PHY 115
Wave and Oscillation
Syllabus:
• Simple Harmonic Motion
• Different types pf Pendulum
• Spring-mass System
• Composition of two SHM
• Damped Harmonic Motion
• Wave Motion
• Forced Harmonic Motion
• Progressive and Stationary waves
Books:
• Waves and oscillation
• Physics for engineering
• Fundamental of Physics
(1) Oscillation is the to and fro motion of the particle about their mean position.
(2) Wave is the disturbance that transfers energy from one place to another.
(3) When the frequency of oscillation is high and amplitude is low is called vibration.
Simple Harmonic Motion (SHM)
Hook's Law:
F∝−xF∝−x
It is called SHM because the power of x is 1. It is a linear equation.
Differential equation for SHM:
From Hook's Law,
F∝−x⟹F=−kx | [k=spring constant]⟹ma=−kx⟹md2xdt2=−kx⟹md2xdt2+kx=0⟹d2xdt2+kmx=0 | [ω2=km]⟹d2xdt2+ω2x=0
The solution of differential equation:
d2xdt2+ω2x=0 Multiply eq. (1) by 2dxdt,2dxdtd2xdt2+ω22dxdtx=0
Integrating with respect to time,
(dxdy2)+ω2x2=CWhen, x=A, dxdt=0 From eq (2), 02+ω2A2=C⟹C=ω2A2
By substituting the value of C,
(dxdt)2+ω2x2=ω2A2⟹(dxdt)2=ω2(A2−x2)⟹dxdt=± ω√A2−x2⟹dx√A2−x2=ω dt
By integrating,
sin−1xA=ωt+δ1⟹xA=sin(ωt+δ1)⟹x=Asin(ωt+δ1)
From equation (3),
−dx√A2−x2=ω dt
By integrating,
cos−1xA=ωt+δ2⟹xA=cos(ωt+δ2)∴ x=Acos(ωt+δ2)
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