Physics 5B Lecture 5 January 20 2012 Chapter 14 Simple Harmonic Oscillator Atherosclerosis Suppose that all of the arteries in your body get reduced d d in radius d bby a hhalf lf d due to plaque l buildup b ld Assuming the blood to be an incompressible fluid yyou would expect p that to maintain the same flow of blood the blood pressure would have to increase by a factor of Thi iis nott a realistic li ti scenario i clearly l l This A 2 But by the same calculation a 10 decrease in radius would require a B 4 pressure increase of 50 C 8 D 16 4 1 1 5 0 9 2 Suppose that the length of an oil pipeline is increased by a factor of 81 Byy what factor should the pipe p p radius be increased in order to have the same flow rate of oil with the same pressure difference A B C D 2 3 4 9 3 Oscillations CHAPTER 14 4 Simple Harmonic Oscillation Results from a Hooke s Law force F k x x0 Good approximation pp for anyy system y that approximately obeys Hooke s law e g p pendulum Many mechanical systems can be modeled p coupled p harmonic as a set of multiple oscillators Tremendous importance in physics and mechanical engineering 5 Finite Element Vibration Analysis Finite d powerful f l engineering i i technique h i i to divide di id a complex l A very common and is shape into a grid of coupled simple harmonic oscillators Jahangir Ansari Virginia State University Finite Element Vibration Analysis and Modal Testing of Bells Proceedings of the 2006 IJME INTERTECH Conference Session ENG 204 113 6 Mass On Spring Idealization we will always assume that the spring itself has negligible mass A d for And f now assume zero friction f i ti http scipp ucsc edu johnson applets SHO 1D htm 7 Newton ss 2nd Law Newton m d 2x dt 2 F k x x0 Linear 2nd order ordinary differential equation equation For notational simplicity we will usually choose our coordinate di system suchh that h the h origin i i is i at the h equilibrium point in which case x0 0 8 Initial conditions x 0 A v 0 0 x t A cos t v t A sin t A cos t 2 a t 2 A cos t 2 A cos t k 2 f m T 1 f http scipp ucsc edu johnson applets SHO 1D htm 9 A mass on a spring is undergoing simple harmonic oscillation When the acceleration of the mass is zero zero oscillation its velocity is A Zero B Maximum C Maximum divided by square root of 2 D Maximum divided by 2 10 A mass on a spring is undergoing simple harmonic oscillation When the force on the mass is maximum maximum oscillation its velocity is A Zero B Maximum C Maximum divided by square root of 2 D Maximum divided by 2 11 Problem 1414 16 The graph of displacement versus time for a small mass m at the end of a spring is shown below below At t 0 t 0 x 0 43 x 0 43 cm cm a If m 9 5g find the spring constant k b Write the equation for displacement x as a function of time 12 Comparing Springs Two identical springs p g have different masses attached to them with mass A being twice as massive as B The masses are displaced equal distances from the equilibrium points and simultaneously released Which mass will return to its equilibrium point first A Mass A will arrive first B Mass B will arrive first B first C They will arrive at the same time 13 Comparing Springs Two identical springs p g have identical masses attached to them The masses are displaced different distances from the equilibrium points with mass A displaced twice as far as mass B B Then they are simultaneously released Which mass will return to its equilibrium point first A Mass A will arrive first B Mass B will arrive first B first C They will arrive at the same time 14 Mass Hanging g g from a Spring p g mg so mg k x0 k ma kx mg x0 m d 2x 2 k x k x0 k x x0 dt so x t x0 A cos t Here the coordinate system is chosen such that x increases in the downward direction The effect of adding the weight is simply to move the equilibrium position downward Oscillation about that equilibrium is still simple harmonic 15 Mass On On Spring Demo Mass http phet colorado edu en simulation mass spring lab http phet colorado edu en simulation mass spring lab 16