Ch0121 - Chapter 21 Vectors: The Cross Product & Torque...

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Chapter 21 Vectors: The Cross Product & Torque 132 21 Vectors: The Cross Product & Torque Do not use your left hand when applying either the right-hand rule for the cross product of two vectors (discussed in this chapter) or the right-hand rule for “something curly something straight” discussed in the preceding chapter. There is a relational operator 1 for vectors that allows us to bypass the calculation of the moment arm. The relational operator is called the cross product. It is represented by the symbol × ” read “cross.” The torque τ h can be expressed as the cross product of the position vector r h for the point of application of the force, and the force vector F h itself: F τ h a a × = r (21-1) Before we begin our mathematical discussion of what we mean by the cross product, a few words about the vector r h are in order. It is important for you to be able to distinguish between the position vector r h for the force, and the moment arm, so we present them below in one and the same diagram. We use the same example that we have used before: in which we are looking directly along the axis of rotation (so it looks like a dot) and the force lies in a plane perpendicular to that axis of rotation. We use the diagramatic convention that, the point at which the force is applied to the rigid body is the point at which one end of the arrow in the diagram touches the rigid body. Now we add the line of action of the force and the moment arm r to the diagram, as well as the position vector r h of the point of application of the force. 1 You are much more familiar with relational operators then you might realize. The + sign is a relational operator for scalars (numbers). The operation is addition. Applying it to the numbers 2 and 3 yields 2+3=5. You are also familiar with the relational operators , , and ÷ for subtraction, multiplication, and division (of scalars) respectively. Axis of Rotation F O Position of the Point of Application of the Force
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Chapter 21 Vectors: The Cross Product & Torque 133 The moment arm can actually be defined in terms of the position vector for the point of application of the force. Consider a tilted x -y coordinate system, having an origin on the axis of rotation, with one axis parallel to the line of action of the force and one axis perpendicular to the line of action of the force. We label the x axis for “perpendicular” and the y axis || for “parallel.” F r Line of Action of the Force The Moment Arm O r h Position Vector for the Point of Application of the Force F O r h ||
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Chapter 21 Vectors: The Cross Product & Torque 134 Now we break up the position vector r h into its component vectors along the and | axes. From the diagram it is clear that the moment arm
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This note was uploaded on 02/29/2012 for the course PHYS 227 taught by Professor Rabe during the Fall '08 term at Rutgers.

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Ch0121 - Chapter 21 Vectors: The Cross Product & Torque...

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