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Notes On Oscillations

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Published in: Physics | Science
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A2 : Oscillations (Notes) A2 : Oscillation (Tutorial)

Fauziah / Kuala Lumpur

13 years of teaching experience

Qualification: Bachelor of Applied Science

Teaches: Physics, Engineering Subjects, Geophysics, Thermodynamics

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  1. speed Of ight In free space permeability of free space permittvity of free space elementary charge the Planck constant unified atomic mass unit rest mass of electron rest mass of proton mlar gas the Avogadro constant the Boltzmann constant gravitational constant acceleration of free tall A2 W — PAV p — pgh StayHomeTut0ring@Tutoring Online LEVEL PHYSICS CHAPTER : OSCILLATIONS To ic : Oscillations 3.00 x 4m x 10—7 H 8.85 x - 8.99 x 109mF-l) e- 1.60 x 10-19C 683 x 10-34Js h 186 x 10-27 kg lu 9.11 x 10-31 kg m 1.67 x 10-27kg 8_31 JK-t mol-t 6.02 x 1023rnol-1 k: 1.38x10-23JK-1 G- 6.67 x g- 9.81ms-2 Name : uniformly accelerated motion work done on/by a gas gravitational potential hydrostatic pressure pressure of an ideal gas simple harmonic motion velocity of particle in s.h.m. Doppler effect electric potential capacitors in series capacitors in parallel energy of charged capacitor electric current resistors in senes resistors in parallel Hall voltage alternating current}voltage radioactive decay decay constant Enquiries : [email protected] O 2020 DATE : = utGat2 s v2 — u2+2as v = vo cos fsv lic - - I/CI * _ c -Gl+G2+. I = Anvq R-R4R2 H — ntq x = xo sin cot x xo Page 1118
  2. Question 1 (a) (b) State, by reference to displacement, what is meant by simple harmonic motion. A mass is undergoing oscillations in a vertical plane. The variation with displacement x of the acceleration a of the mass is shown in Fig. 3 1 Fig. 3.1 State two reasons why the motion of the mass is not simple harmonic. [2] Enquiries : [email protected] O P age 2 | 18
  3. (c) A block of wood is floating in a liquid, as shown in Fig. 3.2 oscillation block of block — — — — — — — — — — —liquid— Fig. 3.2 The block is displaced vertically and then released. The variation with time t Of the displacement y Of the block from its equilibrium shown in Fig. 3.3. 2.0 position is y/ cm 1.5 0.5 -1.5 -2.0 Fig. 3.3 Use data from Fig. 3.3 to determine (i) the angular frequency of the oscillations, Enquiries : i.fauziah@gmaiLcom (O -rads-I Page 3118
  4. — ms-2 (2] (iii) the maximum vertical acceleration of the block. maximum acceleration = . The bock has mass 120g. The oscillations of the block are damped. Calculate the loss in energy of the oscillations of the block during the first three complete periods of its oscillations. energy loss = Enquiries : [email protected] O [Total: 11] Page 4118
  5. Question 2 A metal block hangs vertically from one end of a spring. The Otler end of the spring is tied to a thread that passes over a pulley and is attached to a vibrator, as shown in Fig. 4.1. (a) pulley spring block Fig. 4.1 The vibrator is switched off. vibrator The metal block of mass 120g is displaced vertically and then released. The variation with time t of the displacement y of the block from its equilibrium position is shown in Fig. 4.2. y/ cm 3 2 1.0 t/s Fig. 4.2 For the vibrations of the block, calculate (i) the angular frequency co, Enquiries : [email protected] O rads-I (21 Page 5118
  6. energy of the vbrations. energy = The vibrator is now switched on. The frequency of vibration is varied from 0.7fto 1.3f where f is the frequency of vibration of the block in (a). For the block, complete Fig. 4.3 to show the variation with frequency of the amplitude of (b) (c) vibration. Label this line A. amplitude 0.71 Fig. 4.3 [3] 1.3f Some light feathers are now attached to the uock in (b) to increase air resistance. The frequency of vibration is once again varied from 0.7f to 1.31. The new amplitude of vibration is measured for each frequency. On Fig. 4.3, draw a line to show the variation with frequency of the amplitude of vibration. Label this line B. Enquiries : [email protected] O P age [2] [Total: 9) 61 18
  7. Question 3 To demonstrate simple harmonic motion, a student attaches a trolley to two similar stretched springs, as shown in Fig. 3.1 _ spnng The trolley has mass m of 810g. trolley Fig. 3.1 The trolley is displaced along the line Of the two springs and then released. The subsequent acceleration a of the trolley is given by the expression 2kx where the spring constant k for each of the springs is 64Nm-1 and x is the displacement of the trolley. (a) Show that the frequency of oscillation of the trolley is 20Hz_ (b) The maximum displacement of the trolley is 1 _6cm_ Calculate the maximum speed Of the trolley. Enquiries : [email protected] O ms-I [2] Page 7118
  8. (c) Themassofthe trolley is increased. The initial displacement ofthetrolley remains unchanged. Suggest the change, if any, that occurs in the frequency and in the maximum speed of the oscillations of the trolley. frequency: maximum speed: Enquiries : [email protected] O [Total: 7] Page 8118
  9. Question 4 A mass hangs vertically from a fixed point by means of a spring, as shown in Fig. 4 1 spring Fig. 4.1 The mass is displaced vertically and then released. The subsequent oscillations of the mass are simple harmonic. The variation with time t of the length t of the spring is shown in Fig. 4.2. Fig. 4.2 1 lcm 18 17 16 15 14 13 12 0.2 0.3 0.4 0.5 0.6 (a) Use Fig. 4.2 to (i) state two values of t at which the mass is moving downwards with maximum speed, ...s and Enquiries : [email protected] O P age 9 | 18
  10. (ii) (iii) determine, for these oscillations, the angular frequency o, show that the maximum speed of the mass is 0.42 ms—I. (b) the _ rads- variation Use data from fig. 4.2 and to sketch, on the axes Of Fig. 4.3, displacement x from the equilibrium position of the velocity v of the mass. 0.5 v/ ms-I 0.3 —0.2 with —0.5 Fig. 4.3 Enquiries : [email protected] C) x/ cm [Total: 81 Page 10 | 18
  11. —1.5 Enquiries : [email protected] O Question 5 A bar magnet Of mass 180g is suspended from the free end Of a spring, as illustrated in Fig. 2 1 magnet coil Fig. 2.1 The magnet hangs so that one pole is near the centre Of a coil of wire. The coil is connected in series with a resistor and a switch. The switch is open. The magnet is displaced vertically and then allowed to oscillate with one pole remaining inside the coil. The other pole remains outside the coil. At time t O, the magnet is oscillating freely as it passes through its equilibrium position At time t 3_Os, the switch in the circuit is closed. The variation with time t of the vertical displacement y of the magnet is shown in fig. 2.2. 2.0 1.5 Y/ cm 1.0 0.5 tis —0.5 —1.0 Fig. 2.2 11 | 18 P age
  12. time t 7.5s. Explain your working. Enquiries : [email protected] O (a) Determine, to two significant figures, the frequency of oscillation of the magnet. frequency = AE = _ Hz [21 (b) State whether the closing of the switch gives rise to light, heavy or critical damping. (c) Calculate the change in the energy AEof oscillation of the magnet between time t = 2.7 s and - J [61 [Total: 9] Page 12 | 18
  13. Question 6 Define the radian State, by reference to simple harmonic motion, what is meant by angular frequency. (b) A thin metal strip, clamped horizontally at one end, has a load of mass M attached to its free end, as shown in Fig. 3.1. clamp x oscillation of load metal strip load mass M Fig. 3.1 The metal strip bends, as shown in Fig. 3.1. When the free end of the strip is displaced vertically and then released, the mass oscillates in a vertical plane. Theory predicts that the variation of the acceleration a of the oscillating load with the displacement x from its equilibrium position is given by c ML3 where L is the effective length of the metal strip and c is a positive constant. (i) Explain how the expression shows that the load is undergoing simple harmonic motion. Enquiries : [email protected] O Page 13 | 18
  14. (ii) For a metal strip of length L — 65cm and a bad of mass M — 240g, the frequency of oscillation is 3_2Hz_ Calculate the constant c. Enquiries : [email protected] O kg m3s-2 [3] [Total: 8] Page 14 | 18
  15. Question 7 (a) Explain what is meant by the natural frequency of vibration of a system. (b) A block of metal is fixed to one end of a vertical spring. The other end of the spring is attached to an oscillator, as shown in Fig_ 4.1. oscillator spring metal block Fig. 4.1 The amplitude of oscillation of the oscillator is constant. The variation of the amplitude xo of the oscillations of the block with frequency f of the oscillations is shown in Fig. 4.2. xo Fig. 4.2 Enquiries : [email protected] O Page 15 | 18
  16. (i) (ii) Name the effect shown in Fig. 4_2_ State and explain whether the block is undergoing damped oscillations. (c) State one example in which the effect shown in Fig. 4.2 is useful. Enquiries : [email protected] O [Total: 51 Page 16 | 18
  17. Enquiries : [email protected] O Question 8 A small frictionless trolley is attached to a fixed point A by means of a spring. A second spring is used to attach the trolley to a variable frequency oscillator, as shown in Fig. 2.1. . s and time = _ . 17 | 18 trolley Fig. 2.1 Both springs remain extended within the limit of proportionality. variable frequency oscillator Initially, the oscillator is switched off. The trolley is displaced horizontally along the line joining the two springs and is then released. The variation with time t of the velocity v of the trolley is shown in Fig. 2.2. 0.3 v/ ms-I 0.2 0.1 0.2 0.4 0.6 0.8 1.0 1.2 (i) -0.2 -o. Fig. 2.2 Using Fig. 2.2, state two different times at which 1. 2. the displacement of the trolley is zero. . .......sandtime = — the acceleration in one direction is maximum. P age
  18. Determine the frequency of oscillation of the trolley. frequency = Hz [21 (iii) The variation with time Of the displacement Of the trolley is sinusoidal. The variation with time Of the velocity Of the trolley is also sinusoidal. State the phase difference between the displacement and the velocity. phase difference (b) The oscillator is now switched on. The amplitude Of vibration of the oscillator is constant. The frequency f Of vibration Of the oscillator is varied. The trolley is forced to oscillate by means Of vibrations Of the oscillator. The variation with f Of the amplitude ao Of the oscillations Of the trolley is shown in Fig. 2.3. Fig. 2.3 By reference to your answer in (a), state the approximate frequency at whZh the amplitude is maximum. frequency Hz [11 (c) The amplitude of the oscillations in (b) may be reduced without changing significantly the frequency at which the amplitude is a maximum. State how this may be done and give a reason for your answer. You may draw on Fig. 2.1 if you wish. Enquiries : [email protected] C) Page 18 | 18

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