Important Notes - JEE - Physics - Simple Harmonic Motion

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Simple Harmonic Motion for JEE Main

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Important Notes - JEE - Physics - Simple Harmonic Motion

  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
  2. 2. 9011041155 / 9011031155 Oscillation (S.H.M) It’s a Periodic Motion It is a motion due to vibration
  3. 3. 9011041155 / 9011031155 Oscillatory Motion It is type of motion in which a body moves to and fro, tracing the same path again and again, in equal intervals of time. What is Simple Harmonic Motion (S.H.M.)? Simple harmonic motion is periodic motion produced by a restoring force that is directly proportional to the displacement and oppositely directed.
  4. 4. 9011041155 / 9011031155 type of S.H.M. 1) If the object is moving along a straight line path, it is called ‘linear simple harmonic motion’ (L.S.H.M.) In S.H.M., the force causing the motion is directly proportional to the displacement of the particle from the mean position and directed opposite to it. If x is the displacement of the particle from the mean position, and f is the force acting on it, then f α -x negative sign indicates direction of force opposite to that of displacement f = -kx k is called force per unit displacement or force constant. The units of k are N/m in M.K.S. and dyne/cm in 1 0 -2 C.G.S. Its dimensions are [M L T ]
  5. 5. 9011041155 / 9011031155 S.H.M. is projection of U.C.M. on any diameter ∠ DOP0 = α ∠ P0OP1 = ωt At this instant, its projection moves from O to M, such that distance OM is x. In right angled triangle OMP1, sin t X a
  6. 6. 9011041155 / 9011031155 ∴ x = a sin (ωt + α) This is the equation of displacement of the particle performing S.H.M. from mean position, in terms of maximum displacement a, time t and initial phase α. The time derivative of this displacement is velocity v ∴v= dx = aω cos(ωt + α) dt Time derivative of this velocity is acceleration. 2 d2 x ∴ accl = 2 = - aω sin (ωt + α) dt n But, a sin (ωt + α) = x, n 2 ∴ accl = - ω x
  7. 7. 9011041155 / 9011031155 The negative sigh indicates that the acceleration is always opposite to the displacement. When the displacement is away from the mean position, the acceleration is towards the mean position and vice versa. Also, its magnitude is directly proportional to the displacement. Hence, S.H.M. is also defined as, ‘ the type of linear periodic motion, in which the force (and acceleration) is always directed towards the mean position and is of the magnitude directly proportional to displacement of the particle from the mean position.’
  8. 8. 9011041155 / 9011031155 Q.1 In the equation F=-Kx, representing a S.H.M., the force constant K does not depends upon (a.) elasticity of the system (b) inertia of the system (c) extension or displacement of the system (d.) velocity of the system Q.2 The suspended mass makes 30 complete oscillations in 15 s. What is the period and frequency of the motion? a) 2s,0.5 Hz Q.3 b) 0.5s, 2Hz c) 2s,2Hz d)0.5s,0.5Hz A 4-kg mass suspended from a spring produces a displacement of 20 cm. What is the spring constant a) 196 N/m b) 500 N/m c) 100 N/m d) 80N/m
  9. 9. 9011041155 / 9011031155 Answers :1. (d.) velocity of the system 2. (b) 0.5s, 2Hz 15 s 0.50 s 30 cylces Period : T 0.500 s 1 1 f T 0.500 s Frequency : f 2.00 Hz T 3. a) 196 N/m F 4 kg (9.8 m / s2) F 39.2 k X 0.2 196 N / m 39.2 N
  10. 10. 9011041155 / 9011031155 Differential Equation of S.H.M. If x is the displacement of the particle performing S.H.M., d2 x accln = 2 , dt d2 x force = m 2 dt But f = - kx d2 x ∴ m 2 = - kx dt d2 x ∴ m 2 + kx = 0 ... (1) dt d2 x k ∴ 2 + x m dt 0 But, k m 2 2 d2 x ∴ + ω x = 0 ... (2) 2 dt These two equations are called differential equations of S.H.M. According to second equation,
  11. 11. 9011041155 / 9011031155 Force = mass × acceleration 2 ∴ f = - mω x but, f = - kx also 2 ∴ - kx = - mω x 2 ∴ k = mω ∴ k/m = ω 2
  12. 12. 9011041155 / 9011031155 Formula for velocity d2 x = - ω2x dt 2 d2 x dt 2 d dx dt dt d dx dv But, dt dt dt dv dv dx x dt dx dt dx v dt d2 x dv v dx dt 2 dv 2 v x dx But, Separating the variables, 2 v dv = - ω x dx integrating both sides, v dv 2 x dx
  13. 13. 9011041155 / 9011031155 2 2 x v2 ∴ =-ω 2 2 C Where C is constant of integration. at x = a, v = 0 ∴0= ∴C= v2 2 v2 v2 v 2 2 a 2 C 2 a2 2 2 x2 2 2 2 a 2 2 2 a 2 2 x2 a2 x 2 a2 x 2
  14. 14. 9011041155 / 9011031155 Formula for displacement dx dt But, v dx dt a2 x2 dx a 2 x dt 2 Integrating both sides, dx a 2 x ∴ sin-1 dt 2 x a t where α is constant of integration. It is the initial phase of motion. x a x sin a sin t t
  15. 15. 9011041155 / 9011031155 Special Cases 1. At t = 0, x = 0 0 = a sin α ∴ sin α = 0 ∴ α = 0 Thus, when the body starts moving from the mean position, the initial phase is zero. 2. At t = 0, x = a a = a sin α ∴ sin α = 1 ∴ α = π / 2 Thus, when the body starts moving from the extreme position, the initial phase angle is π / 2
  16. 16. 9011041155 / 9011031155  Ask Your Doubts  For inquiry and registration, call 9011041155 / 9011031155.

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