Fizyx for Felines: A Physics Textbook for the Curious Cat

Chapter 2 – Excerpt 3

Posted in Uncategorized by skonabrittain on 2 April, 2010

This short excerpt is just about cross products, including the well-known Right Paw Rule, which humans call the Right Hand Rule.    




Cross products

Actually, the torque vector is the cross product of the radial vector and the force vector. Since cross products are typically studied in 2nd year calculus, you probably are not familiar with them yet. If tensors made you feel tense, cross products may make you feel cross, but don’t worry – they’re much simpler.

There are two ways to multiply vectors – the dot product and the cross product. The operator symbols • and x are written in bold to indicate that they are operations on vectors. The dot product yields a scalar, i.e. a number, whereas the cross product yields another vector.

The magnitude of the cross product is the product of the magnitudes of the two vectors and the sine of the angle between them. This English mouthful may be easier to take in as an equation:
| V1 x V2 | = | V1 | x | V2 | x sin θ (see Figure 2-8).


Figure 2-8.The vector cross product.


The direction of the resultant vector is perpendicular to the plane of the two vectors being crossed. Of course, there are always two perpendicular directions, for example up and down. Which of the two possible directions is given by the canonical Right Paw Rule, as follows:

Mathematical Definition
The Right Paw Rule
for
Vector Cross Products

Clench your right front paw as if to knead something. If you curl your toes in the direction from V1 to V2 , then the direction of V1 x V2 will be away from the carpal pad or towards the dewclaw. See Figure 2-9.

 

 

 

For those of you who are polydactyl3, there is an easier version of the right paw rule: As you move from V1 to V2, V1 x V2 will be pointing in the direction of your extra claw.

Figure 2-9. A right forepaw, bottom view.


Note that this rule means that the direction depends on the order of the operands. The cross product is not a commutative operation. In fact, V2 x V1 = V1 x V2.

Q. What are î x j, j x k and î x k ?
A. Since î, j and k point in the positive x, y and z directions, respectively, of a rectangular coordinate system (see Figure 1-? or 2-9½), they’re all perpendicular to each other, so sinθ is always 1. And being unit vectors, they all have magnitude 1. So we just apply the right paw rule to get î x j = k, j x k = î, and î x k = – j. (Actually, the way a three-dimensional Cartesian coordinate systems is set up is that the direction of the z-axis is always taken to be the one that makes î x j = k hold.)



Figure 2-9½.
The unit vectors in a 3-dimensional rectangular coordinate system.


Thus, recalling expression (3) above, we have

τ = r x F


Note that the torque τ is perpendicular to both r and F, and hence it is perpendicular to the entire plane of motion. So the direction of the torque is always along the axis of rotation. Assuming that that axis is vertical, the torque points upward when the induced rotation or revolution is counterclockwise, and downward when it’s clockwise.

Although these quantities are vectors, we can often just use scalars instead, because the rotation can be in one of only two directions – clockwise or counterclockwise – with respect to the axis of rotation. By convention, the counterclockwise direction is considered positive and the clockwise direction negative, as mentioned above.


3 – Polydactyl is the technical term for the colloquial “double-pawed”, which is obviously a misnomer, since you are all most likely quadruple-pawed. Polydactyl cats are also known as “six-finger cats”, “thumb cats”, or “mitten kittens”. <return-to-text>



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2 Responses

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  1. Ms. Mica said, on 4 April, 2010 at 08:57

    That’s cool to write physics for cats, because cats are warm, furry, and enjoyed by both men and women. In some women-in-physics lunch, I heard about physics involving such things as food falling off a baby’s high chair tray. While I can imagine learning more physics with examples such as those, I don’t picture that physics translating as well into real jobs in the physics world. Cats are for everyone, not just moms! Keep it up. You could have cats falling off the roof, too, as long as you pointed out that dogs were involved in the accident. I actually like dogs better than cats, so those are the warm furry ones that would attract me to physics more. Maybe dogs could be in there, too, as villains, if you want. Keep it up!

    • skonabrittain said, on 5 April, 2010 at 03:36

      Thanks for your comments and the encouragement.
      I always think it’s funny that some people think Physics for Felines is going to be soft and gentle, i.e. not mathematical. It’s true that cats are intuitive but they’re also very precise.
      Re cats falling off the roof – see my reply to Janet D. in the Comments to the Preface section!
      Since you like dogs better than cats, here are some places dogs appear in Physics for Felines:
      – in the Practical Application (green box) in Chapter 1 – Excerpt 4
      – in an Exercise in Chapter 1 – Excerpt 5
      – in the Historical Note (beige box) about Isaac Newton at the beginning of Chapter 1 – Excerpt 1
      – in the Aside to Human Readers (blue box) at the end of the Preface
      Also, there’s a new physics book for dog lovers – How to Teach Physics to Your Dog by Chad Orzel.


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