Fizyx for Felines: A Physics Textbook for the Curious Cat

Chapter 1 – Excerpt 5

Posted in Uncategorized by skonabrittain on 5 March, 2010

Newton’s Third Law, Linear Momentum, and Unusual Feline Recreation.

Newton’s Third Law

Newton’s Third Law of Motion states “For every action, there is an equal and opposite reaction”.

A force is the result of an interaction between bodies. When two objects interact, they each exert a force on the other. For example, if you are curled up on a sofa while reading this, you are exerting a downward force on the sofa and it is exerting an upward force on you. Such pairs of forces are called action and reaction pairs, but this is somewhat misleading terminology. There is no physical reason why one should be considered the primary force and the other merely a reaction; it’s just a matter of perspective. From your perspective, the sofa is reacting to you. As an analogy, you may think you’re rubbing against your person, while [s]he thinks [s]he’s patting you.

The third law states that the magnitudes of the forces in such a pair are equal, and the directions of the forces are opposite. In other words, whenever there is an interaction between two objects, if the first object exerts a force F on the second, then the second exerts a force –F on the first.

Previously, we considered the force of gravity due to the earth on an object at its surface. From Newton’s third law, as well as from the symmetry of equation (1), we see that the earth is also experiencing a force due to the object. Although that force has the same magnitude, it has a negligible effect on the earth’s motion, since it is being divided by a much larger mass in a = F/m.

As a more perceptible example, suppose the person tossing the ball above is actually playing ice basketball, and is skating at a uniform velocity, friction between metal blades and ice being negligible. As he tosses the ball into the basket, the reaction force that the ball exerts on him will cause him to accelerate in the opposite direction. This deceleration will not result in motion in the opposite direction, but rather just in a reduction of speed in the forward direction. An observer would not notice him slowing down. But if instead he were a figure skater tossing his partner, who despite being svelte is much more massive than a ball, the effect would definitely be noticed by the skater, and probably also by observers, unless they were all staring at her instead.

For those of you who are unfortunate enough to live in towns with no skatable ice, such as Santa Barbara8, here’s an example you may have had the misfortune of encountering instead. Suppose, as shown in Figure 1-10, your person’s progeny mischievously places you on a float in a pool. You would, of course, immediately attempt to jump off the float to land on the side of the pool. To do so, you exert a force on the float, causing it to very observably move far in the opposite direction, perhaps toward the head of the aforementioned progeny. Its recoil force on you is what allows you to escape your predicament.

 Figure 1-10.

If your locale has neither ice skating rinks nor swimming pools, consider instead this universal example analogous to that of the skater. Suppose you and a friend are engaging in some pre-prandial play with a mouse, and as you are running toward your friend you toss the mouse toward him. From the reaction force that the mouse exerts on you, your speed slightly decreases.

Whenever you pounce toward a mouse, you do so by applying a force to the ground. It is the ground’s reaction force applied to you that causes you to move upward and forward. But what about the effect of your force on the ground – how does it exert a force on you without it being simultaneously affected? Although its change may not be perceptible to you, especially when you are intent on the object of your pounce, as you push on it you are compressing it, and it is that spring-like decompression force that is being exerted on you. If the ground is a carpeted floor, you can see some of the compression, and if it is a wood floor, you may hear a resultant squeak as you begin the pounce. (You may also hear a squeak as you end the pounce, but that would be from the mouse.) Note that during the pounce, your horizontal velocity doesn’t change, and your vertical velocity decreases steadily (at the rate of 9.8 m/s2) due to gravity. Thus, the height, length and duration of the pounce have all been completely determined by the time your paws leave the ground, when the contact force terminates.

Metaphysics for meta-felines

If you push hard on the world, … the world pushes hard on you. If you touch the world gently,
the world will touch you gently in return. The way you touch others is the way others touch you.

Paul Hewitt, Conceptual Physics, ©2002, Prentice Hall

Q. If a cat collides into a large dog, which animal experiences the greater impact force, and which one experiences the greater acceleration?

A. The forces are the exact same in magnitude, though opposite in direction. But the cat will experience a much greater acceleration due to its smaller mass.

Q. If a cat collides into a mouse, which animal experiences the greater impact force, and which one experiences the greater acceleration?

Momentum

The quantity mass times velocity is called momentum.

Note that since velocity is a vector quantity, so is momentum, and since mass is a positive scalar, the direction of the momentum vector is always the same as the direction of the velocity vector.

The symbol used for momentum is p. The ‘p’ is probably derived from the Latin petere, meaning "to go to”, the root of “impetus”, or possibly from the Latin pellere, meaning “to push”, which led to both “push” and “pull”, as well as “impel”. During Newton’s lifetime, the term "impetus” was used to describe an object moving independent of any observed force.

So we have the definition p = mv. The SI units of momentum are kg-m/s.

Q. How much momentum does a 1000 kg truck moving at 15 m/s have? How much momentum does a 5 kg cat moving at 15 m/s have? What is your maximum momentum? The maximum speed of an average cat is about 30 mph, which is about 13 m/s. The fastest land animal is a cheetah, a big cat, which can achieve speeds of about 30 m/s9. The average mass of an adult cheetah is about 50 kg. How much momentum could such a cat achieve?

The reason the quantity mv is given a special name is that it is a conserved quantity. That means that in an isolated system, the total amount of momentum doesn’t change. This important principle, called the Conservation of Momentum, was also developed by Isaac Newton, with some help from his cat of course.

To use this principle, we must first understand what is meant by an isolated system. As you probably realze, nothing is truly isolated, physically as well as metaphysically, but, as usual, we can make use of appropriate simplifying assumptions.

A system merely means a set of at least two objects. An isolated system is one for which the net external force acting on it is zero. External forces are forces due to objects outside the system. There may be external forces acting on the system, but, if so, they must be exactly balanced by other external forces.

Within the system, if the objects exert forces on each other and change each other’s momentum, the sum of their momentums doesn’t change. For example, suppose two objects collide. If the collision causes one of the objects to gain momentum, then the other object must experience a loss of exactly that much momentum.

The canonical example of a collision is that of billiard balls. If we neglect friction with the billiard table and the air, the balls can be considered to be an isolated system. When a ball previously hit by a cue collides with another ball, it transfers some of its momentum.

Since you are probably not familiar with the game of billiards, consider instead the well-known cat toy10 shown in Figure 1-11. When the two leftmost balls hit the other three, this causes only the two rightmost balls to move. (Note that this is a rather expensive version of a ball-on-string toy, so be careful not to get too entangled in its strings, lest you provoke an unequal opposite reaction from your person.)

For a more practical example, suppose you are running with a large bird in your mouth, which you have just successfully stalked, but unfortunately it manages to wrench itself free and pushes off from you, flying in the direction you were running, faster than you can run of course. The reason it gets away, before it picks up speed by interacting with the air, is that it slowed you down slightly when it pushed off. Initially you and the bird are an isolated, albeit unfortunately enlarging, system. When it starts flying away with increased speed, your speed decreases, but it is a lesser effect since you are heavier than the bird.

Since the letters ‘p’ and ‘q’ are mirror images, we take them to be the momentums of two objects in a recoil situation, to produce the visually aesthetic symmetric formulation of Newton’s Third Law: p = q.

In any isolated interaction, the changes in momentum are equal and opposite because the forces are equal and opposite. In fact, the Conservation of Momentum principle is actually just a mathematical consequence of Newton’s Laws:

 F1 = –F2 by the Third Law m1a1 = –m2a2 by applying the Second Law m1(dv1/dt) = –m2(dv2/dt) since acceleration is the derivative of velocity d(m1v1)/dt = –d(m2v2)/dt by the constant multiple rule of differentiation dp1/dt = –dp2/dt by the definition of momentum d(p1 + p2)/dt = 0 by the addition rule of differentiation p1 + p2 is constant by antidifferentiation

Ironically, while it turns out that Newton’s Laws of Motion are not quite correct, as we’ll see in Part II when we discuss relativity and quantum mechanics, this conservation principle that he derived from them is a fundamental law, which holds true both relativistically and quantum mechanically.

8 – In Santa Barbara, home of the venerable Kavli Institute of Theoretical Physics and N Physics Nobel Laureates, as well as the author and her cats, the most prevalent frost occurs on the rims of margarita glasses. <return-to-text>

9 – Assuming you realized that the unit of speed is not meters divided by seconds to the ninth power, and have hence made it to this ninth footnote, note that the cheetah can maintain that speed only over short distances, less than 200 meters. Thus it succeeds in capturing its prey, of which its minimum daily requirement is 2 kilograms, only about half the time, when it starts the chase within about 30 meters. Although you have much less momentum, you are no less momentous to your person, and hence you probably have a much better success rate for obtaining food. <return-to-text>

10 – Your person may consider this a human toy, perhaps even an adult executive toy, but if there’s one in her home office, you have probably played with it more than she has. <return-to-text>