times its mass, Newton's law of gravity says that the 
acceleration due to gravity on the moon's surface is about (0.0123)/
(0.272)^2 = 0.166 times that on earth. Thus the astronaut's weight is 
1/6 what it is on earth.

Here is a quick summary. Mass is an intrinsic property of an object,
something it takes with it wherever it goes. But weight depends on 
where an object is - its altitude above the earth, or what planet it 
is on. Mass tells us something about the object itself; weight tells 
us how the earth is interacting with the object.

You didn't say what grade level your students are, so I haven't tried 
to aim this at them. I leave it to you to digest it and translate it 
for their understanding. I hope it helps a little.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/