It turns out that a vector equation governing the rate of rotation of a solid body can be obtained by writing Newton's second law for each particle in the body, multiplying it vectorially with the position vector of that particle, and summing over all particles in the system under consideration. To present this procedure in more detail, we introduce a brief synopsis of the vector multiplication.
Vectors may be multiplied with other vectors in two distinct ways. The scalar product (also called the dot product) of a and b is defined as the projection of one of the vectors onto the other:
where a • b is the dot product, and ||a|| and ||b|| are the magnitudes of a and b. The symbol 0 represents the angle between a and b. Note that the dot product can be positive or negative but has no direction, and therefore it is a scalar.
An important example of dot product is the mechanical work done by a force. When a force acting on a particle is multiplied scalarly with the displacement of the particle, the product W is called the work done by that force. When a person resists a large force to remain at rest, the work done by this force is equal to zero because there is no displacement. Resisting a force statically requires caloric expenditure but produces no mechanical work. In a pendulum, the tensile force exerted by the cord on the bob does no work because this force is always perpendicular to the path traversed by the bob. Work done by a force is positive if the projection of the force on the displacement vector is in the same direction as the displacement. When a particle falls toward earth, gravity does positive work on the particle. On the other hand, when an object is raised vertically the work done by gravity on the object is negative.
According to Eqn. 3.15, the dot product of two vectors that are perpendicular to each other is zero. The following results then hold for the unit vectors e1, e2, and e3:
ei • e2 = ei • e3 = e2 • e3 = e2 • ei = e3 • ei = e3 • e2 = 0 (3.16a)
Scalar product of two vectors can also be written in terms of the projections of vectors on a Cartesian coordinate system. Let a and b be two vectors whose components with respect to a Cartesian coordinate system E are given by the following equations:
Using Eqns. 3.16 and 3.17, the scalar product of a and b can be shown to obey the following relationship:
Figure 3.6a-c. Two vectors a and b and the angle between them (6) are shown in (a). In multiplying vectors, we can bring their tails together and proceed from there. The dot product a • b is a scalar. If a and b are perpendicular to each other, the dot product is zero. The vector product a X b is a vector that is perpendicular to both a and b. The number of vectors (shown with dashed lines) that have the same vector product as rp/o X F when multiplied with F reach infinity (b). Two forces of equal magnitude and opposite direction form a couple (c). The magnitude of the couple is equal to the product of the magnitude of one of the forces and the shortest distance between the lines of action of these forces. The direction of the couple is perpendicular to the plane created by the two parallel forces.
Two vectors a and b can be multiplied vectorially to give a third. The vector product of a and b is called the cross product and is shown as a X b. The resulting product is a vector that is perpendicular to both a and b:
where 6 is the angle between a and b, and e is a unit vector that is normal to both a and b (Fig. 3.6a). The magnitude of a X b is the area of the parallelogram traversed by a and b. The sense of direction of e is determined by the right-hand rule: point the fingers of the righthand in the direction of a, then turn the fingers toward b, and the thumb will point to the right sense of direction of e. The vector product plays a most important role in mechanics. The lever systems of the human body discussed in Chapter 1 are direct consequences of the vector product of force and the lever arm.
According to Eqn. 3.18a, if two vectors are perpendicular to each other, the magnitude of the cross product is equal to the product of the magnitudes of the vectors. Also, if two vectors are parallel to each other, their cross product is equal to zero. This leads to the following vector products between the unit vectors e1, e2, and e3:
The vector product a and b can then be expressed as a X b = (a2b3 — a3b2) e1 — (a1b3 — a3b1) e2 + (a1b2 — a2b1) e3
where a1, a2, a3 and b1,b2, b3 are the projections of the vectors a and b along e1, e2, and e3 directions, respectively.
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The use of dumbbells gives you a much more comprehensive strengthening effect because the workout engages your stabilizer muscles, in addition to the muscle you may be pin-pointing. Without all of the belts and artificial stabilizers of a machine, you also engage your core muscles, which are your body's natural stabilizers.