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Unformatted text preview: Page 1 Chapter 11 CHAPTER 11 – General Rotation 1. ( a ) For the magnitudes of the vector products we have i × i = i i sin 0° = 0; j × j = j j sin 0° = 0; k × k = k k sin 0° = 0. ( b ) For the magnitudes of the vector products we have i × j = i j sin 90° = (1)(1)(1) = 1; i × k = i k sin 90° = (1)(1)(1) = 1; j × k = j k sin 90° = (1)(1)(1) = 1. From the right hand rule, if we rotate our fingers from i into j , our thumb points in the direction of k . Thus i × j = k . Similarly, when we rotate i into k , our thumb points along – j . Thus i × k = – j . When we rotate j into k , our thumb points along i . Thus j × k = i . 2. ( a ) We have A = – A i and B = B k . For the direction of A × B we have – i × k = – (– j ) = j , the positive yaxis . ( b ) For the direction of B × A we have k × (– i ) = – ( k × i ) = – ( j ) = – j , the negative yaxis . ( c ) For the magnitude of A × B we have A × B = A B sin 90° = AB . For the magnitude of B × A we have B × A = B A sin 90° = AB . This is expected, because B × A = – A × B . 3. The magnitude of the tangential acceleration is a tan = α r . From the diagram we see that α , r and a tan are all perpendicular, and rotating α into r gives a vector in the direction of a tan . Thus we have a tan = α × r . The magnitude of the radial acceleration is a R = ϖ 2 r = ϖ r ϖ = ϖ v . From the diagram we see that ϖ , v and a R are all perpendicular, and rotating ϖ into v gives a vector in the direction of a R . Thus we have a R = ϖ × v . 4. When we use the component forms for the vectors, we have A × ( B + C ) = [ A y ( B z + C z ) – A z ( B y + C y )] i + [ A z ( B x + C x ) – A x ( B z + C z )] j + [ A x ( B y + C y ) – A y ( B x + C x )] k = ( A y B z – A z B y ) i + ( A z B x – A x B z ) j + ( A x B y – A y B x ) k + ( A y C z – A z C y ) i + ( A z C x – A x C z ) j + ( A x C y – A y C x ) k = A × B + A × C ....
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This note was uploaded on 03/16/2008 for the course PHYS phys230 taught by Professor Hadley during the Spring '08 term at A.T. Still University.
 Spring '08
 Hadley

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