Vibrations and Waves

To this point, most of the forces we have considered have been constant forces. In this chapter we want to look at the behavior of objects that are experiencing non-constant forces. Specifically, a force that depends on the position of the object.

Oscillatory Motion

• Describing the motion
• equations of kinematics apply to constant acceleration
• non-constant forces



• Simple example
• constant force, but changes direction at the origin
• the position will follow a parabolic path as a function of time
• the velocity will be linear with time
• the acceleration will be constant over ½ interval
• the motion will repeat itself or be periodic
• Amplitude
• maximum distance from equilibrium
• Period
• time to complete one cycle
• Frequency
• number of cycles in a time interval





• Linear Force (Hooke's Law)
• Newton's Second Law
• F = ma
• 



• motion that occurs from a restoring force is called Simple Harmonic Motion (SHM)
• We can solve this equation to determine how x depends on t.
• We can get a qualitative picture by comparing this motion to circular motion
• SHM and Circular Motion - Video: Mechanical Universe, Vol 16, Ch. 26.
Flash Simulation
• the x-projection of an object undergoing circular motion mimics the motion of an object under the influence of a restoring force.



• 
• but for an object moving in a circle, q is equal to wt and thus we can write the position of an object undergoing SHM as
• 



• directly solving the equation:
  • need function that when we take the derivative twice we get the original function
  • 
  • a solution of the form will work if as would . We can also multiply by any constant
  • Most general solution is
  • 
  • Physical significance of the constants in the above equations
  • the maximum value of x will occur when cos or sin equals one.
  • at this point x = A. A is the amplitude.
  • let t increase by 2 p/w
  • 
  • But adding 2p to the argument of the cosine does not change the value.
  
  
  
  • 
  
  
  
  • T = period of the motion
  • f = n = frequency
  
  
  
  • 
  
  
  
  • We can also determine the velocity and acceleration as a function of time
  • ; v_max =wA
  • ; a_max = w² A