Sound Waves Exercise - I

SOUND WAVES Page # 24 Exercise - I UNSOLVED PROBLEMS Q.1. Which of the following examples represent periodic motion ? (a) a swimmer completion one (return) trip from one bank of a river to the other and back. (b) a freely suspended bar magnet displaced from its N-S direction and released. (c) a hydrogen molecule rotating about its center of mass. (d) an arrow released from a bow. Q.2. Which of the following examples represent (nearly) simply harmonic motion and which represent periodic but not simple harmonic motion? (a) the rotation of earth about its axis. (b) motion of an oscillating mercury column in a UTube. (c) motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower most point. (d) general vibrations of a poly atomic molecule about its equilibrium position x x (a) (b) -3 -1 0 1 3 t(s) t(s) 2 7 10 13 t(s) x (d) -3 -2 -1 0 1 2 Q.6. Which of the following relationship between the acceleration a and the displacement x of a particle involve harmonic motion. (a)  =0.7x (b) =200x (C)  =10x (d)  = 100 x2 Q.7. The motion of a particle execution simple harmonic motion is described by the displacement function. x(t) = A cos (t + ) If the initial ( t =0) position of the particle is 1 cm and its initial velocity is  cm/s, what are its amplitude and initial phase angle ? The angular frequency of the particle is  s-1. If instead of the cosine function, we choose the sine function to describe the SHM : x = B sin ( t + ), what are the amplitude and initial phase of the particle with the above initial conditions. Q.8. A spring balance has a scale that reads from 0 to 50 kg. The length of the scale is 20 cm. A body suspended from this balance, when displaced and released oscillates with a period of 0.6 s. What is the weight of the body? (c) 1 Q.5. A particle is inlinear simple harmonic motin between tow points, A and B, 10 cm apart. Take the directin from A to B as the positive direction and give the signs of velocity, accelerating and force on the particle when it is (a) at the end A (b) at the ent B, (c) at the mid-point of AB going towards A. (d) at 2 cm away from b going towards A, (e) at 3 cm away from A going towards B, and (f) at 4 cm away from A going towards A. 3 t(s) Q.3. Figure 14,27 depicts four x-t plots for linear motion of a particle. Which of the plots represent periodic motion? What is the period of motion (in case of periodic motion )? Q.4. Which of the following function of time represent (a) harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion ? Give period for each case of periodic mouton :( is any positive constant). (a) sint–cost (b)sin3 t (c)3cos(4–2t) (d) cost+ cos3t+ cos 5t (e)exp (–2t2) (f) 1 + t + 2t2 Q.9. A spring having with a spring constant 1200 N m-1 is mounted on a horizontal table as shown in fig. 14.28. A mass of 3 kg is attached to the free end of the spring. The masses then pulled sideways to a distance of 2.0 cm and released. Determine (i) the frequency of oscillations, (ii) maximum acceleration of the mass and (iii) the maximum speed of the mass. Q.10. In Exercise 14.9, let us take the position of mass when the spring is un stretched as x = 0, and the direction from left to right as the positive direction of x-axis. Give x as a function of time t for the SOUND WAVES Page # 25 oscillating mass if at the moment we start the stopwatch (t = 0 ), the mass is (a) at the mean position (b) at the maximum stretched position, and (c) at the maximum compressed position. In what way do these function for SHM differ from each other, in frequency, in amplitude or the initial phase? Q.11. Figures 14.29 correspond to two circular motions. The radius of the circle, the period of revolution, the initial position and the sense of revolution (i.e. clockwise or anti- clockwise) are indicated on each figure. Obtain the corresponding simple harmonic motions of the x-projection of the radius vector of the revolving particle P, in each case. Q.12. Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial ( t = 0 ) position of the particle, the radius of the circle, and the angular speed of the rotating particle. For simplicity, the sense of rotation may be fixed to be anticlockwise in every case: (x is in cm and t is in s). (a) x = –2 sin (3 t + / 3) (b) x = cos (/ 6 – t) (c) x = 3 sin ( 2t + / 3) (d) x = 2 cos t Q.13 Figure (i) shows a spring of force constant k clamped rigidly at one end and a mass m attached to its free end. A force F applied at the free end stretches the spring. Figure (ii) shows the same spring with both ends free and attached to a mass m at either end. Each end of the spring in figure (ii) is stretched by the same force F. m k F (i) m F k m F (ii) (a) What is the maximum extension of the spring in the two cases? (b) If the mass in figure (i) and the two masses in Figure (ii) are released free, what is the period of oscillation in each case? Q.14. Figure 14.31 shows four different spring arrangement. If the mass in each arrangement is displaced form its equilitbrium position and released , what is the resulting frequency of vibration in each case? Neglect the mass of the spring. [Figs. (a) and (b) represent an arrangement of springs in parallel, and (c) and (d) represent ‘springs in series’ ] k1 m k1 k2 k2 (a) (c) k1 k1 k2 (c) k2 (d) m m Q.15. A tray or mass 12 kg. is supported by two indentical springs as shown in Fig. 14.32. When the tray is pressed down pressed down slightly and then released, it executes SHM with a time period of 1.5 s. What is the spring constant of each spring ? When a block of mass m is placed on the tray, the period of SHM changes to 3.0 s. What is the mass of the block m 12kg Q.16. The piston in the cylinder head of a locomotive has a stroke ( twice the amplitude of 1.0 m. If the piston moves with simple harmonic motion with angular frequency of 200 rev/ min, what its maximum speed? Q.17. The acceleration due to gravity on the surface of moon is 1.7 m s-2. What is the time period of a simple pendulum on the surface of moon if its time period on the surface of earth is 3.5 s ? (g on the surface of earth is 9.8 m s-2 ) Q.18. Answer the following questions: (a)Time period of a particle in SHM depends on the force constant k and mass m of the pariticle: T= 2 m . A simple pendulum k executes SHM approximately. Why then is the time period of a pendulum independent of the mass of the pendulum? SOUND WAVES Page # 26 (b) The motin of a simple pendulum is approximately simple harmonic for small angle oscillations. For larger angles of oscillation, a more involved analysis shows that T is greater than 2 l g  Think of a qualitative argument to appreciate this result. (c) A man with a wristwatch on his hand falls from the top of a tower. Does the watch give correct time during the free fall? (d) What is the frequency of osculation of a simple pendulum mounted in a cabin that is freely falling under gravity? Q.19. A simple pendulum of lenght l and having a bob of mass m is suspended in a car. The car is moving on a circular track of radius R with a uniform speed If the pendulum makes small oscillations in a radial direction about its equilibrium position, what will be its time period ? Q.20. A cylindrical piece of cork of base area A and height h floats in a liquid of density  the cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with h a period T = 2  g where  is the density of cork. ( 1 Ignore damping due to viscosity of the liquid). Q.21. A trolley of mass 3.0 kg. as shown in Fig. 14.33, is connected to two springs, each of spring constant 600 n m-1If the trolley is displaced from its euilibrium position by 5.0 cm and released, what is (a) the period of ensuing oscillations, and (b) the maximum sped of the trolley ? How much energy is dissipated as heat by the time the trolley comes to rest due to damping forces ? Q.22. One end of a U-tube containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is maintained between the two column. Show that, when the suctinpump is removed, the column of mercury in the U-tube executes simple harmonic motion . Q.23 An air chamber of volume V has a neck area of cross section a into which a ball of mass m can move up and down without any friction (figure). Show that when the ball is pressed down a little and released, it executes SHM. Obtain an expression for the time period of oscillations assuming pressurevolume variations of air to be isothermal (see figure). m a V Air Q.24. You are riding in an automobile of mass 3000 kg. Assuming that you are examining the oscillation characteristics of its suspension system. The amplitude of oscillation 15 cm when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by 50% during one complete oscillation . Estimate the values of (a) the spring constant k and (b) the damping constant b for the spring and shock absorber system of one wheel, assuming that each wheel supports 750 kg. Q.25. A 1500 kg car carrying four 75 kg people move over a ‘washboard’ dirt road with corrugating 4.0 m apart, which cause the car to bounce on its spring suspension. The car bounces with maximum amplitude when its speed is 20 km h-1 . The car now stops and the four people get out. By how much the car body rise on its suspension owing to this decrease in mass Q.26. Show that for a particle inlinear SHM the average kinetic energy over a period of oscillation equals the average potential energy over the same period. Q.27. A circular disc of mass 10 kg is suspended by a wire attached to its centre. The wire is twisted by rotating the disc and released. the period of torsional oscillations is found to be 1.5 s. The radius of the disc is 15 cm. Determine the torsional spring constant of the wire. (Torsional spring constat  is defined by the relation j = –, where J is the restoring couple and the angle of twist).