physics7A-spring03-final-Zettl-exam

physics7A-spring03-final-Zettl-exam - Physics 7A Final...

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Unformatted text preview: Physics 7A Final Examination. Section 3. A. Zettl, Spring 2003. All problems worth 20 points each. - . , _ one Atmenstunql l. A thin rod of length L is cast from a metal alloy in such a way that It has a non-unifomg mass density 9» along its length: A: A + Bx4, where A and B are constants and x is measured from the middle of the rod. The rod is rotated at constant angular speed to about an axis perpendicular to the rod and passing through the center of the rod. Determine the magnitude of the angular momentum the rotating rod. Kaf- AI‘CA o‘c Co 'ha. : 9 w 2o.) / Area. [4‘ 25) 2;", 4c DMS‘J‘} S" a ew— 2. 7. . Densif-f 9; Large KARL 2. A cylindrical block of wood (density p1) of cross sectional area A1 and height L floats in a tank of fluid (density p2). The tank has cross—sectional area A2. You push down a bit on the block and then let go, and the block is observed to bob up and down on its own. Determine the angular frequency a) of the block's up and down oscillations, assuming in turn the following conditions: a) that the tank holding the fluid has an extremely large cross-sectional area (i.e. A2>>A1) 33gb) that the cross—sectional area of the tank is not much larger than A1. This means that the fluid level is not constant as the block oscillates. 3. A model for a one-dimensional solid is a series of identical point masses in (representing the atoms) connected by identical springs of spring constant k and length L (representing the bonds between atoms). a) Estimate the speed of a longitudinal (compressional) wave travelling through such a solid. [Hint Your answer will contain m, L, and k. Recall the method we sometimes used in class to attack rather difficult problems] A m .It m ,k - . ' 1% M Jl M k M C . . - .. . V?) vT—J v V‘— ""’ a}: ' 4. You (of mass m) have just climbed to the top of a tall, thin, uniform flagpole (of length L and mass M) when it cracks at its base and begins to topple over (the cracked base serving as a frictionless hinge). You decide to either a) let go immediately and "free—fall" to the ground, or b) hang on to the tip and ”ride" the pole all the way to the ground. Assuming the criterion for minimum injuries is minimum speed v with which you hit the ground, determine which is the safer choice, a) or b). Do not make any assumptions about the relative ma nitudes of m and M i.e. the ma be com arable . g ( y y p ) “VI/Y6 v=lm/Sec A-VB All) m I: ((4) 4%) \ S) . "MA/s“ BaC t, {i \ 94:92" c-=7A M M l l 6 {fl A ”'7'”? ’3‘ 5. A nervous monkey races around inside his cage, over and over. The path the monkey takes is indicated in the figure—— all motion is in the x-y plane. At time t=0 the monkey starts at the origin at A and runs with a constant speed v=1m/s across the floor to point B (at x=1m, y=0). Then he immediately climbs the curved front of the cage (a quarter-circle, with radius R =lm), again moving at constant speed v=1m/s up and over to C (at x=0, y=1m) where he instantly stops and then immediately "free falls", dropping to the ground at A, whereupon he immediately runs from A to B again, etc. He is bored out of his mind and would happily trade places with you right now. a) Determine the round—trip time T for the monkey in going from A back to A. b) Make quantitative plots of the velocity vector components vx and vy for the monkey, from t=0 to t=T. Keep track of positive and negative quantities. c) Make quantitative plots of the acceleration vector components ax and ay for the monkey, from t=0 to t=T. Keep track of positive and negative quantities. ...
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