6.7. Weight

I’ve been toying with you over units so let’s clean this up. Since we’re Earth-bound we tend to mix up the units of weight and mass at the grocery store. You’ll appreciate that in a bit.

A gram is a unit of mass, (1000 grams in a kilogram) while an ounce is a unit of force (16 ounces in a pound). A kilogram still a measure of mass and a pound is a measure of weight. But we get away with using both systems since we tend to buy things and compare them on Earth. If we had a Mars colony, well then there would be trouble. If you put that 5 Earth-pound bag of Gold Bond flour from Kroger on your bathroom scale which you transported to Mars, it would read back about 2 pounds. But in each location, the bag of flour would still have a mass of 2.27 kg. How can that be?

Weight is a special kind of force—still subject to being the “\(F\)” in Newton’s 2nd law, but called out as a unique characteristic of massive objects on the Earth.

What Galileo showed was that the acceleration due to gravity near the surface of the Earth is a particular value—he presumed it was a constant. If the ground disappeared beneath your feet, then because you have a mass, you’d start to fall towards the Earth’s center with that acceleration of g. But happily the ground pushes back and you are stable on the surface. (It’s a little confusing since you’re not moving but you’re still accelerated) So from Newton’s Second law, when we have a mass and we have an acceleration, we can calculate a force and we define that particular force of attraction by the Earth as the the object’s weight.

  Pens out!

Let’s call it w and we can write it out:

(6.3)\[\begin{equation} w=mg. \end{equation}\]

Notice that this is just Newton’s 2nd law, but with a particular acceleration, \(g\) which leads to a particular force, \(w\).

Weight is the force that a planet exerts on an object on its surface.

We can measure an object’s weight by making use of the fact that the Earth pushes back with a force that’s the same value as the weight. When you “weigh” something, probably a spring is doing the pushing-back. There’s one in your bathroom scale which is calibrated in the U.S. to read that push-back in pounds. In France, that reading would be in kilograms (but it’s still a spring, so really measuring weight and converting to mass)! Basically everyone (else) in the world deals in mass terminology. We’ll see how the Earth does this in a bit when we get to Newton’s other law, that of Gravitation.

By the way, the acceleration due to Mars gravity, its “little \(g_M,\)” is about 4 m/s\(^2\), to compare with our “little \(g_M,\) of about 10 m/s\(^2\). That explains why that bag of flour appears to be light using your Earth-calibrated scale.

I have bad news: unfortunately in the English system, the unit of mass is “slugs.” Now I have good news. We’ll not use slugs for anything in QS&BB.

Wait. Seriously. Slugs.

Wish you’d not asked. I confess I don’t know the origin of the “slug.” One of life’s mysteries.

So we can collect our units appropriate to Newton’s Second law in this table below. In the last column, the standard abbreviations are shown as well.

6.7.1. Mass, acceleration, and force conversions

English - MKS
acceleration	ft/s\(^2\)	m/s\(^2\)	1 ft/s\(^2\) = 0.305 m/s\(^2\)
	\(g =\) 32 ft/s\(^2\)	\(g = 9.8\) m/s\(^2\)
mass	slugs	kilograms	1 slug = 14.59 kg
on earth	mass of 1 slug	mass of 1 kg
	= weight of 32.2 lbs	= weight of 9.8 N
force	pounds (lb)	Newtons (N)	1 lb = 4.45 N

Weighty conversions.

  Please study Example 6:

my weight

  Please answer Question 3 for points:

the BIKING EXAMPLE!!