Proble Set 14: Oscillations AP Physics C Suppleentary Probles 1 An oscillator consists of a bloc of ass 050 g connected to a spring When set into oscillation with aplitude 35 c, it is observed to repeat its otion every 050 s Find (a) the period, (b) the frequency, (c) the angular frequency, (d) the spring constant, (e) the axiu speed, and (f) the axiu force exerted on the bloc A 0 N weight is hung fro the botto of a vertical spring, causing the spring to stretch 0 c (a) Calculate the spring constant This spring is now placed horizontally on a frictionless table One end of it is held fixed and the other end is attached to a 50 N weight The weight is then oved, stretching the spring an additional 10 c, and released fro rest (b) Calculate the period of oscillation 3 A body oscillates with the siple haronic otion according to the equation x = ( 60)cos[(3π rad / s) 3 Calculate a) the displaceent, (b) the velocity, (c) the acceleration, and (d) the phase at the tie t = 0 s Find also (e) the frequency and (f) the period of otion 4 A particle executes linear haronic otion about the point x = 0 At t = 0, it has displaceent x = 037 c and zero velocity The frequency of the otion is 05 Hz Deterine a) the period, (b) the angular frequency, (c) the aplitude, (d) the displaceent at tie t, (e) the velocity at tie t, (f) the axiu speed, (g) the axiu acceleration, (h) the displaceent at t = 30 s, and (i) the speed at t = 30 s 5 A 010 g bloc slides bac and forth along a straight line on a sooth horizontal surface Its displaceent fro the origin is given by x = ( 10c) cos[(10 rad / s) (a) Calculate the oscillation frequency (b) Calculate the axiu speed acquired by the bloc At what value x does this occur? (c) Calculate the axiu acceleration of the bloc At what value of x does this occur? (d) What force ust be applied to the bloc to give it this otion? Probles selected fro Halliday, D, & Resnic, R (1993) Fundaentals of Physics (4 th ed) New Yor: John Wiley & Sons, Inc
Oscillations 6 Two blocs ( = 10 g and M = 10 g) and a spring ( = 00 N/) are arranged on a horizontal, frictionless surface as shown below The coefficient of static friction between the two blocs is 040 Calculate the axiu possible aplitude of the siple haronic otion if no slippage is to occur between the blocs M 7 An oscillator consists of a bloc attached to a spring ( = 400 N/) At soe tie t, the position (easured fro the equilibriu location), velocity, and acceleration of the bloc are x = 010, v = -136 /s, a = -13 /s Calculate (a) the frequency, (b) the ass of the bloc, and (c) the aplitude of the oscillation 8 Two particles oscillate in siple haronic otion along a coon straight line segent of length A Each particle has a period of 15 s but they differ in phase by π/6 radians (a) How far apart are they (in ters of A) 050 s after the lagging particle leaves one end of the path? (b) Are they oving in the sae direction or in opposite directions at this tie? 9 Two identical springs are attaced to a bloc of ass and to fixed supports as shown below Show that the frequency of oscillation on the frictionless surface is 1 f = π
Oscillations 3 10 A bloc weighing 14 N, which slides without friction on a 40 o incline, is connected to the top of the incline by a light spring of unstretched length 045 and force constant 10 N/, as shown below (a) How far fro the top of the incline does the bloc rest in equilibriu? (b) If the bloc is displaced slightly down the incline, what is the period of the ensuing oscillations? 40 o 11 An oscillating boc-spring syste has a echanical energy of 10 J, an aplitude of 010, and a axiu speed of 1 /s Find (a) the force constant of the spring, (b) the ass, and (c) the frequency of oscillation 1 When the displaceent is one-half the aplitude x, what fraction of the total energy is (a) inetic and (b) potential in siple haronic otion? (c) At what displaceent, in ters of the aplitude, is the energy half inetic and half potential? 13 A 30 g particle is in siple haronic otion in one diension and oves according to the equation x = ( 50) cos[( π rad / s) t π rad] 3 4 (a) At what value of x is the potential energy equal to half the total energy? (b) How long does it tae the particle to ove to this position fro the equilibriu position? 14 The balance wheel of a watch vibrates with an angular aplitude of π rad and a period of 050 s Find (a) the axiu angular speed of the wheel, (b) the angular speed of the wheel when its displaceent is π/ rad, and (c) the angular acceleration of the wheel when its displaceent is π/4 rad 15 Calculate the length of a siple pendulu that ars seconds by copleting a full cycle every s
Oscillations 4 16 Two oscillating systes that you have studied are the bloc-spring and the siple pendulu There is an interesting relation between the Suppose that you have a weight on the end of a spring, and when the weight is in equilibriu, the spring is stretched a distance h Show that the frequency of this bloc-spring syste is the sae as that of a siple pendulu whose length is h h h 17 A siple pendulu with length L is swinging freely with sall angular aplitude As the pendulu passes its central or equilibriu position, its cord is suddenly and rigidly claped at its idpoint In ters of the original period of the pendulu T, what will the new period be? 18 A stic of unifor density and length L oscillates as a physical pendulu, pivoted about point O (a) Derive an expression for the period of the pendulu in ters of L and x, the distance fro the point of support to the center of gravity of the pendulu (b) Show that, if L = 100, the period will have a iniu value for x = 887 c (c) Show that, at a site where g = 9800 /s, this iniu value is 155 s O x CM
Oscillations 5 Answers: 1 a) 050 s b) 0 Hz c) 16 rad/s d) 794 N/ e) 44 /s f) 78 N a) 100 N/ b) 045 s 3 a) 30 b) -49 /s c) -66 /s d) 0 rad e) 15 Hz f) 067 s 4 a) 4 s b) 157 rad/s c) 037 c d) ( 037c)cos[( π rad / s) t] e) ( 058c / s)sin[( π rad / s) t] f) 059 c/s g) 091 c/s h) 0 c i) 058 c/s 5 a) 16 Hz b) 100 c/s at 0 c c) 1000 c/s at ± 10 c d) ( 1N)cos[(10rad / s) 6 0 7 a) 558 Hz b) 035 g c) 040 8 a) 018A b) sae direction (both having negative velocity) 9 proof 10 a) 055 b) 069 s 11 a) 00 N/ b) 14 g c) 19 Hz 1 a) 75% b) 5% c) 071A 13 a) 35 b) 074 s 14 a) 395 rad/s b) -34 rad/s c) -14 rad/s 15 099 16 proof 17 0707T 18 a) L + 1x π b) proof c) proof 1gx