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Physics 3221 Spring Term 2007 Test 1, February 7, 2007 • This is an open notes/text book/Homework test lasting 50 minutes. • There are 4 problems, divided into subsections. The points for each part are marked. • Begin each problem on a fresh sheet of paper. Use only one side of a sheet of paper. • Put your name, the problem number, and the page number in the upper right hand corner of each sheet. • To receive partial credit you must explain what you are doing. Carefully labeled figures are important. Randomly scrawled equations aren't helpful. • Draw a box around important results. There are 2 pages including this page. Do not forget to look at all parts of the problems. 1. (4 points) A guitar string has mass m, length l and the tension T. (Tension has the same dimension as force.) Wave travels with velocity v ∝ m a l bT c on the string. Find a, b and c. 2. a) (3 points) The following matrix is a valid rotation matrix. Find the values and signs of a and b. ⎛ 0 .6 a 0 .8 ⎞ ⎜ ⎟ ⎜ 0 1 0 ⎟ ⎜ 0. 8 0 b ⎟ ⎝ ⎠ b) (1 point) Point P is at position x=1, y=2, z=3 in the original coordinate system. Find its new position after the axes are rotated according to the rotation matrix in part (a). 3. Find a normal for the following surfaces a) (2 points) x2y2-z3 = 4 at the point (1,2,0). b) (3 points) z = x2-y2 at the point (2,1,3). 4. In the following figure, Points A and C are vertices of the unit cube. Point B is the midpoint of one of the edges. z 1 A 1 C y B x 1 1/2 a) (2 points) Find the angle between AB and BC. b) (2 points) Find the unit normal to plane ABC. c) (3 points) Find the distance from the origin to plane ABC.