## 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 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 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 Practical Application From the second law, we can see that the Newton unit of force is equal to one kg-m/s So the units of the quantity g, defined earlier as 9.8 N/kg, can actually be more basically expressed as m/s 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 » 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/s (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
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 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, 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 A groundbreaking study by some veterinarians in New York City, originally published in 1987 and more highly publicized the following year Another specially named velocity, in the opposite direction, is the _{esc}= sqrt(2GM/R)=sqrt(2gR)≈sqrt(2×9.8×6.4×10^{6})m/s =11,200 m/s ≈ 25,000 mph.7 – Mehlhaff, Cheryl and Wayne Whitney. "High-Rise Syndrome in Cats." © All Contents Copyright 2008-2010, Skona Brittain |

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