1. Springs & Simple Harmonic Motion

2. Simple Harmonic Motion
An oscillation is a back and forth
motion over the same path. Requires a restoring force.
Periodic: each cycle of the motion takes
place in an equal period of time.
Simple Harmonic Motion (SHM): a projection of circular motion on one axis.
Examples of SHM:
Mass on a spring
simple pendulum
a particle in a bowl
Restoring Force

3. SHM and Circular Motion
An object moving with constant speed in a circular path observed from a distant point will appear to be oscillating with simple harmonic motion.
The shadow of a pendulum bob moves with s.h.m. when the pendulum itself is either oscillating or moving in a circle with constant speed.

4. SHM and Circular Motion
For any SHM there is a corresponding circular motion.
the radius of the circle is equal to the amplitude of the SHM 
the time period of the circular motion is equal to the time period of the SHM
The relationship of circular
motion and SHM is
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.

5. SHM and Circular Motion
Rearranging, 
Recall, therefore
The constant of proportionality between acceleration and displacement for an object moving with SHM is equal and opposite to the square of the angular velocity of the corresponding circular motion.
So, to find the value of the constant for a given oscillation, we simply measure the time period and then use the relation

6. Characteristics of SHM
A particle is disturbed away from a fixed equilibrium position and experience an acceleration that is proportional and opposite to its displacement.
Requirements:
Fixed equilibrium position
Particle moved away from equilibrium position, acceleration is both proportional to the displacement and in the opposite direction.
Characteristics
Period and amplitude are constant
Period is independent of the amplitude
Displacement, velocity, and acceleration are sine or cosine functions over time.
𝒂=− 𝝎 𝟐 𝒙

7. Spring Review Fs=-kx Hooke’s Law Fs= restoring force of a spring
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
Hooke’s Law
Fs= restoring force of a spring
k = spring constant
x = displacement of the spring
The friction free motion shown above is known as Simple Harmonic Motion (SHM)
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.

8. Spring Review SHM graph Sinusoidal motion
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
SHM graph
Sinusoidal motion
max. stretching distance (x) from the equilibrium position is equal to the amplitude of the graph.
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.

9. ConcepTest 13.1a Harmonic Motion I
1) 0
2) A/2
3) A
4) 2A
5) 4A
A mass on a spring in SHM has amplitude A and period T. What is the total distance traveled by the mass after a time interval T?

10. ConcepTest 13.1a Harmonic Motion I
1) 0
2) A/2
3) A
4) 2A
5) 4A
A mass on a spring in SHM has amplitude A and period T. What is the total distance traveled by the mass after a time interval T?
In the time interval T (the period), the mass goes through one complete oscillation back to the starting point. The distance it covers is: A + A + A + A (4A).

11. SHM and Circular Motion
SHM has displacement, velocity and acceleration.
Displacement:
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
Note: r = amplitude (A)

12. SHM and Circular Motion
Period (T) – time required to complete one cycle.
Recall, for 1 cycle &
Therefore,
Frequency (f) – number of cycles per second
Units – Hertz (Hz)
Phase difference (φ) – amount by which one curve is shifted relative to another curve.
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
𝜑= 𝑠ℎ𝑖𝑓𝑡 𝑇 ×360° (𝑑𝑒𝑔𝑟𝑒𝑒𝑠)
𝜑= 𝑠ℎ𝑖𝑓𝑡 𝑇 ×2𝜋 (𝑟𝑎𝑑𝑖𝑎𝑛𝑠)

13. SHM Terms
Copywrited by Holt, Rinehart, & Winston

14. SHM and Circular Motion
Velocity
Velocity of the shadow is the vx of vT.
Recall
At x =0 m v  vmax
Therefore, θ = 0° or
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
Note: r = amplitude (A)

15. SHM and Circular Motion
Acceleration
Acceleration of the shadow is the ax of ac.
Recall
At x = r, a  amax
Therefore, θ = 90° or
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
Note: r = amplitude (A)

16. Graphs Describing SHM Displacement against time x = rsin(ωt)
Velocity against time v = rωcos(ωt)
Acceleration against time a = -rω²sin(ωt)
Note: All these graphs assume that, at t = 0, the body is at the equilibrium position. We defined our equations where t=0 the displacement was the amplitude.

17. SHM (IB Formulation) Alternative form x = Acos(ωt+ϕ) v = -Aωsin(ωt+ϕ)
a = Aω²cos(ωt+ϕ)
where A and ϕ
are both constants
- A is the amplitude
- ϕ is the phase angle
IB shows both
x = x0sin(ωt) or x = x0cos(ωt)
v = v0cos(ωt) or v = -v0sin(ωt)
Physics for the IB Diploma 5th Edition (Tsokos) 2008
𝜑= 𝑠ℎ𝑖𝑓𝑡 𝑇 ×2𝜋 (𝑟𝑎𝑑𝑖𝑎𝑛𝑠)
Sine form of displacement when at t=0 the displacement is zero (particle at equilibrium)
Cosine form of displacement
when at t=0 the displacement is the amplitude.
Physics for the IB Diploma 5th Edition (Tsokos) 2008

18. SHM of Springs Derivation of Frequency of SHM Hooke’s Law: F=-kx
F = ma = -kx 
Frequency,
Period,
Copywrited by Holt, Rinehart, & Winston

19. SHM of Springs

20. Elastic Potential Energy (PEelastic)
Energy that a spring contains by being stretched or compressed.
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.

21. SHM of Springs

22. ConcepTest 13.6a Period of a Spring I
A glider with a spring attached to each end oscillates with a certain period. If the mass of the glider is doubled, what will happen to the period?
1) period will increase
2) period will not change
3) period will decrease

23. ConcepTest 13.6a Period of a Spring I
A glider with a spring attached to each end oscillates with a certain period. If the mass of the glider is doubled, what will happen to the period?
1) period will increase
2) period will not change
3) period will decrease
The period is proportional to the square root of the mass. So an increase in mass will lead to an increase in the period of motion.
T = 2p (m/k)
Follow-up: What happens if the amplitude is doubled?

24. ConcepTest 13.7a Spring in an Elevator I
A mass is suspended from the ceiling of an elevator by a spring. When the elevator is at rest, the period is T. What happens to the period when the elevator is moving upward at constant speed?
1) period will increase
2) period will not change
3) period will decrease

25. ConcepTest 13.7a Spring in an Elevator I
A mass is suspended from the ceiling of an elevator by a spring. When the elevator is at rest, the period is T. What happens to the period when the elevator is moving upward at constant speed?
1) period will increase
2) period will not change
3) period will decrease
Nothing at all changes when the elevator moves at constant speed. The equilibrium elongation of the spring is the same, and the period of simple harmonic motion is the same.

26. ConcepTest 13.7c Spring on the Moon
A mass oscillates on a vertical spring with period T. If the whole setup is taken to the Moon, how does the period change?
1) period will increase
2) period will not change
3) period will decrease

27. ConcepTest 13.7c Spring on the Moon
A mass oscillates on a vertical spring with period T. If the whole setup is taken to the Moon, how does the period change?
1) period will increase
2) period will not change
3) period will decrease
The period of simple harmonic motion only depends on the mass and the spring constant and does not depend on the acceleration due to gravity. By going to the Moon, the value of g has been reduced, but that does not affect the period of the oscillating mass-spring system.
Follow-up: Will the period be the same on any planet?

28. SHM Energy
In SHM the total energy possessed by the oscillating body does not change with time.
Recall,
Total Mechanical Energy (E) = Kinetic Energy (Ek) + Potential Energy (Ep)
An oscillation in which the total energy decreases with time is described as a damped oscillation.
Due to air resistance or other similar causes

29. SHM Ek & Ep Kinetic energy against time ½m[rωcos(ωt)]²
Since, Ep= E- Ek
Potential Energy graph has the same form as the Kinetic Energy graph but is inverted.

30. SHM Ek & Ep When, Ep is a maximum, Ek is a minimum (Ek =0)
Copywrited by Holt, Rinehart, & Winston
When,
Ep is a maximum, Ek is a minimum (Ek =0)
Ek is a maximum, Ep is a minimum (Ep =0)
E = Ep + Ek

31. Energy in SHM (IB Formulation)
Elastic Potential Energy (Ep)
Kinetic Energy (Ek)
Total Mechanical Energy (E)
Total energy is conserved
𝐸 𝑝 = 𝑘 𝑥 2
𝐸 𝑘 = 𝑚 𝑣 2
𝐸= 𝐸 𝑝 + 𝐸 𝑘 = 𝑘 𝑥 𝑚 𝑣 2
𝐸= 𝑘 𝑥 𝑚 𝑣 2 =𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

32. Energy in SHM (IB Formulation)
When mass is released from rest and the extension is the amplitude of the motion A
Solving for velocity
For this motion, 𝜔 2 = 𝑘 𝑚
So,
𝐸= 𝑘 𝑥 𝑚 𝑣 2 = 𝑘 𝐴 2
𝑣=± 𝑘 𝑚 𝐴 2 − 𝑥 2
𝑣=±𝜔 𝐴 2 − 𝑥 2

33. Energy in SHM (IB Formulation)
Maximum velocity occurs at x = 0
At the extremes of the motion, x = ± A, an v = 0
𝑣=±𝜔 𝐴 2 − 𝑥 2
𝑣 𝑚𝑎𝑥 =𝜔𝐴

34. ConcepTest 13.2 Speed and Acceleration
A mass on a spring in SHM has amplitude A and period T. At what point in the motion is v = 0 and a = 0 simultaneously?
1) x = A
2) x > 0 but x < A
3) x = 0
4) x < 0
5) none of the above

35. ConcepTest 13.2 Speed and Acceleration
A mass on a spring in SHM has amplitude A and period T. At what point in the motion is v = 0 and a = 0 simultaneously?
1) x = A
2) x > 0 but x < A
3) x = 0
4) x < 0
5) none of the above
If both v and a were zero at the same time, the mass would be at rest and stay at rest! Thus, there is NO point at which both v and a are both zero at the same time.
Follow-up: Where is acceleration a maximum?

36. ConcepTest 13.5a Energy in SHM I
A mass oscillates in simple harmonic motion with amplitude A. If the mass is doubled, but the amplitude is not changed, what will happen to the total energy of the system?
1) total energy will increase
2) total energy will not change
3) total energy will decrease

37. ConcepTest 13.5a Energy in SHM I
A mass oscillates in simple harmonic motion with amplitude A. If the mass is doubled, but the amplitude is not changed, what will happen to the total energy of the system?
1) total energy will increase
2) total energy will not change
3) total energy will decrease
The total energy is equal to the initial value of the elastic potential energy, which is PEs = 1/2 kA2. This does not depend on mass, so a change in mass will not affect the energy of the system.
Follow-up: What happens if you double the amplitude?

38. ConcepTest 13.5b Energy in SHM II
If the amplitude of a simple harmonic oscillator is doubled, which of the following quantities will change the most?
1) frequency
2) period
3) maximum speed
4) maximum acceleration
5) total mechanical energy

39. ConcepTest 13.5b Energy in SHM II
If the amplitude of a simple harmonic oscillator is doubled, which of the following quantities will change the most?
1) frequency
2) period
3) maximum speed
4) maximum acceleration
5) total mechanical energy
Frequency and period do not depend on amplitude at all, so they will not change. Maximum acceleration and maximum speed do depend on amplitude, and both of these quantities will double (you should think about why this is so). The total energy equals the initial potential energy, which depends on the square of the amplitude, so that will quadruple.
Follow-up: Why do maximum acceleration and speed double?

40. Total Mechanical Energy
What forms of Mechanical Energy have we discussed?
Combining of all these

41. Energy Problem
An object of mass m = kg is vibrating on a horizontal frictionless table as shown. The spring has a spring constant k = 545 N/m. It is stretched initially to xo =4.50 cm and released from rest. Determine the final translational speed vf of the object when the final displacement of the spring is (a) xf = 2.25 cm and (b) xf=0 cm.
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.

42. The Simple Pendulum
Copywrited by Holt, Rinehart, & Winston
When displaced from its equilibrium position by an angle θ and released it swings back and forth.
Plotting the motion reveals a pattern similar to the sinusoidal motion of SHM
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.

43. The Simple Pendulum The gravitational force (Fgx) provides the torque.
Copywrited by Holt, Rinehart, & Winston
The gravitational force (Fgx) provides the torque.
This restoring force (Fgx):
Since the displacement and restoring force act in opposite directions.
The torque of the pendulum is
Copywrited by Holt, Rinehart, & Winston

44. The Simple Pendulum For small angles (10°or smaller) θ = sinθ
Copywrited by Holt, Rinehart, & Winston
For small angles (10°or smaller)
θ = sinθ
Where k‘ is a constant independent of θ
This form is similar to F=-kx (Hooke’s Law)
So, 
Copywrited by Holt, Rinehart, & Winston

45. The Simple Pendulum
For small angles ONLY
(10°or smaller)

46. Energy of a Pendulum TME = Constant TME = PE + KE PEmax  KE = 0
TME = Constant
TME = PE + KE
PEmax  KE = 0
KEmax  PE = 0

47. Swinging Problem
Determine the length of a simple pendulum that will swing back and forth in SHM with a period of 1.00 s.

48. ConcepTest 13.8a Period of a Pendulum I
Two pendula have the same length, but different masses attached to the string. How do their periods compare?
1) period is greater for the greater mass
2) period is the same for both cases
3) period is greater for the smaller mass

49. ConcepTest 13.8a Period of a Pendulum I
Two pendula have the same length, but different masses attached to the string. How do their periods compare?
1) period is greater for the greater mass
2) period is the same for both cases
3) period is greater for the smaller mass
The period of a pendulum depends on the length and the acceleration due to gravity, but it does not depend on the mass of the bob.
T = 2p (L/g)
Follow-up: What happens if the amplitude is doubled?

50. ConcepTest 13.8b Period of a Pendulum II
Two pendula have different lengths: one has length L and the other has length 4L. How do their periods compare?
1) period of 4L is four times that of L
2) period of 4L is two times that of L
3) period of 4L is the same as that of L
4) period of 4L is one-half that of L
5) period of 4L is one-quarter that of L

51. ConcepTest 13.8b Period of a Pendulum II
Two pendula have different lengths: one has length L and the other has length 4L. How do their periods compare?
1) period of 4L is four times that of L
2) period of 4L is two times that of L
3) period of 4L is the same as that of L
4) period of 4L is one-half that of L
5) period of 4L is one-quarter that of L
The period of a pendulum depends on the length and the acceleration due to gravity. The length dependence goes as the square root of L, so a pendulum 4 times longer will have a period that is 2 times larger.
T = 2p (L/g)

52. ConcepTest 13.9 Grandfather Clock
A grandfather clock has a weight at the bottom of the pendulum that can be moved up or down. If the clock is running slow, what should you do to adjust the time properly?
1) move the weight up
2) move the weight down
3) moving the weight will not matter
4) call the repairman

53. ConcepTest 13.9 Grandfather Clock
A grandfather clock has a weight at the bottom of the pendulum that can be moved up or down. If the clock is running slow, what should you do to adjust the time properly?
1) move the weight up
2) move the weight down
3) moving the weight will not matter
4) call the repairman
The period of the grandfather clock is too long, so we need to decrease the period (increase the frequency). To do this, the length must be decreased, so the adjustable weight should be moved up in order to shorten the pendulum length.
T = 2p (L/g)

54. ConcepTest 13.10c Pendulum in Elevator III
A swinging pendulum has period T on Earth. If the same pendulum were moved to the Moon, how does the new period compare to the old period?
1) period increases
2) period does not change
3) period decreases

55. ConcepTest 13.10c Pendulum in Elevator III
A swinging pendulum has period T on Earth. If the same pendulum were moved to the Moon, how does the new period compare to the old period?
1) period increases
2) period does not change
3) period decreases
The acceleration due to gravity is smaller on the Moon. The relationship between the period and g is given by:
therefore, if g gets smaller, T will increase. g L T = 2p
Follow-up: What can you do to return the pendulum to its original period?

56. ConcepTest 13.12 Swinging in the Rain
You are sitting on a swing. A friend gives you a push, and you start swinging with period T1. Suppose you were standing on the swing rather than sitting. When given the same push, you start swinging with period T2. Which of the following is true?
1) T1 = T2
2) T1 > T2
3) T1 < T2 T1
[CORRECT 5 ANSWER]

57. ConcepTest 13.12 Swinging in the Rain
You are sitting on a swing. A friend gives you a push, and you start swinging with period T1. Suppose you were standing on the swing rather than sitting. When given the same push, you start swinging with period T2. Which of the following is true?
1) T1 = T2
2) T1 > T2
3) T1 < T2 L1 T1 L2 T2
Standing up raises the Center of Mass of the swing, making it shorter !! Since L1 > L2 then T1 > T2 g L T = 2p

58. Particle in a Bowl
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
The gravitational force (Fgx) provides the restoring force.
This restoring force (Fgx):
So the acceleration is:
The displacement is:
For small angles (10°or smaller)
Physics for the IB Diploma 5th Edition (Tsokos) 2008
Physics for the IB Diploma 5th Edition (Tsokos) 2008
𝜃= 𝑥 𝑟
𝑠𝑖𝑛 𝑥 𝑟 ≈ 𝑥 𝑟

59. Particle in a Bowl So, In the form, Therefore: 𝑎=− 𝜔 2 𝑥
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
So,
In the form,
Therefore:
The period is:
𝑎=−𝑔 𝑥 𝑟 =− 𝑔 𝑟 𝑥
𝑎=− 𝜔 2 𝑥
Physics for the IB Diploma 5th Edition (Tsokos) 2008
𝜔 2 = 𝑔 𝑟
𝑇=2𝜋 𝑟 𝑔
Note: only for small angles (10°or smaller)

60. Simple Harmonic Motion
Copywrited by Holt, Rinehart, & Winston

61. Damped Simple Harmonic Motion
In real life, energy is lost causing each oscillation to be less than the previous.
Damping – reduction of an oscillation motion by an external force.
Modeled below where damping
force is
The net force is
Halliday, Wiley Publishing, Physics 6th ed
Halliday, Wiley Publishing, Physics 6th ed
Halliday, Wiley Publishing, Physics 6th ed
Pendulum Applet

62. Damping Damped oscillations – resistance forces 3 types of damping
Under-damping
Critical damping
Over-damping

63. Damping Under-damping Small resistance force
Amplitudes gradually decrease
Light Damping
Heavy Damping
Oscillations die out quicker
Larger period than lighter damping
Q factor – determines how quickly the oscillations will die out in an underdamped system.
Physics for the IB Diploma 5th Edition (Tsokos) 2008
Physics for the IB Diploma 5th Edition (Tsokos) 2008
𝑄=2𝜋 𝑒𝑛𝑒𝑟𝑔𝑦 𝑠𝑡𝑜𝑟𝑒𝑑 𝑖𝑛 𝑎 𝑐𝑦𝑐𝑙𝑒 𝑒𝑛𝑒𝑟𝑔𝑦 𝑑𝑖𝑠𝑠𝑖𝑝𝑎𝑡𝑒𝑑 𝑖𝑛 𝑎 𝑐𝑦𝑐𝑙𝑒
Alternative form 𝑄=2𝜋 𝐸 𝑠𝑡𝑜𝑟𝑒𝑑 𝑃

64. Damping Critically damped Over-damping Ideal damping
Returns the system to the equilibrium position as fast as possible
No oscillations
Over-damping
System returns to equilibrium without oscillations
Much slower than critically damped case.

65. Resonance Natural frequency (f0) – frequency at which a
Halliday, Wiley Publishing, Physics 6th ed
Natural frequency (f0) –
frequency at which a
system will vibrate after an
external disturbance
Resonance – condition when the amplitude increases due to a periodic external force where
frequency of driving force (fd)
Resonance
Non-Resonance

66. Resonance
For a small degree of damping, the peak of the curve occurs at the natural frequency, f0
The lower the degree of damping, the higher and narrower the curve
As the amount of damping increases, the peak shift to lower frequencies
At very low frequencies,
the amplitude is essentially constant.
Tacoma Narrows

67. greater. the same. smaller.
A person swings on a swing. When the person sits still, the swing oscillates back and forth at its natural frequency. If, instead, two people sit on the swing, the new natural frequency of the swing is
greater.
the same.
smaller. T1

68. ConcepTest 13.11 Damped Pendulum
After a pendulum starts swinging, its amplitude gradually decreases with time because of friction.
What happens to the period of the pendulum during this time ?
1) period increases
2) period does not change
3) period decreases

69. ConcepTest 13.11 Damped Pendulum
After a pendulum starts swinging, its amplitude gradually decreases with time because of friction.
What happens to the period of the pendulum during this time ?
1) period increases
2) period does not change
3) period decreases
The period of a pendulum does not depend on its amplitude, but only on its length and the acceleration due to gravity. g L T = 2p
Follow-up: What is happening to the energy of the pendulum?