PHY PHYSICS 231 Lecture 31: Oscillations & Waves Remco Zegers Question hours:Tue 4:00-5:00 Helproom Period T Frequency f 1/6 1/3 ½  (m/k) 6/(2  ) 3/(2  ) 2/(2  )  (2  )/6 (2  )/3 (2  )/2

PHY Harmonic oscillations vs circular motion t=0 t=1t=2 t=3 t=4 v 0 =  r=  A v0v0 =t=t =t=t A  v0v0 vxvx 

PHY time (s) A -A -kA/m kA/m velocity v a x A  (k/m) -A  (k/m) x harmonic (t)=Acos(  t) v harmonic (t)=-  Asin(  t) a harmonic (t)=-  2 Acos(  t)  =2  f=2  /T=  (k/m)

PHY Another simple harmonic oscillation: the pendulum Restoring force: F=-mgsin  The force pushes the mass m back to the central position. sin  if  is small (<15 0 ) radians!!! F=-mg  also  =s/L (  tan  =s/L) so: F=-(mg/L)s

PHY pendulum vs spring parameterspringpendulum restoring force F F=-kxF=-(mg/L)s period TT=2  (m/k) T=2  (L/g) * frequency ff=  (k/m)/(2  )f=  (g/L)/(2  ) angular frequency  =  (k/m)  =  (g/L) *

PHY example: a pendulum clock The machinery in a pendulum clock is kept in motion by the swinging pendulum. Does the clock run faster, at the same speed, or slower if: a)The mass is hung higher b)The mass is replaced by a heavier mass c)The clock is brought to the moon d)The clock is put in an upward accelerating elevator? LL mm moonelevator faster same slower

PHY example: the height of the lecture room demo

PHY damped oscillations In real life, almost all oscillations eventually stop due to frictional forces. The oscillation is damped. We can also damp the oscillation on purpose.

PHY Types of damping No damping sine curve Under damping sine curve with decreasing amplitude Critical damping Only one oscillations Over damping Never goes through zero

PHY Waves The wave carries the disturbance, but not the water Each point makes a simple harmonic vertical oscillation position x position y

PHY Types of waves Transversal: movement is perpendicular to the wave motion wave oscillation Longitudinal: movement is in the direction of the wave motion oscillation

PHY A single pulse velocity v time t o time t 1 x0x0 x1x1 v=(x 1 -x 0 )/(t 1 -t 0 )

PHY describing a traveling wave While the wave has traveled one wavelength, each point on the rope has made one period of oscillation. v=  x/  t= /T= f : wavelength distance between two maxima.

PHY example 2m A traveling wave is seen to have a horizontal distance of 2m between a maximum and the nearest minimum and vertical height of 2m. If it moves with 1m/s, what is its: a)amplitude b)period c)frequency 2m

PHY sea waves An anchored fishing boat is going up and down with the waves. It reaches a maximum height every 5 seconds and a person on the boat sees that while reaching a maximum, the previous wave has moved about 40 m away from the boat. What is the speed of the traveling waves?

PHY Speed of waves on a string F tension in the string  mass of the string per unit length (meter) example: violin LM screw tension T v= /T= f=  (F/  ) so f=(1/ )  (F/  ) for fixed wavelength the frequency will go up (higher tone) if the tension is increased.

PHY example A wave is traveling through the wire with v=24 m/s when the suspended mass M is 3.0 kg. a)What is the mass per unit length? b)What is v if M=2.0 kg?

PHY bonus ;-) The block P carries out a simple harmonic motion with f=1.5Hz Block B rests on it and the surface has a coefficient of static friction  s =0.60. For what amplitude of the motion does block B slip?