Fizyx for Felines: A Physics Textbook for the Curious Cat

Chapter 2 – Excerpt 8

Posted in Uncategorized by skonabrittain on 14 May, 2010

This is the final excerpt of the chapter, which finally explains the chapter’s title (Newton’s Cat’s Kittens).    


Summary and Application

We summarize the rotational ideas discussed in this chapter alongside their translational analogs of the previous chapter in Figure 2-27, an expansion of the table in Figure 2-1.

Figure 2-27.
[Linear] Displacement (s) Angular Displacement (θ)
[Linear] Velocity (v) Angular Velocity (ω)
[Linear] Acceleration (a) Angular Acceleration (α)
Force (F) Torque (τ )
Mass (m) Rotational Inertia (I)
[Linear] Momentum (p = mv) Angular Momentum (L = Iω)
Kinetic Energy: ½mv2 Rotational Kinetic Energy: ½Iω2
Newton’s 2nd Law: F = ma τ = Iα
Reformulation of 2nd law: F = dp/dt τ = dL/dt
Conservation of [Linear] Momentum Conservation of Angular Momentum

Finally, we will bring all these concepts together by applying them to a situation that you probably encounter daily – the cat flap door. (We realize we are assuming that the reader is neither homeless nor captive. The first assumption is justified by the price of this book; but for those of you whose loss of freedom is captured by the euphemistic label “indoor”, we suggest that you request your person buy you a cat flap door for the noble purpose of conducting physics experiments. )

~  Historical Note  ~


Not only did Isaac Newton’s cat inspire his musings about angular momentum conservation, but she played a pivotal role (pun intended) in one of his most enduring inventions – the cat flap door.

After completing his development of classical mechanics, Newton began focusing (pun not intended) on optics. His resultant theories of the properties of light will be discussed in Chapter 4. Here we are more interested in how he managed to conduct his explorations of light in a dark room. To his dismay, his cat kept pushing the door open in the middle of an experiment, letting in light that interfered with the results. A lesser person might have firmly shut the door to prevent such occurrences, but Isaac had more respect for his cat. He understood that the mere presence of a closed door is a source of frustration. Thus, necessity being the mother of invention, he created the world’s first cat flap door, by cutting a hole in the door and hanging a piece of dark felt over it.

Speaking of motherhood, when his cat had kittens during this time period, she naturally brought them into the optics room, too. A lesser scientist might have allowed the kittens to go through their mom’s large door, but Isaac dotingly cut out several smaller cat flap doors next to the original one. Although British professional animal writer Pauline Dewberry has claimed this was indicative of the typical genius’s blind spot – that “it didn’t occur to him that they could use the existing one”7, there is no evidence of this lack of thought. If it hadn’t occurred to him, one small door would have sufficed; the fact that he built several indicates that he was just trying to make their lives easier.

Rather, we suspect that it didn’t occur to Pauline Dewberry that rotational physics shows it would actually be much easier for the kittens to use the smaller ones! In fact, as we will derive in the text, they would have had to work four times as hard if Isaac hadn’t made them their own small flaps. Furthermore, his work probably made them feel special!

To this day, cat flaps are more popular in Great Britain, the homeland of their inventor, than in the United States. Between 88% and 92% of British cats have access to the outdoors.8 Interestingly, this corresponds quite precisely to the approximately 90% percent of pet cats that are non-pedigreed, although they are not necessarily the same set. Whereas in the U.S., many more pet cats are forced to be indoor cats, and many more are pedigreed. Apparently the pilgrims’ quest for increased liberty did not apply to their pets.

Your cat flap door may be a lot fancier than Newton’s cats’ doors, but the basic principle is the same. Whether the hinges are metal or the hinging effect is achieved by an adhesive, the axis of rotation is the one through the top of the flap. You probably know from experience that the further away from the axis you push, the easier it is to rotate the door. So your door should be mounted high enough that the bottom is at nose-height, not paw-height.



Figure 2-28.


It is most effective to apply the force perpendicular to the door. In this case, the moment arm is the distance from the axis of rotation to the point where the force is applied. However, if the force is being applied tangentially, only the component perpendicular to the door will cause any rotation. Then the moment arm is the shorter distance from the axis of rotation to the line along the direction of the force.

Even when you start pushing the door with your nose at the efficient right angle to the door, if you continue to push straight ahead while you walk through, that angle changes, because the door rotates to different positions. See Figure 2-28. Thus the torque decreases as you proceed with constant force.

Note that after you have passed through the door, the flap continues to rotate back and forth, in accordance with Newton’s Third Law, like a pendulum. Friction dampens the motion, causing it to eventually halt, which is good because that keeps the raccoons out.

Now let’s calculate the effect of flap size, as alluded to in the historical note above. Figure 2-29 illustrates a large flap door for a mother cat and a small cat door for her kitten, similar to the situation in Newton’s household.


Figure 2-29.


If the original felt piece was rectangular with height H and mass M, the moment of inertia of the mother cat door would be (1/3)MH2, as was derived earlier in this chapter. Assuming that the kittens’ flaps were half the height and half the width of their mom’s flap, their mass would be a quarter of its mass, and hence their rotational inertia would be only one-sixteenth that of their mom’s flap:
Ikitten= (1/3)(Mkitten)(Hkitten)2 =(1/3)(¼M)(½H)2 = (1/16) (1/3)MH2 = (1/16) Icat

Although twice the angular displacement would be required for the same linear displacement at the bottom, and twice the force is required to achieve the same torque, there is still a factor of four remaining. This shows why the kittens would have had to work four times as hard if they’d had to use their mom’s door.

In later chapters we will see how these simple flap doors have been greatly enhanced by applications from other areas of physics, such as magnetism and optics.


7 -ORIGIN OF THE CAT FLAP by Pauline Dewberry, © 2001-2006, TheDailyMews.com, http://www.thedailymews.com/articles/originofcatflap.htm<return-to-text>

8 -THE INDOOR OUTDOOR DEBATE by Sarah Hartwell © 1995, 2003, http://www.messybeast.com/indooroutdoor.htm<return-to-text>

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