961 Chapter 10 Angular Momentum Conceptual Problems 1 • True or false: (a) If two vectors are exactly opposite in direction, their cross product must be zero. (b) The magnitude of the cross product of 2 vectors is at a minimum when the two vectors are perpendicular. (c) Knowing the magnitude of the cross product of two nonzero vectors and their individual magnitudes uniquely determines the angle between them. Determine the ConceptThe cross product of vectorsAGand BGis defined to be nABBAˆsinφ=×GGwhere nˆ is a unit vector normal to the plane defined byAGand BG.(a) True. If AGand BGare in opposite direction, then sin= sin(180°) = 0. (b) False. If AGand BGare perpendicular, then sin= sin(90°) = 1 and the cross product of AGand BGis a maximum. (c) False. ⎟⎟⎠⎞⎜⎜⎝⎛×=−ABBAGG1sin, because of the magnitude ofBAGG×, gives the reference angle associated withBAGG×. 2 •Consider two nonzero vectors GA and GB . Their cross product has the greatest magnitude if GA and GB are (a) parallel, (b) perpendicular, (c) antiparallel, (d) at an angle of 45° to each other. Determine the Concept The cross product of the vectors AGand BGis defined to be nABBAˆsin=×GGwhere nˆ is a unit vector normal to the plane defined by AGand BG. Hence, the cross product is a maximum when sin= 1. This condition is satisfied provided AGand BGare perpendicular. )(bis correct. 3 •What is the angle between a forceFGand a torque vector τGproduced byFG? Determine the Concept Because nrFFrˆsin=×=GGG, where nˆ is a unit vector normal to the plane defined by rGand FG, the angle between FGand Gis.90°
Chapter 10 962 4 •A particle of mass mis moving with a constant speed v along a straight line that passes through point P. What can you say about the angular momentum of the particle relative to point P? (a) Its magnitude is mv. (b) Its magnitude is zero. (c) Its magnitude changes sign as the particle passes through point P. (d) It varies in magnitude as the particle approaches point P. Determine the Concept LGand pGare related according to .prLGGG×=Because the motion is along a line that passes through point P, r= 0 and so is L. )(bis correct. 5 •[SSM]A particle travels in a circular path and point P is at the center of the circle. (a) If the particle’s linear momentum Gp is doubled without changing the radius of the circle, how is the magnitude of its angular momentum about P affected? (b) If the radius of the circle is doubled but the speed of the particle is unchanged, how is the magnitude of its angular momentum about P affected? Determine the Concept LGand pGare related according to .prLGGG×=(a) Because LGis directly proportional topG,Lis doubled.
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