APPLICATIONS OF VECTOR MULTIPLICATION

2. APPLICATIONS OF VECTOR MULTIPLICATION

In Chapter 3, Section 4, we defined the scalar or dot product of vectors A and B, and the vector or cross product of A and B as follows, where θ is the angle (≤ 180 ◦ ) between the vectors:

A · B = AB cos θ = A x B x +A y B y +A z B z .

(2.2) A×B = C, where |C | = AB sin θ, and the direction of C is perpendicular to the plane of A and B and in the sense of the rotation of A to B through the angle θ (Figure 2.1).

Figure 2.1

Section 2 Applications of Vector Multiplication 277

Let us consider some applications of these definitions. Work In elementary physics you learned that work equals force times displace-

ment. If the force and displacement are not parallel, then the component of the force perpendicular to the displacement does no work.

Figure 2.2 Figure 2.3 The work in this case is the component of the force parallel to the displacement,

multiplied by the displacement; that is W = (F cos θ) · d = F d cos θ (Figure 2.2). This can now conveniently be written as

W = F d cos θ = F · d.

If the force varies with distance, and perhaps also the direction of motion d changes with time, we can write, for an infinitesimal vector displacement dr (Figure 2.3)

dW = F · dr.

We shall see later (Section 8) how to integrate dW in (2.4) to find the total work W done on a particle which is pushed along some path by a variable force F.

Figure 2.4

Figure 2.5 Torque In doing a seesaw or lever problem (Figure 2.4), you multiply force times

distance; the quantity F d is called the torque or moment ∗ of F, and the distance d

∗ If the force F is due to a weight w = mg, then the torque about O in Figure 2.4 is mg · d = g · (md); the moment of inertia (Chapter 5, Section 3) of m about O is md 2 . The quantity md is

called the moment (or first moment) of m about O, and the quantity md 2 is called the moment of inertia (or second moment) of m about O. By extension, we call mgd the moment of mg, or F d the moment of F. For an object which is not a point mass, the quantities md and md 2 become integrals (Chapter 5, Section 3).

278 Vector Analysis Chapter 6

from the fulcrum O to the line of action of F is the lever arm of F. The lever arm is by definition the perpendicular distance from O to the line of action of F. Then in general (Figure 2.5) the torque (or moment) of a force about O (really about an axis through O perpendicular to the paper) is defined as the magnitude of the force times its lever arm; in Figure 2.5 this is F r sin θ. Now r × F has magnitude rF sin θ, so the magnitude of the torque is |r × F|. We can also use the direction of r × F in describing the torque, in the following way. If you curve the fingers of your right hand in the direction of the rotation produced by applying the torque, then your thumb points in a direction parallel to the rotation axis. It is customary to call this the direction of the torque. By comparing Figures 2.5 and 2.1, we see that this is also the direction of r × F. With this agreement, then, r × F is the torque or moment of F about an axis through O and perpendicular to the plane of the paper in Figure 2.5.

Angular Velocity In a similar way, a vector is used to represent the angular velocity of a rotating body. The direction of the vector is along the axis of rotation in the direction of progression of a right-handed screw turned the way the body is rotating. Suppose P in Figure 2.6 represents a point in a rigid body rotating with angular velocity ω. We can show that the linear velocity v of point P is v = ω × r. First of all, v is in the right direction: It is perpendicular to the plane of r and ω and in the right sense. Next we want to show that the magnitude of v is the same as |ω × r| = ωr sin θ. But r sin θ is the radius of the circle in which P is traveling, and ω is the angular velocity; thus (r sin θ)ω is |v|, as we claimed.

Figure 2.6