Giancoli Chapter 7 Guide & Recommended Problems
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Chapter 7 introduces the concept of work, defines kinetic energy, and states the first (very primitive) form of what will become one of the most fundamental of all physical laws: the conservation of energy. Section 5 deals with relativistic motion, and will not be covered in this course.
The ultimate expression for work is Equation 7-7 on page 169. This expression, however, involves two concepts that need explanation: the "dot-product" for multiplying two vectors, and (from calculus) the integral. Most of Section 2 is devoted to explaining the dot-product (also called the "scalar product"), while Section 3 reviews the meaning of integration.
Section 1 introduces the idea of work: the work done by a force F is just the force multiplied by the displacement along the direction of F. If the object on which work is being done moves along a straight line (say, the x-axis) a distance 
x and the force is also in the x-direction, then the work is just the simple product W = Fx. But, of course, the displacement doesn't have to be along the same direction as F, and if it is not, then the work is equal to
W = Fx cos(
), where is the angle between the force vector F and the displacement vector.
In Section 2 the only new thing is notation: we define the dot-product (or scalar product) of two vectors to be just exactly what is written above for F and d: the scalar product two vectors A and B is:
A
B = A B cos()
where is the angle between the vectors A and B. The result is a SCALAR, not a vector. Thus, we see that work is a scalar. Note also that, if F is perpendicular to x, (i.e., = 90
) then W = 0 [since cos(90) = 0], while if F is parallel to x, then W = Fx [since cos(0) = 1]. The units of work are Joules, where 1 Joule is defined as (1 Newton)x(1 meter); (see p. 164).
On p. 168 there is a very important result, Eqn. 7-4, which gives us an alternative way (and often an easier way) of computing the dot-product of two vectors. I will explain in class where this equation comes from, and will expect you to understand both how to use this formula and why it works.
Finally, in Section 3, we see what to do when F is not just a constant, but changes as the object on which it acts moves. In this case we must integrate: divide the object's path into small steps of length dx, compute the contributions dW = Fdx for each step, then add all the contributions as we let dx approach zero (but the number of steps approach infinity). This is (if you remember from Calculus I) a "Riemann Sum," and is just exactly what an integral is defined to be. The first example of a variable force is the force exerted by a "spring," which is discussed on p. 170. Example 7-5 is important.
Section 4 then defines kinetic energy: KE = (1/2)mv2. Note that this is also a scalar [the "v2" is to be interpreted as the dot-product vv = (v)(v) cos(0)]. Note also that KE has the same dimensions as W, and thus can be expressed in Joules.
The highlight of this Section (and of the Chapter) is the "Work-Energy Principle." This principle claims that the NET work done on a object (i.e., the sum of the work done by all the forces acting on it) equals the change in the object's kinetic energy.
We must be very careful when applying the work-energy principle. I promise that I will only ask you to apply it in circumstances where the "internal" motion of the moving object can be ignored. Actually, this is the only circumstance where it works! So far in this course we have always tacitly assumed that an object was represented by a "point" in space. When considering an automobile, for example, we have "located" it by a point (e.g., x = 4) and all the forces we have considered acting on it have been "external" forces exerted by other objects (e.g., the roadbed) outside the car. In drawing our free-body diagrams for a mass, we have treated the mass as a "point." In Chapter 9 we will begin to build models of "systems of particles." Until then, just be aware that our models are over-simplified, and that the work-energy principle especially can only be applied in the case of an object when we can ignore the "internal" workings of the object's parts.
Recommended Questions and Problems for Chapter 7:
Questions: Q3 (no), Q6 (yes, if > 90), Q7 (no), Q8 (no), Q9 (yes), Q10 (#2, #1), Q11(9)
Section 1: P9, P11, P15
Section 2: P16, P17, P18, P21, P23, P27, P31
Section 3: P35, P40, P41
Section 4: P51, P57, P59, P63, P65
General Problems: P72, P79, P87
