ch10f - CHAPTER 10 Conservation of Angular Momentum 1 ...

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Unformatted text preview: CHAPTER 10 Conservation of Angular Momentum 1* · True or false: ( a ) If two vectors are parallel, their cross product must be zero. ( b ) When a disk rotates about its symmetry axis, ϖ is along the axis. ( c ) The torque exerted by a force is always perpendicular to the force. ( a ) True ( b ) True ( c ) True 2 · Two vectors A and B have equal magnitude. Their cross product has the greatest magnitude if A and B are ( a ) parallel. ( b ) equal. ( c ) perpendicular. ( d ) antiparallel. ( e ) at an angle of 45 o to each other. ( c ) 3 · A force of magnitude F is applied horizontally in the negative x direction to the rim of a disk of radius R as shown in Figure 10-29. Write F and r in terms of the unit vectors i , j , and k , and compute the torque produced by the force about the origin at the center of the disk. F = - F i ; r = R j ; τ = r × F = FR j ×- i = FR i × j = FR k . 4 · Compute the torque about the origin for the force F = - mg j acting on a particle at r = x i + y j , and show that this torque is independent of the y coordinate. Use Equs. 10-1 and 10-7 τ = - mgx i × j- mgy j × j = - mgx k 5* · Find A × B for ( a ) A = 4 i and B = 6 i + 6 j , ( b ) A = 4 i and B = 6 i + 6 k , and ( c ) A = 2 i + 3 j and B = -3 i + 2 j . Use Equ. 10-7; Note that i × i = j × j = k × k = 0 ( a ) A × B = 24 i × j = 24 k . ( b ) A × B = 24 i × k = -24 j . ( c ) A × B = 4 i × j- 9 j × i = 13 k . 6 · Under what conditions is the magnitude of A × B equal to A ⋅ B ? A × B = AB sin θ = A ⋅ B = AB cos θ if sin θ = cos θ or tan θ = ±1; θ = ±45 o or θ = ±135 o . 7 · A particle moves in a circle of radius r with an angular velocity ϖ . ( a ) Show that its velocity is v = ϖ × r . ( b ) Show that its centripetal acceleration is a c = ϖ × v = ϖ × ( ϖ × r ) . (a) Let r be in the xy plane. Then if ϖ points in the positive z direction, i.e., ϖ = ϖ k , the particle’s velocity is in the j direction when r = r i (see Figure) and has the magnitude r ϖ . Thus, v = ϖ × r = r ϖ j . Chapter 10 Conservation of Angular Momentum ( b ) a = d v / dt = ( d ϖ / dt ) × r + ϖ × ( d r / dt ) = ( d ϖ / dt ) × r + ϖ × v = a t + ϖ × ( ϖ × r ) = a t + a c , where a t and a c are the tangential and centripetal accelerations, respectively. 8 ·· If A = 4 i , B z = 0, B = 5, and A × B = 12 k , determine B . B = B x i + B y j ( B z = 0); write A × B B x 2 + B y 2 = B 2 ; solve for B x 12 k = 4 B y i × j = 4 B y k ; B y = 3 B x = 4 ; B = 4 i + 3 j 9* · If A = 3 j , A × B = 9 i , and A ⋅ B = 12, find B . Let B = B x i + B y j + B z k ; write A ⋅ B and find B y Write A × B and determine B x and B y A ⋅ B = 3 B y = 12; B y = 4 9 i = 3 B x j × i + 3 B z j × k = -3 B x k + 3 B z i ; B x = 0, B z = 3 B = 4 j + 3 k 10 · What is the angle between a particle’s linear momentum p and its angular momentum L ?...
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ch10f - CHAPTER 10 Conservation of Angular Momentum 1 ...

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