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Honors Physics 1 Class 05 Fall 2013 Newton’s Laws Applications 1 Activity: Application The Spinning Terror ride The spinning terror is a large vertical drum which spins so fast that everyone stays pinned to the wall when the floor drops out. For a typical ride the radius of the drum is 2 m. What is the minimum angular velocity if the coefficient of friction between the patron and the wall is 0.3? 2 Spinning Terror v2 The radial acceleration that must act on the rider is R 2 . R The normal force is therefore N Ma MR 2 . The maximum static friction that can act on the rider is therefore f max N . Since we require M to be in vertical equilibrium, we need f max Mg . So N MR 2 Mg . g and the smallest value of that R allows the ride to run is: Thus 2 min g 10 rad 4 R 0.3 2 s which is about 0.6 turns per second. 3 Application example: Mass on a spring Equilibrium position x 0; Starting point=x0 F kx Write the F ma equation. Assume a solution of the form: x(t ) Ae t . See what conditions have to be met by A and to solve the relation F ma and satisfy initial conditions. 4 Mass on a spring F kx and F ma m d 2x dt 2 kx d 2x k which gives 2 x 0, a common differential equation. m dt We will try a solution of the form: x(t ) Ae t k t 2 t Ae Ae 0 and assuming A 0, m k k 2 i i m m So a general solution is: x Aeit Beit (each one works independently) 5 Mass on a spring: initial conditions x Aeit Be it (Initial conditions: x(0) x0 and v(0) 0) dx (0) 0 i A i B so A B dt x(t ) A eit e it 2 A cos t x(0) x0 2 A k The mass oscillates with amplitude x0 at frequency . m 6 Application example: Falling through a viscous fluid Assume that the density of the fluid is very small compared to the density of the falling object. (e.g – a human body in air) Assume that the body falls under the action of constant gravity and drag force only. Assume that the drag force is linear in speed: FD Cv Is there a terminal velocity? If there is, find the terminal velocity. 7 Falling... The falling object is acted on by buoyant force, gravity, and drag. F Mg FB Cv Ma If the body is in equilibrium (not accelerating) then Ma=0. Mg FB Cvterminal Mg FB so vterminal C Mg and for a body falling in air: vterminal C 8 Taylor series ( x a)2 f ( x) f (a ) f '(a )( x a ) f ''(a ) ... 2! 1 3 1 5 sin( x) x x x ... 3! 5! x2 1 4 cos( x) 1 x ... 2 4! 9