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:

x

• dt

x d

2 2 2

 

Then it must be oscillating in simple harmonic form with the solution

) t ( cos A x    

Horizontal Spring as Simple Harmonic Motion

L (natural length) x F= -kx Extension x=0 x F= -kx Compression

Solution: Equation of motion:

kx dt x d m dt x d m kx

2 2 2 2

     m k ) t ( cos A x      

(natural frequency)

A and  are integration constants to be determined by x and v at t=0. A is called amplitude and f = /(2) is the frequency of the

• scillation.

Pendulum as Simple Harmonic Motion

Solution: Equation of motion:

x L g dt x d dt x d m sin mg

2 2 2 2

      L g ) t ( cos A x      

(natural frequency)

A and  are integration constants to be determined by x and v at t=0. A is called amplitude and f = /(2) is the frequency of the

• scillation.

L m x x=0 

For small angle , sin  tan    and x  L

Vertical Spring

L (natural length)

Solution: Equation of motion:

kx dt x d m dt x d m kx

2 2 2 2

     m k ) t ( cos A x      

(natural frequency)

A and  are integration constants to be determined by x and v at t=0. A is called amplitude and f = /(2) is the frequency of the

• scillation.

Equilibrium position L (natural length) x=0 d

Use the equilibrium position as the origin

Conservation of energy

L (natural length) x F= -kx Extension x=0 x F= -kx Compression

v=0 at x=A: Conservation of energy:

constant kx 2 1 mv 2 1

2 2

 

2 max

kA 2 1 U and K  

U=0 at x=0:

U and mv 2 1 mv 2 1 K

2 max 2 x

  

 2 2 2 2

x

mv 2 1 kA 2 1 kx 2 1 mv 2 1



  

Simple Harmonic Motion ‐ Energy

Simple harmonic motion is the

• scillating interchange

between the two kinds of mechanical energy:

2 2 2

x m 2 1 U energy Potential mv 2 1 K energy Kinetic   