1. Chapter 15 Periodic Motion
© 2015 Pearson Education, Inc.

2. Chapter 15: Periodic Motion
Chapter Goal: Study the kinematics and dynamics of periodic motion, i.e., motion that repeats itself at regular intervals.
© 2015 Pearson Education, Inc.

3. Section 15.1: Periodic motion and energy
Any motion that repeats itself at regular time intervals is called periodic motion.
The figure shows the periodic motion of a spring-cart system.
Hooke’s law: If a spring is stretched (or compressed) by a small distance x – x0 from its unstretched length x0, the x component of the force it exerts on the load is (Fby spring on load)x = –k(x – x0) where k is called the spring constant of the spring.
© 2015 Pearson Education, Inc.

4. Checkpoint 15.1
(a) List the forces exerted on the spring-cart system of Figure 15.1 right after it is released, and draw a free-body diagram for each object in the system. (b) Which of these forces do work on the system as it oscillates? (c) As the hand pushes on the cart and compresses the spring, is the work done by the cart on the spring positive, negative, or zero?
(d) How does the work done by the cart on the spring compare with the work done by the spring on the cart?
Fsc
© 2015 Pearson Education, Inc.

5. Section 15.1: Periodic motion and energy
The time interval it takes to complete a full cycle of the motion is the period T.
The inverse of the period is called the frequency: f = 1/T and has units of Hz = 1/sec.
The angular frequency is ω = 2πf
and has the same unit (s–1) as frequency.
The object’s maximum displacement from the equilibrium position is called the amplitude A.
© 2015 Pearson Education, Inc.

6. Section 15.1: Periodic motion and energy
In practice, periodic motion in mechanical systems will die out due to energy dissipation.
If we ignore these damping effects, we find that
Periodic motion is characterized by a continuous conversion between potential and kinetic energy in a closed system.
© 2015 Pearson Education, Inc.

7. Section 15.1: Periodic motion and energy
The figure shows examples of oscillating systems.
© 2015 Pearson Education, Inc.

8. Section 15.2: Simple harmonic motion
Simple harmonic motion is closely related to circular motion.
The figure shows the shadow of a ball projected onto a screen.
As the ball moves in circular motion with constant rotational speed , the shadow moves with simple harmonic motion.
The ball sweeps out at an angle  =  t in time t.
Then the position of the ball’s shadow is described by Asin( t), where A is the radius of the circle.
© 2015 Pearson Education, Inc.

9. Section 15.2: Simple harmonic motion
As illustrated in the figure, the correspondence between circular motion and simple harmonic motion can be demonstrated experimentally.
© 2015 Pearson Education, Inc.

10. Section 15.6: Simple harmonic motion and springs
Consider the spring-cart system shown. Let x0 = 0.
The force exerted by the spring on the cart is
Using we can find the equation of motion for the car to be
© 2015 Pearson Education, Inc.

11. Section 15.6: Simple harmonic motion and springs
define
The motion of the cart is given by
The figure shows four different solutions that satisfy the equation of motion of the spring-cart system.
© 2015 Pearson Education, Inc.

12. ax = –ω2x (simple harmonic motion)
Section 15.5: Energy of a simple harmonic oscillator
Comparing equations for x(t) and a(t), we can write
ax = –ω2x (simple harmonic motion)
Using Newton’s 2nd law, we get
© 2015 Pearson Education, Inc.

13. Checkpoint 15.2
(a) In Figure 15.2e, the cart’s displacement from the equilibrium position is maximum. Is the x component of the cart’s acceleration at that instant positive, negative, or zero? (b) At which instant(s) in Figure is the magnitude of the cart’s acceleration greatest? At which instant(s) is it smallest? (Use the blue position curve to answer these questions.)
15.2
© 2015 Pearson Education, Inc.

14. Section 15.2: Simple harmonic motion
Investigation of oscillating systems reveal that, when the amplitude is not too large, the period is independent of the amplitude.
An oscillating system that exhibits this property is called isochronous.
© 2015 Pearson Education, Inc.

15. Section 15.2: Simple harmonic motion
The x(t) curve of an isochronous oscillation are sinusoidal.
Periodic motion that yields a sinusoidal x(t) curve is called a simple harmonic motion (SHM):
A object executing simple harmonic motion is subject to a linear restoring force that tends to return the object to its equilibrium position and is linearly proportional to the object’s displacement from its equilibrium position.
© 2015 Pearson Education, Inc.

16. Section 15.3: Fourier’s theorem
Fourier’s theorem: Any periodic function with period T can be written as a sum of sinusoidal functions of frequency fn = n/T, where n ≥ 1 is an integer.
f1 = 1/T is called the fundamental frequency or the first harmonic.
All other components, called higher harmonics, are integer multiples of the fundamental frequency:
fn = nf1 = n/T
According to Fourier’s theorem, any periodic motion can be treated as a superposition of simple harmonic motions.
© 2015 Pearson Education, Inc.

17. Section 15.3: Fourier’s theorem
Fourier’s theorem: Any periodic function with period T can be written as a sum of sinusoidal functions of frequency fn = n/T, where n ≥ 1 is an integer.
By adjusting the amplitude An of each harmonic, we can make the sum of harmonics fit the original nonsinusoidal periodic function.
© 2015 Pearson Education, Inc.

18. Section 15.3: Fourier’s theorem
The breaking down of a function into harmonic components is called Fourier analysis.
The resulting sum of sinusoidal functions is called a Fourier series.
The figure shows a graphical representation of the Fourier series.
The plot of An2 as a function of the frequency is called the spectrum of the periodic function.
© 2015 Pearson Education, Inc.

19. Section 15.3: Fourier’s theorem
Fourier analysis has many applications, ranging from electronic signal processing to chemical analysis and voice recognition.
The figure shows the spectrum analyzer of a stereo system, which displays the frequency content of the sound being played.
© 2015 Pearson Education, Inc.

20. Checkpoint 15.6
(a) What does the spectrum of a single sinusoidal function of period T look like? (b) As T is increased, what change occurs in the spectrum?
15.6
© 2015 Pearson Education, Inc.

21. Chapter 15 Periodic Motion
© 2015 Pearson Education, Inc.

22. Section 15.4: Restoring forces in simple harmonic motion
Periodic motion requires a restoring force that tends to return the object to the equilibrium position.
A consequence of the restoring force about a stable equilibrium position is
In the absence of friction, a small displacement of a system from a position of stable equilibrium causes the system to oscillate.
© 2015 Pearson Education, Inc.

23. Checkpoint 15.7
(a) Are there any equilibrium positions in the force-versus- distance graph in Figure 15.13? If so, is the equilibrium stable, unstable, or neutral? (b) Compare the magnitude of the restoring force for equal displacements on either side of x0 in Figure
15.7
© 2015 Pearson Education, Inc.

24. Section 15.4: Restoring forces in simple harmonic motion
As illustrated in the figure:
For sufficiently small displacements away from the equilibrium position x0, restoring forces are always linearly proportional to the displacement.
Consequently, for small displacements, objects execute simple harmonic motion about a stable equilibrium position.
© 2015 Pearson Education, Inc.

25. Section 15.5: Energy of a simple harmonic oscillator
The work done by the forces exerted on the harmonic oscillator as it moves from the equilibrium position toward the positive x direction is
This work causes a change in kinetic energy, given by
For a closed system E = K + U = 0, which gives us
© 2015 Pearson Education, Inc.

26. Objects move to the minimum potential energy
Section 15.5: Energy of a simple harmonic oscillator
Objects move to the minimum potential energy
The negative derivative of the potential energy is the force
© 2015 Pearson Education, Inc.

27. Section 15.4: Restoring forces in simple harmonic motion
Periodic motion requires a restoring force that tends to return the object to the equilibrium position.
In the absence of friction, a small displacement of a system from a position of stable equilibrium causes the system to oscillate.
© 2015 Pearson Education, Inc.

28. Example 15.8 The simple pendulum as a one-dimensional oscillator.
Section 15.7: Restoring torques
Example 15.8 The simple pendulum as a one-dimensional oscillator.
❶ The bob is displaced a horizontal distance x from its equilibrium position, (Figure 15.35a). The free-body diagram for the bob (Figure 15.35b), showing with the component vectors of parallel to and perpendicular to the string in its displaced orientation. The bob is constrained to move on a circular path.
© 2015 Pearson Education, Inc.

29. Example 15.8 The simple pendulum as a one-dimensional oscillator.
Section 15.7: Restoring torques
Example 15.8 The simple pendulum as a one-dimensional oscillator.
❸ From the free-body diagram,
For small angles, x ~ s
© 2015 Pearson Education, Inc.

30. Example 15.8 The simple pendulum as a one-dimensional oscillator.
Section 15.7: Restoring torques
Example 15.8 The simple pendulum as a one-dimensional oscillator.
❸ In the small-angle approximation, Thus, or
© 2015 Pearson Education, Inc.

31. Example 15.8 The simple pendulum as a one-dimensional oscillator.
Section 15.7: Restoring torques
Example 15.8 The simple pendulum as a one-dimensional oscillator.
❸ Rewrite in form similar to spring, or
define
© 2015 Pearson Education, Inc.

32. Section 15.4: Restoring forces in simple harmonic motion
Oscillations arise from interplay between inertia and a restoring force.
For springs, strings, and vibrating masses
The period of an oscillating object increases when its mass is increased and decreases when the magnitude of the restoring force is increased.
For pendulums:
The period of a pendulum is independent of the mass of the pendulum.
© 2015 Pearson Education, Inc.

33. Checkpoint 15.9
(a) What effect, if any, does increasing the length ℓ of a simple pendulum have on its period? (b) A pendulum is taken to the Moon, where the acceleration due to gravity is smaller than that on Earth. How does the period of the pendulum on the Moon compare with its period on Earth?
15.9
© 2015 Pearson Education, Inc.

34. Section 15.4 Clicker Question 4
A child and an adult are on adjacent swings at the playground. Is the adult able to swing in synchrony with the child?
No, this is impossible because the inertia of the two are different.
Yes, as long as the lengths of the two swings are adjusted.
Yes, as long as the lengths of the swings are the same.
© 2015 Pearson Education, Inc.

35. Section 15.4 Clicker Question 4
A child and an adult are on adjacent swings at the playground. Is the adult able to swing in synchrony with the child?
No, this is impossible because the inertia of the two are different.
Yes, as long as the lengths of the two swings are adjusted.
Yes, as long as the lengths of the swings are the same.
© 2015 Pearson Education, Inc.

36. Chapter 15: Self-Quiz #3
In the cart-spring system shown in the figure, is the cart’s speed greater at A or at B?
Once released from the stretched position shown, the cart speeds up as it moves left toward the equilibrium position, so its speed is greater at A than at B.
© 2015 Pearson Education, Inc.

37. Chapter 15: Self-Quiz #3
At which of these two positions is the restoring force acting on the cart greater?
The restoring force is proportional to the displacement from the equilibrium position, so it is greater at B than at A.
© 2015 Pearson Education, Inc.

38. Example 15.5 Vertical oscillations
Section 15.6: Simple harmonic motion and springs
Example 15.5 Vertical oscillations
A block of mass m is suspended from a spring of spring constant k. (a) How far below the end of the relaxed spring at x0 is the equilibrium position xeq of the suspended block (Figure 15.29a)? k m
© 2015 Pearson Education, Inc.

39. Example 15.5 Vertical oscillations
Section 15.6: Simple harmonic motion and springs
Example 15.5 Vertical oscillations
When the suspended block is in translational equilibrium at xeq (which lies below x0), the vector sum of these forces must be zero. With the block at xeq, the spring is stretched such that the end attached to the block is also at xeq.
© 2015 Pearson Education, Inc.

40. Vertical oscillations
Section 15.6: Simple harmonic motion and springs
Vertical oscillations m
(b) Is the frequency f with which the block oscillates about this equilibrium position xeq greater than, smaller than, or equal to that of an identical system that oscillates horizontally about x0 on a surface for which friction can be ignored (Figure 15.29b)? k k
© 2015 Pearson Education, Inc.

41. Section 15.6: Simple harmonic motion and springs
❶ When the block is below xeq so the spring is stretched more than when the block is at xeq, causing the magnitude of to increase above the equilibrium value, and so the vector sum of the forces exerted on the block points upward.
© 2015 Pearson Education, Inc.

42. Example 15.5 Vertical oscillations
Section 15.6: Simple harmonic motion and springs
Example 15.5 Vertical oscillations m
EVALUATE RESULT The two oscillations take place about different equilibrium positions (x0 for the horizontal case, xeq for the vertical case), but the effect of the combined gravitational and elastic forces in the vertical arrangement is the same as that of just the elastic force in the horizontal arrangement because the force exerted by the spring is linear in the displacement. k k
© 2015 Pearson Education, Inc.

43. Section 15.6 Clicker Question 6
If you know the mass of an object hanging from a spring in an oscillating system, what else do you need to know to determine the period of the motion? Answer all that apply.
The spring constant
The initial velocity
The angular frequency
The amplitude
© 2015 Pearson Education, Inc.

44. Section 15.6 Clicker Question 6
If you know the mass of an object hanging from a spring in an oscillating system, what else do you need to know to determine the period of the motion? Answer all that apply.
The spring constant
The initial velocity
The angular frequency
The amplitude
© 2015 Pearson Education, Inc.

45. Section 15.7 Clicker Question 7
What approximation is necessary in order for a pendulum to execute simple harmonic motion? Answer all that apply.
No approximations are necessary; a pendulum always executes simple harmonic motion.
The mass must be very large.
The mass must be very small.
The length must be very large.
The length must be very small.
The amplitude of the angular displacement must be much less than 1 radian.
© 2015 Pearson Education, Inc.

46. Section 15.7 Clicker Question 7
What approximation is necessary in order for a pendulum to execute simple harmonic motion? Answer all that apply.
No approximations are necessary; a pendulum always executes simple harmonic motion.
The mass must be very large.
The mass must be very small.
The length must be very large.
The length must be very small.
The amplitude of the angular displacement must be much less than 1 radian.
© 2015 Pearson Education, Inc.

47. Section 15.8: Damped oscillations
Section Goals
You will learn to
Integrate the concept of friction to oscillatory motion and represent the combined effects graphically.
Model damped harmonic motion mathematically.
Define the time constant for damped harmonic motion.
© 2015 Pearson Education, Inc.

48. Section 15.8: Damped oscillations
Mechanical oscillators always involve some friction that causes the energy of the oscillator to convert to thermal energy.
This will cause the oscillator to slow down.
Such a system is said to execute a damped oscillation.
The figure shows examples of two damped oscillations.
© 2015 Pearson Education, Inc.

49. Section 15.8: Damped oscillations
The drag force exerted by air and liquids at slow speeds can be modeled as where b is called the damping coefficient.
In presence of a drag, the equation of motion becomes
The solution of this equation takes the forms
© 2015 Pearson Education, Inc.

50. Section 15.8: Damped oscillations
The figure shows oscillations for various values of the damping coefficient b.
© 2015 Pearson Education, Inc.

51. Section 15.8: Damped oscillations
The ratio m/b has units of time and is called the time constant:   m/b.
Amplitude of the damped oscillation at any given time is given by
xmax(t) = Ae–t/2.
The mechanical energy of the oscillator can be expressed as
© 2015 Pearson Education, Inc.

52. Checkpoint 15.18
A tuning fork that sounds the tone musicians call middle C oscillates at frequency f = 262 Hz. If the amplitude of the fork’s oscillation decreases by a factor of 3 in 4.0 s, what are (a) the time constant of the oscillation and (b) the quality factor?
15.18
© 2015 Pearson Education, Inc.

53. Section 15.8 Clicker Question 8
During a time interval of one time constant, by what factor does the mechanical energy of an oscillating system decrease?
By a factor of 1/e (0.3679)
By a factor of ln(2) (0.6931)
By a factor of (2)1/2
By a factor of 2
By a factor of e (2.718)
By a factor of pi
By a factor of 10
© 2015 Pearson Education, Inc.

54. Section 15.8 Clicker Question 8
During a time interval of one time constant, by what factor does the mechanical energy of an oscillating system decrease?
By a factor of 1/e (0.3679)
By a factor of ln(2) (0.6931)
By a factor of (2)1/2
By a factor of 2
By a factor of e (2.718)
By a factor of pi
By a factor of 10
© 2015 Pearson Education, Inc.

55. Concepts: Fundamental characteristics of periodic motion
Chapter 15: Summary
Concepts: Fundamental characteristics of periodic motion
Periodic motion is any motion that repeats itself at regular time intervals. Oscillation (or vibration) is back-and-forth periodic motion.
The period T is the minimum time interval in which periodic motion repeats, and the amplitude A of the motion is the magnitude of the maximum displacement of the moving object from its equilibrium position.
© 2015 Pearson Education, Inc.

56. Quantitative Tools: Fundamental characteristics of periodic motion
Chapter 15: Summary
Quantitative Tools: Fundamental characteristics of periodic motion
The frequency f of periodic motion is the number of cycles per second and is defined as
The SI unit of frequency is the hertz (Hz):
1 Hz  1 s–1.
© 2015 Pearson Education, Inc.

57. Concepts: Simple harmonic motion
Chapter 15: Summary
Concepts: Simple harmonic motion
Simple harmonic motion is periodic motion in which the displacement of a system from its equilibrium position varies sinusoidally with time. A system moving in this way is called a simple harmonic oscillator.
© 2014 Pearson Education, Inc.

58. Concepts: Simple harmonic motion
Chapter 15: Summary
Concepts: Simple harmonic motion
A restoring force that is linearly proportional to displacement tends to return a simple harmonic oscillator to its equilibrium position. For small displacements, restoring forces are generally proportional to the displacement and therefore cause objects to execute simple harmonic motion about any stable equilibrium position.
© 2015 Pearson Education, Inc.

59. Concepts: Simple harmonic motion
Chapter 15: Summary
Concepts: Simple harmonic motion
A phasor is a rotating arrow whose component on a vertical axis traces out simple harmonic motion. The reference circle is the circle traced out by the tip of the phasor, and the length of the phasor is equal to the amplitude A of the simple harmonic motion.
© 2015 Pearson Education, Inc.

60. Quantitative Tools: Simple harmonic motion
Chapter 15: Summary
Quantitative Tools: Simple harmonic motion
The angular frequency  of oscillation is equal to the rotational speed of a rotating phasor whose vertical component oscillates with a frequency f. Angular frequency is measured in s–1 and is related to the frequency (measured in Hz) by
 = 2f.
For a simple harmonic oscillator of amplitude A, the displacement x as a function of time is
x(t) = A sin(t + i),
where the sine argument is the phase (t) of the periodic motion,
(t) = t + i,
and i is the initial phase at t = 0.
© 2015 Pearson Education, Inc.

61. Quantitative Tools: Simple harmonic motion
Chapter 15: Summary
Quantitative Tools: Simple harmonic motion
The x components of the velocity and acceleration of a simple harmonic oscillator are
© 2015 Pearson Education, Inc.

62. Quantitative Tools: Simple harmonic motion
Chapter 15: Summary
Quantitative Tools: Simple harmonic motion
Any object that undergoes simple harmonic motion obeys the simple harmonic oscillator equation:
where x is the object’s displacement from its equilibrium position.
The mechanical energy E of an object of mass m undergoing simple harmonic motion is
© 2015 Pearson Education, Inc.

63. Concepts: Fourier series
Chapter 15: Summary
Concepts: Fourier series
Fourier’s theorem says that any periodic function with period T can be written as a sum of sinusoidal simple harmonic functions of frequency fn = n/T, where n is an integer. The n = 1 term is the fundamental frequency or first harmonic, and the other components are higher harmonics.
© 2015 Pearson Education, Inc.

64. Quantitative Tools: Oscillating springs
Chapter 15: Summary
Quantitative Tools: Oscillating springs
For an object attached to a light spring of spring constant k, the simple harmonic oscillator equation takes the form
and the angular frequency of the oscillation is
The motion of the object is described by
© 2015 Pearson Education, Inc.

65. Concepts: Rotational oscillations
Chapter 15: Summary
Concepts: Rotational oscillations
A horizontal disk suspended at its center by a thin fiber forms a type of torsional oscillator.
A pendulum is any object that swings about a pivot. A simple pendulum consists of a small object (the bob) attached to a very light wire or rod.
© 2015 Pearson Education, Inc.

66. Quantitative Tools: Rotational oscillations
Chapter 15: Summary
Quantitative Tools: Rotational oscillations
If a torsional oscillator of rotational inertia I is twisted through a small angle from its equilibrium position 0 to position , the restoring torque  is
 = –( – 0),
where  is the torsional constant. When 0 = 0, the simple harmonic oscillator equation for the torsional oscillator is
© 2015 Pearson Education, Inc.

67. Quantitative Tools: Rotational oscillations
Chapter 15: Summary
Quantitative Tools: Rotational oscillations
The rotational position  of the torsional oscillator at instant t is given by
 = max sin(t + i),
where max is the maximum rotational displacement and
© 2015 Pearson Education, Inc.

68. Quantitative Tools: Rotational oscillations
Chapter 15: Summary
Quantitative Tools: Rotational oscillations
For small rotational displacements, the simple harmonic oscillator equation of a pendulum is
and its angular frequency is
where is the distance from the center of mass of the pendulum to the pivot.
The period of a simple pendulum is
© 2015 Pearson Education, Inc.

69. Concepts: Damped oscillations
Chapter 15: Summary
Concepts: Damped oscillations
In damped oscillation, the amplitude decreases over time due to energy dissipation. The cause of the dissipation is a damping force due to friction, air drag, or water drag.
A damped oscillator that has a large quality factor Q keeps oscillating for many periods.
© 2015 Pearson Education, Inc.

70. Quantitative Tools: Damped oscillations
Chapter 15: Summary
Quantitative Tools: Damped oscillations
At low speeds, the damping force tends to be proportional to the velocity of the object:
where b, the damping coefficient, has units of kilograms per second.
For small damping, the position x(t) of a damped spring is
x(t) = Ae–bt/2m sin(dt + i)
and its angular frequency d is
© 2015 Pearson Education, Inc.