Giancoli Chapter 10 Guide & Recommended Problems

From Eckerd Academic Wiki

A long chapter: 27 pages, 10 sections, and 106 problems! This chapter is a review of everything we have done so far, but with rotation as well as translation. Through the first 8 chapters, we pretended that all our "objects" were in fact point particles. In Chapter 9 we dealt with "systems" of particles. Now, in Chapter 10, we look at particle systems which are rigid bodies: they can rotate. (And: if they are not quite totally rigid, they can deform, bend, and stretch, as we will see in future chapters). In chapter 10 we extend our study of motion by just one step, and include rotation as well as translation. It is a theorem (proved in an advanced course) that any possible motion of a rigid body (one that holds its shape) can be understood as a combination of translation plus rotation.

The "saving grace" of this chapter is that almost everything new that we will encounter will be "similar to" something we have already studied. We can even set up a "table of correspondence" between the new rotational variables and the previously encountered variables which we used to describe straight-line motion. For example:

position x --> (theta) angle

velocity v --> (omega) angular velocity

acceleration a --> (alpha) angular acceleration

momentum p --> L angular momentum

mass m --> I rotational inertia (or "moment of inertia")

force F --> (tau) torque

kinetic energy mv2/2 --> I²/2 rotational kinetic energy

Newton's 2nd law F=dp/dt --> = dL/dt Newton's 2nd law for rotation

and so on. If mass is constant, the second law becomes F = ma, and if the axis is fixed and we are speaking of a rigid body, the second law for rotation becomes = I . And so on.

So, although the chapter summary (pp. 274-2675) lists many important "new" equations, most of them are recognizable. Of course they will require practice to be usable. Section 1 introduces the three basic rotational variables: angle, angular speed/velocity, and angular acceleration. Read these carefully. Get used to dealing with angles measured in radians. Some of the most important equations (e.g., Eqn 10.1, 10. 4, & 10.5) are valid ONLY if the the angular units are radians (or rad/sec, etc.). Section 2 explains how angular speed and acceleration are interpreted as vectors (but this will not be important in the problem-solving).

Section 3 gives a list of 4 equations (Eqns 10.9 - analogs of Eqns 2-12, p. 29) to use when the angular acceleration is constant. Most students find the problems in the first 3 sections doable.

Section 4 introduces torque, and now the more difficult material begins. Keep in mind that you cannot compute the torque until you know what rotational AXIS is being used. Diagrams become absolutely essential to figuring out the torque for a given case - the formula itself (Eqn 10-10c) is simple if you use the right "R." Then Section 5 gives us Newton's second law for rotation. Again, this is just like F = ma, except that now it is = I. Usually the hard part is being aware of the rotational axis, the CW and CCW torques, and the "rotational inertia" I. Section 6 follows with "problem-solving" using this law. The list of formulas for I on page 260 is essential, and will be provided if needed on tests (you don't need to memorize these, but know where to find them). Section 7 explains where these mysterious formulas for I come from. We will not do any of the complicated integrals, but I expect you to understand the concept of how to compute I. The Parallel-Axis Theorem (p. 264) is important: I will not ask you to "prove" it but you must know how to use it.

Section 8 derives the expression for the kinetic energy an object possesses due to its rotationalmotion: KE = I²/2 (we already have mv²/2 for the KE due to linear motion). Then we get the equations for rotational work done by a torque: W = (analogous to W = Fx in Chapter 7), and the corresponding work-KE principle.

Section 9 is a key section: we now can do energy-conservation problems where we take into account both translational KE and rotational KE. Pay really close attention to the worked-out examples here. Be sure when using Eqn 10-23 that the I is computed about an axis through the center of mass! Examples 16, 17 & 18 are crucial.

Section 10 is worth reading, but I will not test you on it.

I will be honest: the material in this chapter, beginning with section 5, is where many students start to have real difficulty with physics. We will not try to cover the whole chapter in one or two class days. You should seek help (from one another, the tutors, me) when you don't understand a concept or problem. Don't give up.

Recommended Questions and Problems for Chapter 10:

Questions: Q1, Q4 (yes), Q5, Q6, Q7 (no, no), Q8, Q9, Q14

Section 1: P1, P2, P3, P6, P8, P11

Section 2: None

Section 3: P16, P17, P19, P21, P23

Section 4: P25, P27, P29, P30

Sections 5 & 6: P31, P38, P40, P41, P44, P46, P51

Section 7: P55, P56, P58, P59

Section 8: P62, P67

Section 9: P71, P73

Section 10: None

General Problems: P93, P94