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Unformatted text preview: homework 20 FISHER, CHRISTIAN Due: Apr 8 2008, 11:00 pm 1 Question 1, chap 13, sect 4. part 1 of 2 10 points A 3 kg bicycle wheel rotating at a 2684 rev / min angular velocity has its shaft supported on one side, as shown in the figure. When viewing from the left (from the positive xaxes), one sees that the wheel is rotating in a clockwise manner. The distance from the center of the wheel to the pivot point is 0 . 6 m. The wheel is a hoop of radius 0 . 4 m, and its shaft is horizontal. Assume all of the mass of the system is located at the rim of the bicycle wheel. The acceleration of gravity is 9 . 8 m / s 2 . y z x . 6 m mg 2 6 8 4 r e v / m i n . 4 m radius Find the change in the precession angle (in degree) after a 1 . 9 s time interval. Correct answer: 14 . 2339 (tolerance 1 %). Explanation: Let : m = 3 kg , = 2684 rev / min = 2 (2684 rev / min) (60 s / min) = 281 . 068 rad / s , b = 0 . 6 m , and R = 0 . 4 m . Basic Concepts: vector = d vector L dt Solution: The magnitude of the angular momentum I of the wheel is L = I = mR 2 , and is along the negative xaxis. From the figure in Part 2, we get = L L . Using the relation, L = t, where is the torque, vector = vector b mvectorg . The precession angle is = L L = t L = mg b t mR 2 = g b t R 2 = (9 . 8 m / s 2 ) (0 . 6 m) (1 . 9 s) (0 . 4 m) 2 (281 . 068 rad / s) = 0 . 248428 rad = 14 . 2339 . Question 2, chap 13, sect 4. part 2 of 2 10 points The direction of precession as viewed from the top is 1. clockwise. correct 2. along the direction of rotation of the wheel. 3. counterclockwise. 4. opposite to the direction of rotation of the wheel. 5. static, since angular momentum is con served. Explanation: vector = vector b mvectorg , where vector b is along the positive yaxis and vectorg is into the page, in the figure below. Therefore the direction of the torque vector is along the positive yaxis. homework 20 FISHER, CHRISTIAN Due: Apr 8 2008, 11:00 pm 2 + x x L + L L L wheel Viewed from Above + y L We can see the direction of precession is clockwise, due to vector = L t , that is L is in the direction of the torque and is producing clockwise motion ( vector L vector produces clockwise motion). Question 3, chap 13, sect 99. part 1 of 2 10 points A thin hollow cylindrical rod has length 1 . 4 m, mass 0 . 09 kg, and diameter 0 . 3 cm . The thin cylindrical rod is free to rotate about a vertical axis that passes through its center and is perpendicular to the cylinders axis. Inside the cylinder are two masses of 0 . 7 kg each, attached to springs of spring constant k and unstretched lengths 0 . 3 m . The springs have negligible mass. The inside walls of the cylinder are frictionless....
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