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

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Unformatted text preview: CHAPTER 10 Conservation of Angular Momentum 1* Â· True or false: ( a ) If two vectors are parallel, their cross product must be zero. ( b ) When a disk rotates about its symmetry axis, Ï– is along the axis. ( c ) The torque exerted by a force is always perpendicular to the force. ( a ) True ( b ) True ( c ) True 2 Â· 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 Â· 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 ; Ï„ = r Ã— F = FR j Ã—- i = FR i Ã— j = FR k . 4 Â· 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 Ï„ = - mgx i Ã— j- mgy j Ã— j = - mgx k 5* Â· Find A Ã— 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 Ã— i = j Ã— j = k Ã— k = 0 ( a ) A Ã— B = 24 i Ã— j = 24 k . ( b ) A Ã— B = 24 i Ã— k = -24 j . ( c ) A Ã— B = 4 i Ã— j- 9 j Ã— i = 13 k . 6 Â· Under what conditions is the magnitude of A Ã— B equal to A â‹… B ? A Ã— B = AB sin Î¸ = A â‹… B = AB cos Î¸ if sin Î¸ = cos Î¸ or tan Î¸ = Â±1; Î¸ = Â±45 o or Î¸ = Â±135 o . 7 Â· A particle moves in a circle of radius r with an angular velocity Ï– . ( a ) Show that its velocity is v = Ï– Ã— r . ( b ) Show that its centripetal acceleration is a c = Ï– Ã— v = Ï– Ã— ( Ï– Ã— r ) . (a) Let r be in the xy plane. Then if Ï– points in the positive z direction, i.e., Ï– = Ï– k , the particleâ€™s velocity is in the j direction when r = r i (see Figure) and has the magnitude r Ï– . Thus, v = Ï– Ã— r = r Ï– j . Chapter 10 Conservation of Angular Momentum ( b ) a = d v / dt = ( d Ï– / dt ) Ã— r + Ï– Ã— ( d r / dt ) = ( d Ï– / dt ) Ã— r + Ï– Ã— v = a t + Ï– Ã— ( Ï– Ã— r ) = a t + a c , where a t and a c are the tangential and centripetal accelerations, respectively. 8 Â·Â· If A = 4 i , B z = 0, B = 5, and A Ã— B = 12 k , determine B . B = B x i + B y j ( B z = 0); write A Ã— B B x 2 + B y 2 = B 2 ; solve for B x 12 k = 4 B y i Ã— j = 4 B y k ; B y = 3 B x = 4 ; B = 4 i + 3 j 9* Â· If A = 3 j , A Ã— B = 9 i , and A â‹… B = 12, find B . Let B = B x i + B y j + B z k ; write A â‹… B and find B y Write A Ã— B and determine B x and B y A â‹… B = 3 B y = 12; B y = 4 9 i = 3 B x j Ã— i + 3 B z j Ã— k = -3 B x k + 3 B z i ; B x = 0, B z = 3 B = 4 j + 3 k 10 Â· What is the angle between a particleâ€™s linear momentum p and its angular momentum L ?...
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ch10f - CHAPTER 10 Conservation of Angular Momentum 1 Â...

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