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HW07_Solutions - Problem Set 7 Giancoli 4th ed 29(a We use...

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Problem Set # 7 Giancoli 4 th ed. 29. ( a ) We use the determinant rule, Eq. 11-3b, to evaluate the cross product. ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 5.4 3.5 0 7.0 10.8 0.49 7.0 11 0.5 8.5 5.6 2.0 ! = " = " " + # " " + " i j k A B i j k i j k ! ! ( b ) Now use Eq. 11-3a to find the angle between the two vectors. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 2 2 2 2 2 2 2 2 1 1 7.0 10.8 0.49 12.88 ; 5.4 3.5 6.435 ; 8.5 5.6 2.0 10.37 12.88 sin sin sin 11.1 or 168.9 6.435 10.37 A B AB AB ! ! " " # = " + " + = = + = " + + = # # = $ = = = ° ° A B A B A B ! ! ! ! ! ! Use the dot product to resolve the ambiguity. ( )( ) ( )( ) ( ) 5.4 8.5 3.5 5.6 0 2.0 26.3 = ! + + = ! A B ! ! i Since the dot product is negative, the angle between the vectors must be obtuse, and so 168.9 170 . ! = ° " ° 36. Use Eq. 11-6 to calculate the angular momentum. ( ) ( ) ( ) ( ) ( ) 2 2 2 ˆ ˆ ˆ 0.075kg 4.4 6.0 0 m s 3.2 0 8.0 ˆ ˆ ˆ ˆ ˆ ˆ 0.075 48 35.2 19.2 kg m s 3.6 2.6 1.4 kg m s m = ! = ! = " " = + + = + + i j k L r p r v i j k i j k ! ! ! ! ! i i 36. Use Eq. 11-6 to calculate the angular momentum. ( ) ( ) ( ) ( ) ( ) 2 2 2 ˆ ˆ ˆ 0.075kg 4.4 6.0 0 m s 3.2 0 8.0 ˆ ˆ ˆ ˆ ˆ ˆ 0.075 48 35.2 19.2 kg m s 3.6 2.6 1.4 kg m s m = ! = ! = " " = + + = + + i j k L r p r v i j k i j k ! ! ! ! ! i i
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41. ( a ) We assume the system is moving such that mass B is moving down, mass A is moving to the left, and the pulley is rotating counterclockwise. We take those as positive directions. The angular momentum of masses A and B is the same as that of a point mass. We assume the rope is moving without slipping, so pulley 0 . v R ! = ( ) A B pulley A 0 B 0 A 0 B 0 0 A B 0 0 v L L L L M vR M vR I M vR M vR I R I M M R v R ! = + + = + + = + + = + + " # $ % & ( b ) The net torque about the axis of the pulley is that provided by gravity, B 0 . M gR Use Eq. 11-9, which is applicable since the axis is fixed. ( ) ( ) ( ) B 0 A B 0 A B 0 0 0 B 0 B A B 2 A B 0 0 0 dL d I I M gR M M R v M M R a dt dt R R M gR M g a I I M M M M R R R ! = " = + + = + + " = = + + + + # $ # $ % & % & ( ( # $ % & ( ) 47. This is a variation on the ballistic pendulum problem. Angular momentum is conserved
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