Giancoli Chapter 11 Guide & Recommended Problems

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We will cover the first six sections. In Section 1 we first encounter angular momentum, defined for the special case of rotation about a fixed axis. Later, in Section 3 we will see the "official" definition of angular momentum. Angular momentum is the third and final (in this course) quantity that is conserved (along with energy and linear momentum). The three of are equal importance, and each is based on a deep symmetry of nature: I'll say more about this in class because it is fascinating - at least to me.

Almost all the "rules" you learn at the introductory level are really just approximations to deeper rules - and even the "deepest" rules we know are almost certainly not "absolutely" accurate. So Newton's second law ( F = ma) is actually only approximately true, and is more accurate for large masses than for small ones. For individual electrons, for example, it is terrible. But the "real" laws (quantum mechanics) are really, really complicated, and for most "practical," non-atomic level case, there is no measurable difference between the results predicted by Q.M. and Newton. So, again and again, we start learning science by describing nature using "approximate" rules, then we delve deeper and learn "better" rules, and then deeper still. . . . . well, you get the idea (I hope).

So in Section 1 we have angular momentum for an object rotating about an axis that does not, itself, change orientation: L = I (the angular analog of the linear momentum, p = mv). Then Newton's second law for rotation, which we learned in Chapter 10 as τ = I (the angular analog of F = ma) is more correctly written as = dL/dt (just as the linear law is actually F = dp/dt). Examples 1, 2 &3 are important. Then, on p. 288, we see that angular momentum is a vector quantity, L pointing along the rotational axis. Examples 4 & 5 are important.

Not only is L a vector; it is a product of two other vectors. In Chapter 6 we learned to multiply two vectors together to obtain a scalar (the "dot-product" e.g., Work = F x). Now we learn a second way to multiply two vectors: the "cross-product" where the result of multiplying the two original vectors is a new vector: C = A×B. The first application of this is to re-state the definition of torque: = r×F. You should be sure you understand that this new statement is equivalent to the earlier definitions of torque in Eqns. 10-10 on p. 257.

In Section 11.3, then, we define angular momentum for a particle moving relative to a fixed point (usually taken as the origin of the coordinate system). One can then prove (as in Sections 4 & 5) all kinds of things, and see clearly why angular momentum is conserved, why L is simply I for rotation of a rigid body about a fixed axis, and more! In the starred part of Section 5 you learn why dynamic balancing of automobile wheels is important. I will not focus on this, or on the derivations, however, but the Examples 8 and 9 are important applications of Newton's second law for rotation.

What I WILL focus on is in Section 6: problems involving the conservation of angular momentum. Examples 11 and 12 are crucial, and I will work lots of problems in class using this principle.

Sections 7, 8, & 9 contain some "interesting" stuff." Section 7 discusses the interesting case of a gyroscope, and why its motion is so stable when it is spinning. Sections 8 & 9 describe what happens when the laws of physics are applied in a rotating (non-inertial) reference frame: you get "fictitious" forces such as the "centrifugal" and "coriolis" forces. Marine geophysics majors especially should read these sections: since the earth is a rotating body, and thus is not an inertial frame, these "fictitious" forces show up. The coriolis force, for example, is responsible for the fact that whirlpools rotate CCW in the northern hemisphere (and CW in the southern hemisphere) of the earth. Hurricanes (typhoons in the pacific) are coriolis storms: the coriolis force is the cause of the rotation in such a storm. I will not test you over sections 8 & 9.

Recommended Questions and Problems for Chapter 11:

Questions: Q1 (a), Q2 (no), Q4, Q5 (speeds up), Q7 (no change), Q8 (either vector zero magnitude, vectors parallel or anti-parallel), Q11 (same as Q7 in Chapter 10), Q17 (all except A.M & torque; displacement & acceleration)

Section 1: P2, P3, P5, P11, P13, P16

Section 2: P22, P23

Section 3: P33, P34

Sections 4 & 5: P38, P39, P41

Section 6: P48, P49

General Problems: P66, P67, P7