The topic for this newsletter is Weight and Mass. This topic may seem inconsequential, but a thorough understanding will keep us from making mistakes in understanding force and lead to the concept of g's, a convenient way to conceptualize force. We will be aided in our understanding by two cartoon characters first used in our Calculus for the Utterly Confused book. The twins, Newt and Liebie, are named for Isaac Newton and Gottfried Leibniz, the originators of calculus.

Weight and Mass

Aren't weight and mass the same? No, mass is a measure of the "quantity of matter" while weight is a force. The confusion arises because the two terms are used to describe the same thing. If you stand on a typical bathroom scale, in some parts of the world it will read in kilograms. A similar scale in primarily English-speaking countries will read in pounds.

What is a scale? The simple view of a scale is that it is a spring. The amount of compression of the spring is a measure of the force on the spring. Your mass times the force due to gravity compresses the spring, and the readout on the scale is in either force (pounds) or mass (kilograms).

What is this force due to gravity? If you drop something near the surface of the earth, it falls; actually it accelerates toward the center of the earth at a fixed rate. That rate, in SI units (Remember, SI units are the meter, kilogram and second) is 9.8 meters/per second per second, and in British units is 32 feet per second per second, or 32 feet per second squared.

The mass of the dropped object times the acceleration (due to gravity) is the force exerted by gravity. In SI units the mass in kilograms times the acceleration in meters per second squared is the force. The force unit has a special name, the Newton.

When you step on that scale, you are applying a force in Newtons to the spring. The readout may be in kilograms, but the force is in Newtons.

Meanwhile in British units a mass in slugs (Remember from the first newsletter when the units of mass were discussed, and the mass unit in the British system was the slug?) times an acceleration in feet per second squared produces a force measured in pounds.

We may not know why scales are as they are, but now we at least understand them. How scales work at measuring forces can be taken a step further. And for this step we will use a thought or "what if" experiment proposed by none other than Albert Einstein.

Suppose you step into an elevator and as the doors close you notice that Dr. Einstein is standing next to you on a scale. Ever the teacher Einstein says: "Notice that the scale reads my weight, 145 pounds. The scale reads my weight due to the force of gravity, BUT since there are no windows in this elevator there is no way for you to know whether the elevator is on the earth or accelerating in outer space. You see accelerations and gravity both produce forces. I call this the equivalence principle. It is part of my general theory of relativity."

Einstein hits the button for the 20th floor. The elevator starts up. "See, when the elevator is accelerating up, the scale now reads 165 pounds." As the elevator reaches a constant speed up, the scale reading goes back to 145 pounds, and as the elevator slows the scale reads 125 pounds. With the elevator at rest, the scale again reads 145 pounds. As Einstein exits the elevator, he says: "In a spare moment you should calculate the curvature of space and time produced by the gravitational field of the earth."

That space-time curvature business is best left to the general relativity people. But, the first person to FAX me a (correct) solution to how fast the elevator is accelerating will receive a Captain Calculus baseball cap - Ed.

Now let's take a look at this thing called g's. Accelerations are often measured in g's because it is a way to conceptualize the magnitude of a force. Take a hypothetical airplane ride. Sit in the seat, seat horizontal and back vertical, without your feet on the floor. A scale (and remember a scale is just a force meter) between you and the seat would read your weight. As the airplane accelerates forward, a scale on your back would go from zero to some number roughly 20% of your weight. Now exchange that airplane for a carrier launched jet, and the scale on your back will go from zero to roughly four times your weight. This is enough to really get your attention.

Amusement park rides are generally designed for maximum g forces less than 3.5 because somewhere around 3.5 people's emotions go from thrill to fear. The NASA accelerator used in astronaut training, sometimes refered to as the BARF Machine, is limited to 10 g's. The pictures you may have seen of people under high g accelerations with their mouths stretched open occurs at a little over 10 g's.

Let's take a look at some typical automobile accident scenarios remembering that each case is different (the vehicles are different, and what they hit is different) and that we're only looking to get a feel for what we might expect.

A typical front end or rear end collision with a very modest amount of damage can get up to the 10 g range. If a person's head ends up being "whipped" in the collision you can roughly double the g-force to the head. Isn't the first complaint of an accident victim a sore neck?

A rear end collision where there is some bumper bending, a little panel wrinkling and some rear deck bending can be in the neighborhood of a 25 g crash.