Chapter 1 – Excerpt 3
This excerpt is short and highly illustrated because it’s about mass and inertia, and cats, while not massive, are experts on inertia. It includes Newton’s First Law, a.k.a. the Law of Inertia.
As mentioned in the previous excerpt, not all forces actually affect motion. The tendency to resist changes to one’s motion is called inertia. Cats are excellent at illustrating inertia. See Figures 1-4 and 1-5.
Mass is a measurement of inertia. In a sense, mass measures the quantity of matter, but not in a spatial sense – that’s volume. Technically, the mass of a body is a measure of its ability to resist force. If an object is sufficiently massive, it can completely resist changes to its state of motion. As we will see when we discuss Newton’s Second Law, mass specifically measures how much a body’s velocity is changed by a force. The SI unit of mass is the kilogram (abbreviated kg).
You have probably noticed, for example, that a can of catfood has the mass of its contents printed on the label, typically about 150 grams. In SI units we would express this as .15 kg. (That same can of catfood will probably also say that its weight is about 5.5 ounces, but that’s not really correct; we will discuss this somewhat confusing issue shortly.)
The force of gravity is an attractive force between any two objects that have mass. It is proportional to their masses, and inversely proportional to the square of the distance between them. In equation form, we say that the amount of the gravitational force is
(1) F = Gm1m2/d2
where m1 and m2 are the masses of the two objects and d is the distance between them. The constant of proportionality, G, is called The Gravitational Constant. Its value in SI units is
G = 6.7×10-11Nm2/kg2.
Since G is such a small number, this attractive force is inconsequential between most pairs of objects we encounter – it only plays a significant role when at least one of the masses is large enough to compensate. Thus, we notice the gravitational attraction between the earth and objects on it, but not between those other objects themselves.
For example, suppose you and a mouse are a few feet apart, staring at each other. In addition to being repelled by fear, the mouse is attracted to you by the force of gravitation. But this force will unfortunately not result in any motion towards you because its magnitude is infinitesimal:
F = Gm1m2 / d2 = (6.7×10-11Nm2/kg2)(5 kg)(.024 kg) / (1 m)2 = 8 × 10-12 Newtons
So if you want that mouse, you are going to have to take steps toward getting it, rather than sit back and rely on the law of gravity.
Now let’s calculate the far more significant, observable gravitational force that the earth exerts on an object at its surface. The quantity d, the distance between the earth and the object, may at first appear to be zero, because the earth and object are touching each other. But the object is touching only a small part of the earth; most of the earth is quite far away.
It turns out that, with respect to gravitation, massive objects behave as if all their mass were concentrated at a single point. (It was actually to prove this point that Newton invented calculus!) This point is called the Center of Mass (or, somewhat less generally, the Center of Gravity) of the object. For a sphere, this point would obviously be right at the center.
For most bodies, however, the distribution of mass is not so symmetric. Being as felines are close to perfectly proportioned, your own body’s center of mass is very close to the geometric center of your torso. For your person, on the other hand, the location of the center of mass is about an inch below the navel, but it will be lower in women than in men, and higher in children than in adults.
How to retaliate if your person ever tries to train you to do tricks: Ask him to stand against a wall and reach down to touch his toes without bending his knees. To your amusement and his chagrin, he will not be able to do this without falling over. (If the wall weren’t there, he would automatically extend his hips backward as he bent over forwards, in order to keep his center of mass directly above his feet.)
Note that there may actually be no mass located at an object’s center of mass. For example, the center of mass of a donut or a coffee cup or Zepto Chan’s favorite toy (see Figure 1-6) would be in the empty space near the middle of the object.
Although the earth is not spherically symmetric, modeling it as a sphere is a very good approximation, as alluded to in the introduction to this text. Thus, d, the distance from the center of mass to the object at the surface, is equal to the radius of the earth, which is about 6.4×106m.
The final fact we need is that the mass of the earth is known to be about 6.0×1024kg.
F = Gm1m2 / d2 = (6.7×10-11Nm2/kg2)(6.0×1024kg)(m kg) / (6.4×106m)2 = 9.8 × m in Newtons
Temporarily, let’s let g = 9.8 N/kg, so we can simply say F = mg.
Newton’s First Law
Newton’s First Law of Motion states that a body at rest stays at rest, and a body in motion stays in motion, with the same speed and same direction, unless acted upon by an unbalanced force. See Figures 1-7 and 1-8.
You may have experienced this latter phenomenon when being taken to the vet in a car. It’s bad enough to be trapped in a box and moving without controlling your own motion. But then when the driver reaches the destination and applies the brake, this causes the road to exert an unbalanced force on the locked wheels, which causes the car to abruptly change its velocity and come to a stop. But you, being a body not under the influence of unbalanced forces, don’t experience a change to your motion, and hence you and your box keep on going, in the direction the car had been traveling, perhaps sliding off the car seat onto the floor. (And then, after all that, you get carried into the veterinarian office and examined and poked while enduring a cacophony of canine barking!) This ignominy doesn’t happen to the people in the car because they are being acted on by the unbalanced forces of seat belts.
Newton’s First Law, also known as the Law of Inertia, is a generalization of Galileo’s Principle of Inertia, which stated "A body moving on a level surface will continue in the same direction at constant speed unless disturbed". This principle was in contradiction with the then prevailing notion of Aristotle, that in the absence of a force, all objects on earth would naturally come to rest.
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