Chapter 1 – Excerpt 4
This excerpt, about Newton’s Second Law, finally contains the long-anticipated application – with possibly life-saving ramifications – that motivated the study of vectors at the beginning of the chapter. You’ll also learn which floor of a building is the most dangerous one to throw a cat out of, which I hope is a much less practical application.
Newton’s Second Law
Newton’s Second Law of Motion states that when there is an unbalanced force acting on an object,
This is our first example of a vector equation replacing an equivalent system of three scalar equations,
Note that the second law subsumes the first law. The first law is just the special case where F=0. F=0 obviously implies a=0, and, since a is the derivative of v, this implies v is a constant.
These laws conflict with our intuition, which tells us that forces cause motion, as Aristotle believed. Forces actually don’t lead to motion, but to changes in motion. A source of this fallacy is the ubiquitous (except in
Now you can see why that mouse you were staring at wasn’t approaching you. Even without any friction or air resistance, its acceleration would have been only 8×10-12 N /.024 kg = 3.3×10-10 m/s2. At that rate, it would have taken over 21 hours to reach you. (We calculated the time by solving equations (2) below for t.)
How you can use Newton’s Second Law to help you escape from a large dog: (This is the application you were promised near the beginning of the chapter!) Recall that velocity is not just speed, but also incorporates direction. So instead of running far away in a straight line, zigzag your path as much as possible. Each change of direction is an acceleration because it is a change of velocity. So each requires the effort of exerting a force to effect it. And according to Newton’s Second Law, the amount of force required is proportional to the mass. As you know, you will be able to easily change your direction as frequently as you want to. But, since it is much more massive, the dog will have a much harder time making such frequent changes to its state of motion. Hence, it won’t be able to chase you for long!
From the second law, we can see that the Newton unit of force is equal to one kg-m/s2.
So the units of the quantity g, defined earlier as 9.8 N/kg, can actually be more basically expressed as m/s2. Thus g appears to be an acceleration quantity, and that’s exactly what it is – it’s the acceleration of an object at the earth’s surface due to the gravitational attraction between the object and the earth. F = mg is just a direct application of F = ma. When we consider g as a vector, its direction is the negative z direction, pointing straight down toward the center of the earth.
Since we live most of our lives, and conduct most of our scientific experiments, at the surface of the earth, we often use the shorthand term “the force of gravity” to refer to the gravitational force exerted by the earth on bodies at its surface, and we say simply that g is the “acceleration due to gravity”.
» Concepts to Sniff At «
Weight is a measure of the force of gravity. There is some popular confusion between weight and mass, but they are two different types of quantities. Mass is measured in SI units in kilograms, whereas weight is a force, so it is measured in Newtons = kg-m/s2. In the non-scientific system of units popular in commercial and everyday use in many English-speaking countries, weight is measured in pounds, and sometimes taken to be a synonym for mass. A pound corresponds to .45 kg at sea level.
(In the older scientific FPS (foot-pound-second) system of measurements, where weight is also measured in pounds, the corresponding unit of mass is called a slug. Presumably, this term originated with those gastropod creatures whose sluggishness convey a vivid sense of inertia. Although they are considered quite a gastronomical delicacy by some, you have probably never tasted one, being as they are no fun to chase and pounce on.)
Your weight depends on the distance from the center of the planet pulling on you. For example, if your mass is exactly 5.5 kg, then your weight is about 12 pounds at sea level. But if your person is a Tibetan monk living near the top of the Himalayas (as was quite likely in a previous life if you are a Burmese cat), your weight would be slightly less. And if your person is an astronaut who takes you to the moon, your weight there would be 2 pounds (whereas in a future life, when you might venture to the much more massive planet Jupiter, your weight there would be about 28 pounds). However, wherever you are, your mass would still be exactly 5.5 kg.
People often say they are on weight-loss diets but, instead of simply making a global move to a location farther from the center of the Earth, they go to great effort to move around locally without actually going anywhere, and to eat less food. What they are really trying to do is to reduce their mass (or even just change their shape by changing the form of some of their mass from fat molecules to muscle molecules).
Henceforth, unless we say otherwise, when we mention weight it will be assumed to be at sea level on Earth.
Q. Suppose your person is on a mass-loss diet that requires him to eat an apple a day. A typical apple weighs about 1 newton. How much does his mass increase when he eats one?
Let’s apply this concept of the acceleration due to gravity being the constant vector g = 9.8 m/sec2 in the negative z direction, with magnitude g, to a study of the motion of objects.
First we will consider a tossed ball. Assume that it is tossed upward and across a field. Thus the initial force on it had both horizontal and vertical components. Once it is let go, however, the only force acting on it is the vertical force of gravity, assuming we can neglect the negligible force of air friction.
We obtain the velocity by integrating the acceleration, and then the displacement by integrating the velocity. In the horizontal direction, the velocity is constant so the displacement is just linear in time. In the vertical direction, we have:
The constants of integration are, respectively, the initial velocity and the initial displacement, i.e. the height of the ball when it is let go.
Thus, the motion is described by a quadratic equation, and hence its graph is a parabola. Note that since the vertical displacement is a quadratic function of time, and the horizontal displacement is just a multiple of time, the vertical displacement is also a quadratic function of the horizontal displacement. So the motion that we observe in space is a parabolic arc, as Figure 1-8 shows.
This type of motion, that of a projected object that has been let go, is called projectile motions. The observation that the horizontal and vertical components of a projectile’s motion are independent was first recorded by Galileo over 400 years ago.
For objects that are more geometrically complex than a ball, the motion can be much more complex, but the object’s center of mass point moves with the same simple parabolic motion. See Figure 1-9.
Q. If it took 1 second for the falling apple to reach Newton’s head, which was 1 meter high, how high was the branch that it fell from, how fast was it going when it hit him, and what was its average speed?
Although we generally ignore friction in our calculations, it does of course have an effect, which puts an upper bound on the acceleration due to gravity. An object in free fall will not accelerate indefinitely – when the frictional force of the air resistance balances the force of gravity, the object reaches its terminal velocity.
A groundbreaking study by some veterinarians in New York City, originally published in 1987 and more highly publicized the following year7, found that injuries suffered by cats falling from high-rise apartments peaked at 7 floors – i.e. falling from either higher or lower is safer. This is because it takes a distance of about 7 floors for a cat to reach its terminal velocity, which is approximately 65 mph. The terminal velocity for humans is about twice as fast, due to the fact that they have a smaller ratio of surface area to mass. It is obvious that when falling from a lower height the impact is lessened, but the reasons why falling from a higher height is also beneficial are more complicated. In the next chapter, we will discuss what happens after terminal velocity is reached during feline free fall.
Another specially named velocity, in the opposite direction, is the escape velocity. When an object is propelled upward with sufficient force to overcome the force of gravity on it, it can actually escape from the clutches of the earth’s gravitational field. The concept of escape velocity is reminiscent of the almost universal kittenhood fantasy of flying. Perhaps you recall wishing you could fly under your own power when you were young and playful. It would sure make it easier to catch butterflies and birds. Due to the amount of force required to achieve this velocity, however, such projectiles tend to be payloads propelled by rockets rather than cats. With the reinstating of our assumption of negligible air resistance, we have vesc= sqrt(2GM/R)=sqrt(2gR)≈sqrt(2×9.8×6.4×106)m/s =11,200 m/s ≈ 25,000 mph.
7 – Mehlhaff, Cheryl and Wayne Whitney. "High-Rise Syndrome in Cats." Journal of the American Veterinary Medical Association. Vol. 191 (1987): 1399-1403.
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