OSCILLATIONS.pptx
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This document discusses oscillations, including different types of motion like simple harmonic motion. It covers topics such as periodic motion, terminologies, linear and angular simple harmonic motion, energies in SHM, examples of SHM like a spring and pendulum, and equations of motion. It also discusses free oscillations, damped oscillations, and forced oscillations, specifically resonance. Forces, displacements, velocities, and accelerations of SHM are examined. Combinations of springs and their effective spring constants are analyzed.
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OSCILLATIONS.pptx
- 1. OSCILLATIONS
- 2. Overview - Types of motion - Periodic motion - Terminologies in Oscillations - Simple Harmonic Motion - Linear SHM Oscillations due to a spring - Combination of springs - Angular SHM - Energies in SHM- Kinetic, Potential - Examples of SHM Oscillations due to spring, Simple pendulum, Oscillation of liquid in a U-tube - Free Oscillation - Damped Oscillation - Forced Oscillation Resonance 2
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- 21. Equations of Periodic motion 21
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- 47. Combination of springs Calculation of Effective spring constant (Ks or Kp) 47 Springs connected in series Total displacement
- 48. Combination of springs Calculation of Effective spring constant (Ks or Kp) 48 Springs connected in Parallel
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- 50. Projection of uniform Circular motion on a diameter of SHM 50
- 51. Displacement, Velocity and Acceleration of SHM Projection on Y-axis Displacement (y) Velocity (V) 51
- 52. Displacement, Velocity and Acceleration of SHM Projection on Y-axis Velocity (V) Accleration (a) 52
- 53. Displacement, Velocity and Acceleration of SHM Projection on Y-axis At mean position T=0 , or t =0属 53
- 54. Displacement, Velocity and Acceleration of SHM Projection on Y- axis At mean position T=T/4 , or t =/2 54
- 55. Displacement, Velocity and Acceleration of SHM Projection on Y- axis At mean position T=2T/4 or T/2 , or t = 55
- 56. Displacement, Velocity and Acceleration of SHM Projection on Y- axis At mean position T=3T/4, or t = 3/2 56
- 57. Displacement, Velocity and Acceleration of SHM Projection on Y- axis At mean position T=4T/4 or T , or t =2 57
- 58. Variation of Displacement, Velocity and Acceleration at different instant of time Projection on y-axis 58
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- 63. Displacement, Velocity and Acceleration at different instant of time Projection on x-axis 63
- 64. Variation of Displacement, Velocity and Acceleration at different instant of time Projection on x-axis 64
- 65. Other examples of Linear SHM Simple harmonic Motion occurs when a particle or object moves back and forth within a stable equilibrium position under the influence of a restoring force proportional to its displacement. Spring Mass system Pendulum Swing Shock absorber of a car Strings of a musical instrument Bungee Jumping 65
- 66. Angular SHM 66
- 67. Angular SHM 67
- 68. Linear versus Angular SHM Linear Displacement Acceleration Mass Moment of Inertia Angular acceleration Angular Displacement Acceleration Angular acceleration Restoring Force Restoring Torque Angular frequency Angular frequency 68
- 69. POTENTIAL ENERGY IN SIMPLE HARMONIC MOTION- based on y-axis Potential energy U (t) POTENTIAL ENERGY IN SIMPLE HARMONIC MOTION- based on x-axis Potential energy U (t) Variation of potential energy (U(t)) 69
- 70. KINETIC ENERGY IN SIMPLE HARMONIC MOTION- based on y-axis Kinetic Energy KE (x) KINETIC ENERGY IN SIMPLE HARMONIC MOTION- based on x-axis Kinetic energy K (x) Variation of Kinetic energy (U(t)) 70
- 71. TOTAL ENERGY IN SIMPLE HARMONIC MOTION- based on y-axis Total Energy E TOTAL ENERGY IN SIMPLE HARMONIC MOTION- based on x-axis Total Energy E 71
- 72. Examples of Linear SHM Simple pendulum Considering the normal component, along the string 72 m is the mass of the bob l length f the string
- 73. Simple Pendulum (Contd..) Derivation of Time for oscillation based on restoring force 73 Considering the Tangential component which is equal to the restoring force
- 74. Simple Pendulum (Contd..) Derivation of Time for oscillation based on restoring force 74
- 75. Simple Pendulum (Contd..) Derivation of Time for oscillation based on restoring torque 75 Considering the tangential component , restoring torque is
- 76. Simple Pendulum (Contd..) Derivation of Time for oscillation based on Restoring Torque 76 Since sin =, when is small, then We know that From the above equations, We know that, moment of inertia of bob oscillating with massless string = 2 = 2= T = 2/ T=2
- 77. Oscillation of a liquid in U-tube Time period for oscillating about a mean position 77 Restoring force on the liquid = pressure x area of the cross-section of the tube Pressure on the liquid = 2 y g A Cross section area of glass tube Density of liquid L Total length of the liquid column in U-tube Restoring force on the liquid = - 2 y g x A Force due to mass of the liquid = mass of the liquid x acceleration Mass of the liquid = x volume = x A x l Force due to mass of the liquid = A l x a
- 78. Oscillation of a liquid in U-tube Time period for oscillating about a mean position 78 Restoring force on the liquid = Force due to mass of the liquid -2 y g x A = A L a -2 y g = L a a = 2 y g / L a = (2g/L)*y We know that, = 2 T = 2/ T=2 2
- 79. Free oscillations The free oscillation possesses constant amplitude and period without any external force to set the oscillation. Ideally, free oscillation does not undergo damping. But in all-natural systems damping is observed unless and until any constant external force is supplied to overcome damping. In such a system, the amplitude, frequency and energy all remain constant. 79
- 80. Damped Oscillation The damping is a resistance offered to the oscillation. The oscillation that fades with time is called damped oscillation. Due to damping, the amplitude of oscillation reduces with time. Reduction in amplitude is a result of energy loss from the system in overcoming external forces like friction or air resistance and other resistive forces. Thus, with the decrease in amplitude, the energy of the system also keeps decreasing. 80
- 81. Damped oscillation Derivation of Displacement and Mechanical energy 81 Damping force Restoring force Total force Applying Newtons second law,
- 82. Damped oscillation Derivation of Displacement and Mechanical energy 82 Expressing the eqn. in the form of PDEs, we get Where,
- 83. Damped oscillation Derivation of Displacement and Mechanical energy 83 Mechanical Energy of undamped oscillator E(t) = 遜 KA2
- 84. Forced Oscillations When a body oscillates by being influenced by an external periodic force, it is called forced oscillation. Here, the amplitude of oscillation, experiences damping but remains constant due to the external energy supplied to the system. For example, when you push someone on a swing, you have to keep periodically pushing them so that the swing doesnt reduce. 84
- 86. Forced Oscillations 86 Expressing the eqn. in the form of PDEs, we get
- 88. Forced Oscillation Small Damping, Driving Frequency far from Natural Frequency i.e. b=0 Driving Frequency Close to Natural frequency - Resonance 88
- 89. Forced Oscillation Example of simple pendulum Pendulum 1 and 4 have the same length When pendulum 4 oscillates, pendulum 1 also oscillates This happens because in this the condition for resonance is satisfied, i.e. the natural frequency of the system coincides with that of the driving force 89
- 90. Thank you 90