Take two arbitrary 3-D vectors A and B . Consider the more complicated product A

Question

Take two arbitrary 3-D vectors A and B . Consider the more complicated product Ax(A xB) with rightwards arrow on top close parentheses . Which of the following statements for this product is true?

A. You can move the parentheses around the double product, so let's put them around A xA, wich is zero. Thus the double product is equal to the zero vector.

B. There is too little information to make a meaningful statement.

C. You can move the parentheses around the double product so let's put them around AxA which is just A^2 where A is the length of vector A . Thus the double product is equal to A^2 xB.

D. The direction of the vector representing the double product is easily determined: It lies in the plane spanned by A and B and furthermore is perpendicular to A .

QUESTION 2 A small flywheel is spinning around a horizontal axle. Neglect any friction. Initially the axle points into the x-direction, but you would like to rotate the axle by 90 degrees so that it points into the y-direction. You think you know how to do this: you grab the axle and apply a force in the x-y-plane to rotate it into the y-direction. What will happen?

A. The rotating flywheel has angular momentum which points along the axle. Angular momentum is conserved, so you can not simply change the direction of the axle. If you want to change the direction of motion you first need to stop the flywheel, then rotate the axle, then spin it up again.

B. Newton's Second Law tells us that if a force is applied to rotate the axle from the x- to the y-axis the axle will follow that force. No surprises here.

C. A force in the x-y-plane means the torque on the axle is perpendicular to the x-y-plane. Thus there is a change in angular momentum which points out of the x-y-plane and hence the axle is moving out of the x-y-plane, perpendicular to the direction of the force.

D. The force on the axle has nothing to do with the rotation of the flywheel around the axle. Therefore angular momentum considerations don't apply. The axle will simply follow the force.

E. Both B and D are correct.

Explanation / Answer

FIRST QUESTION: for A X (A X B)

D. The direction of the vector representing the double product is easily determined: It lies in the plane spanned by A and B and furthermore is perpendicular to A

SECOND QUESTION:

E. Both B and D are correct.

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