10 Oscillations Tutorial

in w hich it is a nuisance . 5 1 1 D escribe two exam ples of resonance, o n e in w hich this phenom enon is useful and the other features of your graphs. varies w ith driving frequency for very light, m o derate and heavy dam ping . E xplain the S ketch a set of graphs, tising the sam e axes, to show how the am plitude of forced oscillation5 10 5 9 W hat do you understand by forced oscillations and resonance? suspension system im portant? 5 8 G ive one practical exam ple of a lightly dam ped oscillation. W hy is critica©¥ dam ping in a car the vibration itse©¥f? fńction©¥ess surface. W hat is the frequency of the energy variation as com pared w ith that of harm onic m otion of a m ass attached to a horizonta©¥ spring . A ssum e the m ass is m oving on a D escribe the variation in betw een kinetic and potential energy w ith tim e during sim p©¥e disp©¥acem ent. D raw graphs to show how the velocity and acceleration of the osci©¥lator in 5 4 vary w ith tim e of the osci©¥©¥ator in 5 4 . D raw graphs to show the changes in displacem ent, v e locity and acceleration w ith respect to equilibrium position. H ow do you express its ve©¥ocity and acce©¥eration In term e of tim e? W rite dow n a solution for a sim ple harm onic oscillator w hich start8 ©©ta m otion from the uefine spie harmon motjon. S late the defining equallon of sim p©¥e harm onic m otion, E xpress the period T ©¥n term s of frequency f and angular frequency a) . angu©¥ar frequency of a slm p©¥e harm onic m otion? W r©¥at do you understand by the term s displacem ent, a m p©¥ilude, period, frequency arxl m « C heck Q uest©¥one = ©¥ - ʨ¤S a$ W . » = W = TTT uttt , ' +++ , " Y r S A P H Y S ©¥C S D E P A R T M E N T R A F F LE S ©¥N S T ©¥TU T ©¥O N

J82/I u9 E 4 00 rad z a 2 - Z D 100 rad ' a J90/©¥/1 1 ; N 9 5/©¥/9 any of 1 , 2 , 3 , 4 and 5 either 2 or 4 ' ' ' P . W B either 1 or 5 D A 3 C num ber of hits? fixed aim in order to score the g reatest A t w hich reg ion should the player take a m otion. from side to side w ith sim ple harm onic gun in a fixed direction, a n d the target m oves random tim es. T he p©¥ayer has to point the a m oving target. T he gun fires by itse©¥f at In a fairground shooting gam e, a gun fires atS P 3 J85/08 ; J9 2/©¥lg E total energy am plitude angu©¥ar frequency D force total energy am plitude C angu©¥ar frequency acceleration force B am plitude angular frequency acceleration A acceleration force tota©¥ energy undam ped sim p©¥e harm onic m otion? ©¥n which of the fo©¥©¥ow ©¥ng lists are a©¥l three quantities constant w hen a partic©¥e m oves inS P 2 C 4 . 0 rada 5 4 B 1 . 0 rad a B Q A 0 . 25 rad a ©¥ a (N ote : ve©¥ocity ¡¤ Ŕi, a c c e ©¥eration - * ) position. W hat is the constant a, ' 7 X is the displacem ent from the ¨¤uilibrium ' A ￠ w ¡¤ ' " " " " " " " h - subsequent m o©¥ion is * . - 1 u 2 x , w here x and is then released. T he equation of the D Q (0 . 0 5 m ) towards Q by a force of 1 0 N T he trolley is dieplaced a sm a©¥l diatance springs under tension as shown In the figure below . A trolley of m ass 2 kg with free - running w heels ©¥©¥ a\©¥ached to tw o r©¥xed poin©¥©¥ P and Q by ©¥w oS P 1 e©¥fpract©¥ceQueationa©¥©¥i ' ' ' , a w' " 4 » . . M m ¢®M & « mwlaM ©¥aa= « M = H + fl! Y r 5 4 P H Y S IC S D E P A R T M E N T R A F F LE S IN S T ©¥T U T IO N

acceleration in opposite directions? A t w hich point are the velocity and perform ing sim p©¥e harm onic m otion . displacem ent against tim e for a body S P 4 T he diagram show s the graph of 3 J2000/1/9 C 4 T he speed is a m axim um at t- ï ¡¤ D T T he restoring force on the m ass increases betw een t- o a n d t = T T he kinetic energy is a m axim um at t- i ¡¤ B T A T he am plitude of the oscillation is 7 0 cm . Į â W hat can be deduced from this graph? = ceiling varies w ith tim e t. graph show s how its distance from theêÅ国ûö sim ple harm onic m otion of period T . T he T he m ass then oscillates vertically w ith from a ceiling is pulled dow n and released . S P 6 A m ass hanging from a spring suspended N 9 2/u9 . ' " y - m - fk D ìé 1 2 V S energy l . k X a '" ' " '" ' " ' " ' " w' " ' " ' " ' " ' " e' " gra . N d po©¥eM al Y r 5 - 6 P H Y S IC S D E P A R T M E N T R A FF L E S ©¥N S T ©¥T U T ©¥O N

4 pulled dow nw ards and released, s u c h that it oscillates vertically . S P 9 D iscuss the energy changes w hich take place w hen a m ass suspended from a spring is W rite an equation to describe how the displacem ent of Q varies w ith tim e. D raw , u s ing the sam e axes, the displacem ent - tim e graphs for m otions of P and Q , the m otion of Q lags that of P by ?c/2 rad and the am plitude of Q is tw ice that of P . A nother particle Q also m oves w ith sim ple harm onic m otion of the sam e * equency . H ow ever, (e) W hat is the m axim um acceleration of the particle during lts m otion? the extrem e end of the sw ing? (d) W hat is the velocity of the particle as it passes through its equilibrium position, a n d a t (c) H ow long does it take for the particle to com plete one oscillation? (b) W nat is the * equency of the m otion? (a) W hat is the am plitude of the m otion? w here x is in m etres and t in seconds. x - (0 . 05) sin 8 nt by the expression S p8 T he displacem ent of a particle P w hich m oves w ith sim ple harm onic m otion can be described J86/08 ; N 9 3lu7 : J 9 9/©¥/9 0 c f O D A C A O į 户 «í «Ï ¡£ disp©¥acem ents x. variation w ith tim e t of their sam e intial disp©¥acem ent and are then : ø¢ "T w o objects P and Q are given theS P 7 Y r 5 4 P H Y S IC S D E P A R T M E N T R A F F L E S ©¥N S T IT U T ©¥O N

©©um tablo screen W t speed (©¥) a s show n in th e fig u re . P a r a E c l r, r o tating w ith a constant angular of a horizontal turntab©¥e of radius A vertical peg is fixed to the rim D 4 [N 9 6nm ] sucH that the cup does not slip . (d) If the am plitude of the m otion is 0 . 050 m , c a lculate the m axim um possible frequency be observed to slip if the frequency of oscillation increases beyond a certain value. G iven that the m axim um value of F is half the w eight of the cup, e xplain w hy the cup will m otion, a n d the displacem ent x of the tray . (b) W ńte an equation for F in term s of the m ass m of the cup, the angular * equency m of the (a) D raw the frictional force F acting on the cup for the instant of tim e show n. X - O by the a rrow show n in the fig u re below _ _ _ _ _ _ I - - ¡¤ _ _ _ _ _ _ W requilibń um position (x - O ) as indicated the tray is displaced to the right of the T raY harm onic m otion. A t one instant of tim e, horizontal©¥y back and forth in sim ple C up D Į A tray, holding an em pty cup, is m oved O tim e before entering? tide. H ow long w ill the boat have to w ait w ater of 1 . 5 m , a pproaches the harbour at low b w tlde A boat w hich requires a m inim um depth of hours. tim e betw een successive low tides is 1 2 ' \, I1 . 0 m at low tide and 3 . 0 m at high tide. T he sim ple harm onic. T he depth varies between , . .. R ise a n d fa ll o f w a te r in a h a rb o u r is T 1 - - - - - - - - - - - - . " hh tloe D 2 0 9 1/u9] Pĺac©©ic : 4 ņ n ¡¤ ©¥ ex©©m d fin om urtp { ©¥ (e) the m agnitude of the m axim um acce©¥eration of the m ass (d) the m agnitude of velocity w hen the m ass is 0 . 0 50 m from equilibrium «c) the m axim um velocity (b) the am plitude of the oscillation «a) the spring constant pulled dow n 0 . 10 0 m below this equilibrium point and released. D eterm ine D 1 A light spring stretches 0 . 150 m w hen a 0 . 30 0 kg m ass is hung from its ©¥ow er end . T he m ass is Y r 5 4 P H Y S IC S D E P A R T M E N T R A F F L E S ©¥N S T ©¥T U T ©¥O N

F ig . 1 clam p m etre ru©¥e in F ig . 1 (b) A horizontal m etre rule is clam ped at one end . T he ï ee end oscillates vertically as ş