Chapter 9: Simple Harmonic Motion (SHM)

University:
California State University, Northridge
Course:
PHYS 100A | General Physics I
Academic year:

2023
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KOLEJ MATRIKULASI SELANGOR physics\akademik\1PN\SF016\ch9 9 Simple Harmonic Motion (SHM) IDEA/ Examples of SHM system: simple pendulum, vertical and 7. Information: (i) Maximum velocity, vmax = A (ii) Maximum displacement/ Amplitude = A horizontal spring oscillations 8.  SHM is about a system that doing a periodic motion/ oscillations WHAT? •The back & forth motion of an object about a fixed point (equilibrium position) WHY? •There is a force, restoring force FS which causes simple harmonic motion to occur •This force is proportional to the displacement from equilibrium & always directed towards equilibrium Oscillation/ Periodic Motion 9.1 Simple Harmonic Motion (SHM) 1. A periodic motion without loss of energy in which the acceleration of a body is directly proportional to its displacement from the equilibrium position (fixed point) and is directed towards the equilibrium position but in opposite direction of the displacement 2. 3. 4. 𝑑2 𝑥 𝑎 = −2 𝑥 = 2 𝑑𝑡 Amplitude, A = maximum magnitude of the displacement from the equilibrium position Period, T = time to make one complete oscillation Frequency, f = number of cycles in one second 1 1 2 𝑇 =  f = , = = 2𝑓 𝑓 𝑇 𝑇 9.2 Kinematics of SHM 5. Solution for SHM 𝑑2 𝑥 𝑎 = 2 = −2 𝑥 𝑑𝑡 x = A sin (t + ) 𝑑𝑥 = 𝐴 cos(𝑡 + ) 𝑑𝑡 𝑑2 𝑥 𝑎 = 2 = −2 𝐴 sin(𝑡 + ) 𝑑𝑡 𝑣= In term of t x = A cos (t + ) 𝑑𝑥 𝑣= = −𝐴 sin(𝑡 + ) 𝑑𝑡 𝑑2 𝑥 𝑎 = 2 = −2 𝐴 cos(𝑡 + ) 𝑑𝑡 In term of x 𝑣 = ±√𝐴2 − 𝑥 2 , 𝑣𝑚𝑎𝑥 = 𝐴 𝑎 = −2 𝑥, 𝑎𝑚𝑎𝑥 = 𝐴 9.3 Graphs of SHM 6. Displacement-time graph (x-t), Velocity-time graph (v-t) & , Acceleration-time graph (a-t) Velocity-displacement graph (v-x) 9. Acceleration-displacement graph (a-x) Information: (i) Maximum acceleration, amax = A2 (ii) Maximum displacement/ Amplitude = A (iii) Gradient of the graph = – 2 Energy-displacement graph (E-x) 1 1 1 Potential energy, 𝑈 = k𝑥 2 = 𝑚2 𝑥 2 = 𝑚2 𝐴2 𝑠𝑖𝑛2 (𝑡 + ) 2 2 2 1 1 1 Kinetic energy, 𝐾 = 𝑚𝑣 2 𝑚2 (𝐴2 − 𝑥 2 ) = 𝑚2 𝐴2 𝑐𝑜𝑠 2 (𝑡 + ) 2 2 2 1 1 2 2 𝑇𝑜𝑡𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦, 𝐸 = 𝐾 + 𝑈 = 𝑚 𝐴 = 𝑘𝐴2 2 2 9.4 Period of SHM