Class 38: Energy and Simple Harmonic Motion General equation of - PowerPoint PPT Presentation

Class 38: Energy and Simple Harmonic Motion

General equation of motion for simple harmonic motion If you can show the equation of motion of a particle is in the form: 2 d x   2 - x 2 dt Then it must be oscillating in simple harmonic form with the solution     x A cos ( t )

Horizontal Spring as Simple Harmonic Motion Equation of motion: F= -kx 2 2 d x d x Extension      kx m m kx 2 2 dt dt x Solution: x=0     x A cos ( t ) k   (natural frequency) L (natural length) m A and  are integration F= -kx Compression constants to be determined x by x and v at t=0. A is called amplitude and f =  /(2  ) is the frequency of the oscillation.

Pendulum as Simple Harmonic Motion Equation of motion: 2 2 d x d x g       mg sin m x 2 2 dt dt L Solution:      x A cos ( t ) L g   (natural frequency) L A and  are integration m constants to be determined x by x and v at t=0. x=0 A is called amplitude For small angle  , and f =  /(2  ) is the sin   tan    frequency of the and x  L  oscillation.

Vertical Spring Equation of motion: 2 2 d x d x      L (natural length) L (natural length) kx m m kx 2 2 dt dt Solution:     x A cos ( t ) k   (natural frequency) d m x=0 A and  are integration Equilibrium position constants to be determined by x and v at t=0. Use the equilibrium A is called amplitude and f =  /(2  ) is the position as the origin frequency of the oscillation.

Conservation of energy Conservation of energy: F= -kx Extension 1 1   2 2 mv kx constant x 2 2 x=0 v=0 at x=A: 1   2 K 0 and U kA max 2 L (natural length) U=0 at x=0: F= -kx Compression 1 1 x    2 2 K mv mv and U 0  x 0 max 2 2 1 1 1 1    2 2 2 2 mv kx kA mv  x 0 2 2 2 2

Simple Harmonic Motion ‐ Energy Simple harmonic motion is the oscillating interchange between the two kinds of mechanical energy: Kinetic energy 1  2 K mv 2 Potential energy 1   2 2 U m x 2