JEE Main 2014 Physics Syllabus - Stationary Waves


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JEE Main 2014 Physics Syllabus - Stationary Waves

  1. 1. 9011041155 / 9011031155 • Live Webinars (online lectures) with recordings. • Online Query Solving • Online MCQ tests with detailed solutions • Online Notes and Solved Exercises • Career Counseling 1
  2. 2. 9011041155 / 9011031155 Stationary Waves Stationary Waves The resultant waves formed due to superposition of two exactly identical progressive waves, having same amplitude, wavelength and speed, travelling in the same medium, along the same path, but in opposite directions are called stationary waves. Characteristics of Stationary waves 1. Stationary waves are formed due to superposition of two exactly identical waves travelling through the same medium, along the same path but, in opposite directions. 2. When stationary waves are set up in a medium, the particles at some points are permanently at rest (i.e. amplitude is zero). Such points are called 2
  3. 3. 9011041155 / 9011031155 ‘nodes’. The distance between two successive nodes is λ/2. 3. When stationary waves are set up in a medium, the particles at some points vibrate with maximum amplitude. Such points are called ‘antinodes’. Distance between two successive antinodes is λ /2. 4. Nodes and antinodes are alternately situated. The distance between any node and successive antinode is λ /4. 5. In stationary waves loops are formed between two successive nodes. All the particles in one loop are 3
  4. 4. 9011041155 / 9011031155 in the same phase, while the particles in the successive loop are out of phase by πc. 6. Stationary waves are periodic in time and space. 7. Stationary waves do not transfer energy through the medium. 8. In stationary waves, all the particles except those at nodes, vibrate with same period as that of the interfering waves. 9. The amplitude of vibration is different for different particles and that increases from node to antinode. 10. A loop is formed between two successive nodes 4
  5. 5. 9011041155 / 9011031155 because of two reasons. Firstly, the amplitude of vibration of the particles increases from node to antinode and decreases from antinode to node. Secondly, all the particles reach their maximum displacement at a time. 11. In case of stationary waves formed due to interference of longitudinal waves, displacement of the particle at the node is zero. But, pressure at the nodes changes between maximum to minimum. At antinodes, the displacement is maximum, but, the pressure remains constant. Hence, displacement nodes are called pressure antinodes and displacement antinodes are called pressure nodes. 5
  6. 6. 9011041155 / 9011031155 Equation of a Stationary Wave y 1 a sin 2 t x T and y a sin 2 2 t T i.e. y = y1 + y2 y a sin 2 t x T a sin 2 t T x We know that, sin A sin B 2 sin A B 2 2 t 2 x y 2 a sin cos T 2 x put, 2a cos A 2 t T But,1 / T n y A sin 2 n t y A sin 6 A B cos 2 x
  7. 7. 9011041155 / 9011031155 Hence, the resultant wave is also an S.H.M. with same period T. But, in this equation the term ± x/λ is absent. Thus, the resultant wave is not a progressive wave. It is travelling neither along + ve X - axis nor along - ve X-axis. Such a steady or localized wave is called Stationary Wave or Standing Wave. Transfer of energy is 0. Amplitude of the resultant wave varies between ± 2a 1. At nodes, i.e. A = 0 2 x 2 a cos 2 x x 2 0 , 3 ,5 ,....... 2 2 2 ,3 , 5 ,........ 2 2 7
  8. 8. 9011041155 / 9011031155 distance between two successive nodes is, 3 2 2 2 OR 5 3 2 2 x x 2 x 0, 0, 2 i.e. A = ± 2a 2. At antinodes, 2a cos 2 and so on 2a i.e. cos 2 x 1 , 2 ,3 , ...... , 2 ,3 ,....... 2 2 2 Distance between two successive antinodes is, 2 0 2 or 2 2 2 2 and so on Hence, distance between successive nodes and antinodes is 2 or the nodes and antinodes are equidistant. Also, the distance between consecutive node and antinode is λ / 4. 8
  9. 9. 9011041155 / 9011031155 M.C.Q. Q.1 The phase difference between the particles in successive loops of a stationary wave is (a.1) 90° Q.9 (b.1) 45° (c.1) 180° (d.1) zero Find the ‘wrong’ statement from the following: The equation of a stationary wave is given by , where y and x are in cm and t is in second. Then for the stationary wave, (a.9) Amplitude=3 cm (b.9) Wavelength=5cm (c.9) Frequency=20 Hz (d.9) Velocity=2 m/s Harmonics and Overtones The fundamental frequency along with its integral multiples are called harmonics. The fundamental frequency itself is the first than the harmonic. Vibrations fundamental of frequencies frequency which present are called overtones. 9 higher are actually
  10. 10. 9011041155 / 9011031155 Free Oscillations The body starts oscillating about its mean position, on its own. Such vibrations are called free oscillations. The frequency of vibrations of the body is called ‘natural frequency of vibrations’ of the material of the body. Examples 1. Simple pendulum 2. Tuning fork 10
  11. 11. 9011041155 / 9011031155 3. Stretched string 11
  12. 12. 9011041155 / 9011031155 Damped oscillations Amplitude of free oscillations decreases because in every oscillation a little amount of energy is utilized to overcome the air resistance. So, the oscillations are called ‘damped oscillations’. Forced oscillations the body is forced to oscillate with a new frequency. Hence, such oscillations are called ‘forced oscillations’. The oscillations in the body due to an external periodic force whose frequency is different from natural frequency of vibration of the body are called forced vibrations. e.g. 1. 12
  13. 13. 9011041155 / 9011031155 2. 13
  14. 14. 9011041155 / 9011031155 Resonance Amplitude of forced vibrations is inversely proportional to the difference between the natural frequency of vibrations of the body and the frequency of the external force (called forcing frequency or driving frequency). Thus, as the driving frequency approaches the natural frequency, the amplitude of vibration goes on increasing. Finally when the driving frequency matches exactly with the natural frequency, we get maximum amplitude of vibration. This effect is called resonance. Getting maximum amplitude of forced vibrations due to synchronization or matching between natural frequency and driving frequency is called resonance. 14
  15. 15. 9011041155 / 9011031155 e.g. 1. 2. 15
  16. 16. 9011041155 / 9011031155 3. 16
  17. 17. 9011041155 / 9011031155 Q.65 In the experiment of a simple pendulum, the oscillations of the simple pendulum are actually (a.65) Free oscillations (c.65) Damped harmonic oscillations (b.65) Forced oscillation (d.65) Resonant oscillations Q.66 When a regiment of soldiers have to cross a suspension bridge, they are ordered to (a.66) March in steps (b.66) Break the steps (c.66) Stand in attention (d.66) Stand at ease Q.67 In the case of forced vibrations, the resonance becomes very sharp, when the (a.67) Restoring force is very small (b.67) Applied periodic force is small (c.67) Damping force is small (d.67) Quality factor is small • Ask Your Doubts • For inquiry and registration, call 9011041155 / 9011031155. 17