Mass versus Weight

Date: 09/09/99 at 22:56:22
From: Scott Parks
Subject: Mass

I'm a grade 5 teacher at a local elementary school. In our science 
unit we refer to mass as "the amount of matter in an object." In math, 
this term is often used as a weight. But, weight is defined as "the 
gravitational pull exerted on a body." Obviously, weight and mass are 
two very different things. In space, the mass of an object would be 
the same as on earth and would be constant, but the weight of an 
object would be very different.

An example we've looked at is a box of cereal. If we are told it is 
450 grams, does that refer to the weight of the cereal, or the amount 
of matter in the box? Can we quantify matter (the mass) using weight?

My question is, on earth do mass and weight refer to the same thing? 
Is mass equal to weight?

Thank you,
Scott

Date: 09/10/99 at 12:13:05
From: Doctor Rick
Subject: Re: Mass

Hi, Scott.

There are people who get very uptight about distinguishing mass from 
weight, and they argue, for example, about whether the pound was 
historically defined as a unit of mass or weight. The fact is, though 
they are very different things as you say, we can get away with using 
the two more or less interchangeably as long as we're on the surface 
of the earth. A pound is a unit of mass, but we can use it to express 
weights.

When we say, "this bag of flour weighs 5 pounds," this is a shorthand 
for "this bag of flour weighs the same as a mass of 5 pounds." If we 
weigh the flour on a two-pan balance or on a triple-beam balance such 
as chemists or doctors use, this is exactly what we are doing: we 
compare the weight of the flour against the weight of a set of 
standard masses.

If you took the balance to the moon, the flour would have a different 
weight. But so would the standard masses. You would find that exactly 
the same masses would be needed to match the weight of the flour. So 
you would still say "this bag of flour weighs [the same as a mass of] 
5 pounds." Though a balance scale requires gravity in order to work, 
it measures mass, not weight.

A spring scale, on the other hand, compares the weight of the flour 
against the force on a standard spring - not a gravitational force. 
Thus it measures the force of gravity on the flour (which we call 
weight), and not its mass. If you took the spring scale to the moon, 
or even to the top of a mountain if it were accurate enough, you would 
find that it gave a different reading: the flour would no longer 
"weigh 5 pounds." More precisely, it would not weigh the same that a 
mass of 5 pounds weighs _on the surface of the earth_ (since that's 
what we mean by a weight of 5 pounds).

Scientists, who need to be precise, always refer to mass when 
measuring the quantity of a substance. They use balances, not spring 
scales, because they do not want the amount of substance that they 
measure to depend on their altitude, so they need an instrument that 
actually measures mass.

Scientists and engineers have also formalized the meaning of something 
"weighing 5 pounds" by defining units such as "pound force" and 
"kilogram force." A pound force (lbf) is the force exerted by gravity 
on a mass of one pound, given a specific, average value for g, the 
acceleration of gravity (9.80665 meters per second per second). This 
allows them to speak of a force of one pound. The traditional unit of 
torque, the pound-foot, is a pound force times a lever arm of one 
foot.

I hope I've answered your questions! In summary: mass does not equal 
weight, but to a good approximation there is a constant 
proportionality factor between them as long as we're on the earth. 
This allows us to determine an object's mass by measuring its weight, 
and to express weight in units related to mass (pounds force or 
kilograms force).

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