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

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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. example: the height of the lecture room
demo
PHY 231

12. 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

13. 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

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

15. 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

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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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