PHYSICS 231 Lecture 34: Oscillations & Waves Period T 6 3 2 Frequency f 1/6 1/3 ½ (m/k) 6/(2) 3/(2) 2/(2)  (2)/6 (2)/3 (2)/2 Remco Zegers Question hours:Monday 9:15-10:15 Helproom PHY 231

Harmonic oscillations vs circular motion v0=r=A =t =t t=3 t=4 v0  vx  A PHY 231

xharmonic(t)=Acos(t) time (s) -A =2f=2/T=(k/m) velocity v A(k/m) vharmonic(t)=-Asin(t) -A(k/m) kA/m a aharmonic(t)=-2Acos(t) -kA/m PHY 231

xharmonic(t)=Acos(t)=0.1cos(22.4t) Example A mass of 0.2 kg is attached to a spring with k=100 N/m. The spring is stretched over 0.1 m and released. What is the angular frequency () of the corresponding circular motion? What is the period (T) of the harmonic motion? What is the frequency (f)? What are the functions for x,v and t of the mass as a function of time? Make a sketch of these. =(k/m)= =(100/0.2)=22.4 rad/s =2/T T= 2/=0.28 s =2f f=/2=3.55 Hz (=1/T) xharmonic(t)=Acos(t)=0.1cos(22.4t) vharmonic(t)=-Asin(t)=-2.24sin(22.4t) aharmonic(t)=-2Acos(t)=-50.2cos(22.4t) PHY 231

0.1 x 0.28 0.56 time (s) -0.1 velocity v 2.24 0.28 0.56 -2.24 50.2 a 0.28 0.56 -50.2 PHY 231

question An object is attached on the lhs and rhs by a spring with the same spring constants and oscillating harmonically. Which of the following is NOT true? In the central position the velocity is maximal In the most lhs or rhs position, the magnitude of the acceleration is largest. the acceleration is always directed so that it counteracts the velocity in the absence of frictional forces, the object will oscillate forever the velocity is zero at the most lhs and rhs positions of the object PHY 231

An object is oscillating as shown in the figure. At which point quiz (extra credit) A E 0.1 B x D 0.28 0.56 time (s) -0.1 C An object is oscillating as shown in the figure. At which point is the velocity of the object largest? A) B) C) D) E) PHY 231

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 (<150) radians!!! F=-mg also =s/L (tan=s/L) so: F=-(mg/L)s PHY 231

pendulum vs spring * parameter spring pendulum restoring force F F=-kx F=-(mg/L)s period T T=2(m/k) T=2(L/g)* frequency f f=(k/m)/(2) f=(g/L)/(2) angular frequency =(k/m) =(g/L) * PHY 231

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: The mass is hung higher The mass is replaced by a heavier mass The clock is brought to the moon The clock is put in an upward accelerating elevator? L m moon elevator faster same slower PHY 231

example: the height of the lecture room demo PHY 231

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 231

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 231

Waves The wave carries the disturbance, but not the water position y position x Each point makes a simple harmonic vertical oscillation PHY 231

Types of waves wave oscillation Transversal: movement is perpendicular to the wave motion oscillation Longitudinal: movement is in the direction of the wave motion PHY 231

A single pulse velocity v time to time t1 x0 x1 v=(x1-x0)/(t1-t0) PHY 231

describing a traveling wave : wavelength distance between two maxima. While the wave has traveled one wavelength, each point on the rope has made one period of oscillation. v=x/t=/T= f PHY 231

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: amplitude period frequency 2m amplitude: difference between maximum (or minimum) and the equilibrium position in the vertical direction (transversal!) A=2m/2=1m v=1m/s, =2*2m=4m T=/v=4/1=4s f=1/T=0.25 Hz PHY 231

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? Period: 5 seconds (time between reaching two maxima) Wavelength: 40 m v= /T=40/5=8 m/s PHY 231

Speed of waves on a string F tension in the string  mass of the string per unit length (meter) screw tension T example: violin L M v= /T= f=(F/) so f=(1/)(F/) for fixed wavelength the frequency will go up (higher tone) if the tension is increased. PHY 231

example A wave is traveling through the wire with v=24 m/s when the suspended mass M is 3.0 kg. What is the mass per unit length? What is v if M=2.0 kg? a) Tension F=mg=3*9.8=29.4 N v=(F/) so =F/v2=0.05 kg/m b) v=(F/)=(2*9.8/0.05)=19.8 m/s PHY 231

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? The block starts to slip if Ffriction<Fmovement sn-maP=0 smg=maP so sg=aP ap= -2Acos(t) so maximally 2A=(2f)2A sg=(2f)2A A= sg/(2f)2=0.066 m PHY 231