1. Chapter 14
Periodic Motion
© 2016 Pearson Education Inc.

2. Learning Goals for Chapter 14
Describe oscillations in terms of amplitude, period, frequency, & angular frequency.
Apply simple harmonic motion to different situations.
Analyze motions of a pendulum.
Determines how rapidly oscillations die out.
Explore how driving force applied to oscillators at a particular frequency can cause very large responses (resonance).

3. Introduction
Why do dogs walk faster than humans? Does it have anything to do with the characteristics of their legs?
Many kinds of motion (such as a pendulum, musical vibrations, and pistons in car engines) repeat themselves. We call such behavior periodic motion or oscillation.

4. What causes periodic motion?
Displace body attached to spring from equilibrium position
Spring exerts a (variable!) restoring force F= kx, which acts to (continuously) restore object to equilibrium position.
Force causes oscillation, or periodic motion.

5. What causes periodic motion?
Restoring force accelerates glider back towards equilibrium position!

6. What causes periodic motion?

7. What causes periodic motion?
Remember! zero acceleration  zero velocity!!!!
The glider can still be *moving* at equilibrium

8. What causes periodic motion?
Glider “overshoots” equilibrium, slowing down as the Spring now pushes in the opposite direction, until it stops

9. Characteristics of periodic motion
Amplitude A = maximum +/- displacement from equilibrium.
Units: distance (m, cm, mm, etc.)
Period T = time for one cycle. (sec)
Frequency f = # cycles per unit time. (Hertz = Hz or sec-1)
Frequency & period are reciprocals: f = 1/T & T = 1/f.
Angular frequency w = 2π times frequency:
Units: radians/sec

10. Simple harmonic motion (SHM)
IF restoring force is directly proportional to displacement from equilibrium…
F = -kx
Resulting motion is simple harmonic motion (SHM).

11. Simple harmonic motion (SHM)
IF restoring force is directly proportional to displacement from equilibrium…
Resulting motion is simple harmonic motion (SHM).

12. Simple harmonic motion (SHM)
In many systems restoring force IS ~ proportional to displacement if displacement is sufficiently small.
IF amplitude is small enough, oscillations modeled with SHM.

13. Simple harmonic motion viewed as a projection

14. Simple harmonic motion viewed as a projection
Projected circle for ball’s motion is reference circle.
Q moves around reference circle @ constant angular speed!
Vector OQ rotates w/ same angular speed.
Rotating vector is called a phasor.

15. Simple harmonic motion viewed as a projection
Projected circle for ball’s motion is reference circle.
Q moves around reference circle @ constant angular speed!
Vector OQ rotates w/ same angular speed.
Rotating vector is called a phasor.

16. Simple harmonic motion viewed as a projection
Projected circle for ball’s motion is reference circle.
Q moves around reference circle @ constant angular speed!
Vector OQ rotates w/ same angular speed.
Rotating vector is called a phasor.

17. Characteristics of SHM
Body mass m vibrating from ideal spring w/ force constant k:

18. Characteristics of SHM
Body mass m vibrating from ideal spring w/ force constant k:

19. Characteristics of SHM
Body mass m vibrating from ideal spring w/ force constant k:

20. Characteristics of SHM
As mass m in tuning fork’s tines increases…
frequency of oscillation decreases…
pitch of sound that tuning fork produces LOWERS

21. Displacement as a function of time in SHM
Displacement as function of time for SHM is:

22. Displacement as a function of time in SHM
Increasing mass with same Amplitude & spring constant k increases period of displacement vs time graph.

23. Displacement as a function of time in SHM
Increasing k with same A & m decreases period of displacement vs time graph.

24. Displacement as a function of time in SHM
Increasing A with same m and k DOES NOT CHANGE period of displacement vs time graph.

25. Displacement as a function of time in SHM
Increasing phase delay ϕ with same A, m, & k only shifts displacement vs time graph to left.

26. Graphs of displacement and velocity for SHM

27. Graphs of displacement and acceleration for SHM

28. E = 1/2 mvx2 + 1/2 kx2 = 1/2 kA2 = constant
Energy in SHM
Total mechanical energy E = K + U conserved in SHM:
E = 1/2 mvx2 + 1/2 kx2 = 1/2 kA2 = constant

29. Energy diagrams for SHM
Potential energy U & total mechanical energy E for a body in SHM as a function of displacement x.

30. Energy diagrams for SHM
Potential energy U, kinetic energy K, & total mechanical energy E for a body in SHM as a function of displacement x.

31. Vertical SHM
If a body oscillates vertically from a spring, restoring force has magnitude kx. Therefore vertical motion is SHM.

32. Vertical SHM
If a body oscillates vertically from a spring, restoring force has magnitude kx. Therefore vertical motion is SHM.

33. Vertical SHM
If a body oscillates vertically from a spring, restoring force has magnitude kx. Therefore vertical motion is SHM.

34. Vertical SHM
If weight mg compresses spring a distance Δl, force constant k = mg/Δl .

35. Angular SHM
Coil springs exert restoring torque is called torsion constant of spring.
Result is angular simple harmonic motion.

36. Potential energy of a two atom system

37. Vibrations of molecules
Two atoms with centers a distance r apart, with equilibrium point at r = R0.
If displaced small distance x from equilibrium…
Restoring force ~
Fr = –(72U0/R02)x
k = 72U0/R02, motion is SHM.

38. The simple pendulum
Simple pendulum consists of point mass (“bob”) suspended by “massless, unstretchable” string.
If pendulum swings with a small amplitude vertically, motion is simple harmonic.

39. The physical pendulum
Physical pendulum is any real pendulum with actual “extended” body (instead of a point-mass bob)
For small amplitudes, its motion is simple harmonic.

40. Tyrannosaurus rex and the physical pendulum
We can model the leg of Tyrannosaurus rex as a physical pendulum.

41. Damped oscillations
Real-world systems have dissipative forces that will decrease amplitude over time.
Decrease in amplitude is called damping
Motion is called damped oscillation.

42. Forced oscillations and resonance
Damped oscillator left to itself will eventually stop moving.
Maintain constant-amplitude oscillation by applying a force that varies with time in a periodic way.
Additional force called a driving force.
Apply periodic driving force with angular frequency ωd to a damped harmonic oscillator, motion that results is called a forced oscillation or a driven oscillation.

43. Forced oscillations and resonance
Start with Newton’s 2nd Law:
F = ma = -kx (without damping)
Add damping term proportional to velocity:
F = ma = -kx – cv
This is a differential equation:
ma + kx + cv = 0
𝑚 𝑥 +𝑐 𝑥 +𝑘𝑥=0
                                                                                                                                                                                                                      
Damped Harmonic Oscillator

44. Forced oscillations and resonance
Solution:
Rewrite as: where
Damped Harmonic Oscillator

45. Forced oscillations and resonance
Solution:
Rewrite and solve as a quadratic!:
See Hyperphysics for more information

46. Underdamped Case: General Solutions: 𝑥 𝑡 =𝐴 𝑒 −𝑏𝑡/2𝑚 cos⁡( 𝜔 ′ 𝑡+𝜑)
𝑤ℎ𝑒𝑟𝑒 𝜔 ′ = 𝑘 𝑚 − 𝑏 2 4 𝑚 2
Cases:
b = 0 (no damping, SHM)
k/m > b2/4m2 and roots are real
Lets put an egg on a spring!

47. Weighing DNA
It has recently become possible to "weigh" DNA molecules by measuring the influence of their mass on a nano-oscillator. The image shows a thin rectangular cantilever etched out of silicon (density 2300 kg/m3) with a small gold dot at the end. If pulled down and released, the end of the cantilever vibrates with simple harmonic motion, moving up and down like a diving board after a jump. When bathed with DNA molecules whose ends have been modified to bind with gold, one or more molecules may attach to the gold dot.

48. Weighing DNA
The addition of the DNA molecule’s mass causes a very slight - but measurable - decrease in the oscillation frequency. A vibrating cantilever of mass M can be modeled as a block of mass 1/3 M attached to a spring. (The factor of 1/3 arises from the moment of inertia of a bar pivoted at one end.)
Neither the mass nor the spring constant can be determined very accurately - perhaps to only two significant figures - but the oscillation frequency can be measured with very high precision simply by counting the oscillations.

49. Weighing DNA
In one experiment, the cantilever was initially vibrating at exactly 12 MHz. Attachment of a DNA molecule caused the frequency to decrease by 50 Hz. The dimensions of the silicon beam are 4000nm, 400nm and 100nm
What was the mass of the DNA?

50. Forced oscillations and resonance
Lady beetle flies by means of a forced oscillation.
Unlike wings of birds, insect’s wings are extensions of its exoskeleton.
Muscles attached to inside of exoskeleton apply a periodic driving force that deforms the exoskeleton rhythmically, causing attached wings to beat up and down.
Oscillation frequency of wings & exoskeleton is same as frequency of driving force.

51. Forced oscillations and resonance
MIT Example: Show that the system 𝑥 + 𝑥 +3 = 0 is underdamped, find its damped angular frequency; graph the solution with initial conditions
x(0)= 1, 𝑥 (0)= 0.
Solution. Characteristic equation is: s2 + s + 3 = 0 Characteristic roots: −1/2 ± i√11/2
Basic real solutions: e−t/2 cos(√11 t/2) or e−t/2 sin(√11 t/2). General solution: x(t)= e−t/2(c1 cos(√11 t/2)+c2 sin(√11 t/2))
x(t)= Ae−t/2 cos(√11 t/2 −φ).
Since the roots have a nonzero imaginary part, the system is underdamped.
The damped angular frequency is ωd = √11/2.

52. Forced oscillations and resonance
MIT Example: Show that the system 𝑥 + 𝑥 +3 = 0 is underdamped, find its damped angular frequency; graph the solution with initial conditions
x(0)= 1, 𝑥 (0)= 0.
The initial conditions are satisfied when c1 = 1 and c2 = 1/√11. So,
x(t)= e−t/2 cos(√11 t/2)+ (1/ √11) sin(√ 11 t/2)
Or x(t) = Ae−t/2 cos(√11 t/2 – f)
And f = tan-1 (1/ √11)