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Simple Harmonic Motion

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Application of Differential Equation in SHM

Application of Differential Equation in SHM

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  • 1. Seminar Report on Applications of Differential Equationsin Simple Harmonic Motion Presented by- Pratheek Manjunath 2nd Semester, EEE RVCE, Bangalore INTRODUCTION The Linear Differential Equation with constant coefficients find their most important applications in the study of electrical, mechanical and other linear systems, especially oscillatory systems.
  • 2. What is Simple Harmonic Motion? When the acceleration of a particle is proportional to tis displacement from a fixed point and is always directed towards it, then the motion is said to be Simple Harmonic. Case 1: Spring-Mass Systems Consider a mass mis attached to the free endof a mass-less spring. x=0 is the mean position; B depicts compression and C, elongation by a distance ‘x’. Let the maximum displacement be ‘A’-Amplitude The spring force is given by: F(x)= -kx where ‘k’ is the Spring constant. By force balance, -kx=ma md2 x + kx = 0 dt2 d2 x+ µ2 x=0 where µ2 = k dt2 m
  • 3. (D2 +µ2 )x=0 Solution of the differential equation is x=c1cos µt + c2sin µt Velocity ‘v’= dx = µ(-c1sinµt+c2cosµt) dt at t=0, x=a => c1=A at t=0, v=0 =>c2=0 Therefore x=A cos µt From the above equation of motion, we can find the following: Time period of oscillation T=2π = 2π√(m) µk frequency = T-1 = 1√(k) 2π√(m) Maximum velocity = Aµ Maximum acceleration= Aµ2