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Unformatted text preview: PHY 7A - Review for Final Exam (Solutions) Dimitri Dounas-Frazer Spring 2008 Strategy for Harmonic Motion Problems 1. Draw a picture. Choose an origin and a define a set of coordinate axes. Make sure to draw the system slightly displaced from equilibrium. If necessary, choose a pivot point as well. 2. Draw an extended free body diagram (FBD). Decompose all forces along the axes. 3. Apply F i = m a and/or i = I . Make approximations if necessary (e.g., sin ). 4. Solve for the equation of motion of the system: d 2 /dt 2 =- 2 . Identify the angular frequency in terms of the parameters of the problem. 5. Solve. Check your answer (units, limits, etc.). Problem 1 1. Draw a picture. Choose an origin and coordinate axes. There are three forces acting on the ball: gravity F , friction f , and the normal force N . Because the ball rolls back and forth along the tunnel, a convenient set of axes is parallel ( x ) and perpendicular ( y ) to the tunnel. With this convention, f points in the x-direction and N points in the positive y-direction. The gravitational force F points from the center the ball to the center of the earth. Choose the origin O at the earths center of mass and the pivot point P at the balls center of mass. 2. Draw an extended FBD. Decompose forces along axes. Because f and N already point along the axes, we only need to decompose F . The magnitude F of F is given by F = GM m r 2 = GMmr R 3 = mg R r, (1-1) where r is the distance from the center of the earth to the center of the ball. Here m is the mass of the ball, M = M ( r 3 /R 3 ) is the mass of sphere underneath the ball, and M and R are the mass and radius of the earth, respectively. (See pages 142-143 of Giancoli.) I have used g = GM/R 2 = 9 . 8 m / s 2 to simplify this expression. Let x and y denote the horizontal and vertical distances from the origin to the center of the ball. After drawing a right triangle, you can see that F x =- mg R x and F y =- mg R y, (1-2) where F x and F y are the components of F along the x- and y- axes, that is, F = F x i + F x j . 1 3. Apply Newtons second law. Since there is no motion in the y-direction, we ignore it com- pletely. In the x-direction, we have X F x,i =- F x + f = ma, (1-3) where a is the acceleration of the balls center of mass. Furthermore, the torque equation is X i = r obj f...
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This note was uploaded on 10/28/2009 for the course PHYSICS 7A/7B taught by Professor All during the Fall '08 term at University of California, Berkeley.
- Fall '08