1. PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves
Lesson 5: Pendulum Motion as SHM

2. Reading Segment #1:
Kinematics and Dynamics of Pendulum Motion
To prepare for this section, please read:
Unit 6: p. 13

3. Kinematics and Dynamics of Pendulum Motion
You will soon discover that pendulum motion can be equivalent to SHM.
However, before you can see this, you need to understand the nature of the forces and acceleration during pendulum motion.

4. Pendulum Motion Applet
To analyze the velocity, acceleration, and forces during pendulum motion, click on the following link:
Instructions:
 Click on mass and move pendulum to an angle  Click on “Play” and it will oscillate  Click on Button 1 to see the forces  Click on Button 2 to see the velocity and acceleration

5. Consider moving a pendulum to a deviation angle  and then releasing it.
rest

6. First, we draw the forces on the mass:  FT
Fg = mg
rest
First, we draw the forces on the mass:
The force of gravity always acts downward.
Tension force acts along the string and towards the pivot (i.e. away from the mass).

7. Next, we establish the y-axis along the string.  FT
rest 
Fg = mg x
Next, we establish the y-axis along the string.
Notice that Fg is the diagonal force here.

8. Next, we draw the x- and y-components of Fg. FT
rest
Fg y 
Fg = mg
Fg x
Next, we draw the x- and y-components of Fg.

9. We can determine each component as follows: FT
rest
Fg y 
Fg = mg
Fg x
We can determine each component as follows:

10.  FT
rest
mg cos  
Fg = mg
mg sin 
The two “vertical” forces are equal but opposite in this position, and so there is no acceleration along the y-axis.

11. That is, the acceleration is tangential. FT
rest a
mg cos  
Fg = mg
mg sin 
The net force is mg sin  , and so the pendulum’s acceleration is entirely along the x-axis.
That is, the acceleration is tangential.

12.  FT
rest x a
mg cos  
Fg = mg
mg sin 
(Fs)
mg sin  acts in a direction opposite (approximately) to the displacement x, much like a spring force.
It is trying to return the mass to the equilibrium position (i.e. a restoring force).

13. Equilibrium: Max speed FT
Fg = mg
mg cos 
mg sin 
(Fs) x
rest a v
Equilibrium: Max speed
The pendulum speeds up until it reaches equilibrium position, where it has achieved maximum speed.

14. Again, we draw the forces on the mass:  FT
Fg = mg
mg cos 
mg sin 
(Fs) x
rest a FT v Fg
Again, we draw the forces on the mass:
The force of gravity always acts downward.
Tension force acts along the string and towards the pivot (i.e. away from the mass).

15. At this moment, the mass is in uniform circular motion.  FT
Fg = mg
mg cos 
mg sin 
(Fs) x
rest a FT v ac Fg
At this moment, the mass is in uniform circular motion.
The acceleration is entirely vertical and towards the centre, and so it is centripetal.

16. It is very similar when it moves back up.  FT
Fg = mg
mg cos 
mg sin 
(Fs) x
rest a ac FT v Fg  FT x
rest a
mg cos  
Fg = mg
mg sin 
(Fs)
It is very similar when it moves back up.
mg sin  acts to slow down the mass until it comes to rest at its maximum displacement.

17. Reading Segment #2:
Pendulum Motion as SHM
To prepare for this section, please read:
Unit 6: pp

18. Consider a pendulum deviated at an angle  .
Pendulum Motion as SHM
Consider a pendulum deviated at an angle  .  x
Fs = mg sin  L
Again, we consider mg sin  as being equivalent to the restoring force Fs.

19. For the triangle shown: x
Fs = mg sin  L

20.  x
Fs = mg sin  L
Combining this with
we discover that or

21. Since m, g, and L are constant for the pendulum, it follows that x
Fs = mg sin  L
Since m, g, and L are constant for the pendulum, it follows that
Thus, the “spring force” has a direct relationship with the displacement x.

22. This is not actually true, so a pendulum does not undergo SHM.
However, technically speaking, for the pendulum to be in SHM, the restoring force must be directly proportional to the displacement along the arc.  L x a
Fs = mg sin 
This is not actually true, so a pendulum does not undergo SHM.

23. Under this condition, Fs  x and the pendulum undergoes SHM.
However, if the angle of deflection is less than 15, then we can safely assume that the displacement is equal to the arclength.
 < 15 L
x  a a
Fs = mg sin 
Under this condition, Fs  x and the pendulum undergoes SHM.

24. Period of Pendulum Motion x
Fs = mg sin  L
As we have seen, for small amplitudes ( < 15), we can consider mg sin  as a restoring force, much like Fs .
As a result, since this is equivalent to a mass-spring system, we can use Hooke’s Law (Fs = k x) to create a formula.

25. Starting with the formula we derived earlier: x
Fs = mg sin  L
Starting with the formula we derived earlier:
We now substitute Hooke’s Law Fs = k x. This leaves us with

26. Thus,  x
Fs = mg sin  L
or,

27. If we substitute this into the period formula for a mass-spring system, we discover the following:
but, or
So,

28. Equation:
The period T of a pendulum is calculated using the formula
SI Units: s
where
L is the length of the pendulum (in m)
g is the magnitude of the acceleration due to gravity (in m/s2)

29. L is measured from the axis of rotation to the centre of the mass. L 
axis
Note:
L is measured from the axis of rotation to the centre of the mass.
For this formula to be used,  < 15

30. Example
A pendulum oscillates at a frequency of 0.42 Hz. Determine its length.
Try this example on your own first.
Then, check out the solution.

31. Find period:
= s

32. Find length:

35. = m

36. Practice Problems
Try these problems in the Physics 20 Workbook:
Unit 6 p. 66 #1 - 4

37. Reading Segment #3:
Energetics of Pendulum Motion
To prepare for this section, please read:
Unit 6: p. 15

38. Energetics of Pendulum Motion
Again, consider moving a pendulum to a deviation angle  and then releasing it. 
rest

39. Pendulum Motion Applet
To analyze the energy of pendulum motion, click on the following link:
Analysis of Pendulum Motion
Note: Click on “show” if you wish to see the acceleration vector and its components.

40. rest
 = A
At the position of maximum displacement, the angle of deviation represents the amplitude of oscillation.
i.e.
max  = A

41. Thus, it has maximum gravitational potential energy.
Max Epg
rest
hmax
Ref h: h = 0
Equilibrium
position
At maximum displacement, the mass is at a maximum height (relative to the equilibrium position).
Thus, it has maximum gravitational potential energy.

42. The mass is also at rest, and so it has no kinetic energy.
Max Epg
Ek = 0
rest
hmax
Equilibrium
position
The mass is also at rest, and so it has no kinetic energy.

43. The total mechanical energy at this position is
Max Epg
Ek = 0
rest
hmax
Equilibrium
position
The total mechanical energy at this position is
The mass only has gravitational potential energy

44. As the pendulum falls, it speeds up.
Max Epg
Ek = 0
rest
hmax
vmax
Equilibrium
position
As the pendulum falls, it speeds up.
When it reaches the equilibrium position, the mass has attained its maximum speed.

45. Since the speed is at a maximum, the mass has maximum kinetic energy.
Max Epg
Ek = 0
rest
hmax
vmax
Equilibrium
position
Max Ek
Since the speed is at a maximum, the mass has maximum kinetic energy.

46. A
Max Epg
Ek = 0
rest
hmax
vmax
Ref h: h = 0
Equilibrium
position
Max Ek
Epg = 0
However, since the mass has reached the equilibrium position, it has no gravitational potential energy.

47. The total mechanical energy at the equilibrium position is
Max Epg
Ek = 0
rest
hmax
vmax
Equilibrium
position
Max Ek
Epg = 0
The total mechanical energy at the equilibrium position is
The mass only has kinetic energy

48. It is similar when it moves up to maximum displacement.
rest
vmax A
hmax
Equilibrium
position
Max Epg
Ek = 0
Max Ek
Epg = 0
Maximum displacement
It is similar when it moves up to maximum displacement.
The mass slows down until it comes to rest at its maximum height. At this position, it has no kinetic energy and maximum gravitational potential energy.

49. Energy Formulas of Pendulum Motion
rest
vmax A
hmax
Equilibrium
position
Max Epg
Ek = 0
Max Ek
Epg = 0
Energy Formulas of Pendulum Motion
To derive an energy formula for pendulum motion, we compare maximum displacement with equilibrium position.

50. Thus, the total mechanical energy must remain constant.
rest
vmax A
hmax
Equilibrium
position
Max Epg
Ek = 0
Max Ek
Epg = 0
If this is an ideal pendulum, then there is no energy lost due to friction or air resistance.
Thus, the total mechanical energy must remain constant.
i.e.
Em (top) = Em (bottom)

51. Em (top) = Em (bottom) A Max Epg Ek = 0 rest hmax vmax Equilibrium
position
Max Epg
Ek = 0
Max Ek
Epg = 0
Em (top) = Em (bottom)

52. Em (top) = Em (bottom) A Max Epg Ek = 0 rest hmax vmax Equilibrium
position
Max Epg
Ek = 0
Max Ek
Epg = 0
Em (top) = Em (bottom)

53. rest
vmax A
hmax
Equilibrium
position
Max Epg
Ek = 0
Max Ek
Epg = 0
Em (top) = Em (bottom)
This is not on the formula sheet, and so you must derive this each time.

54. Example
On a different planet, an 80 cm long pendulum oscillates with a period of 2.9 seconds.
If this pendulum reaches a maximum height of 1.5 cm (above its equilibrium position), then determine its maximum speed.
Assume no air resistance or friction.
Try this example on your own first.
Then, check out the solution.

55. Find the acceleration due to gravity:

58. = m/s2

59. Find the maximum speed:
rest
vmax A
hmax
Equilibrium
position
Max Epg
Ek = 0
Max Ek
Epg = 0
Find the maximum speed:
Em (top) = Em (bottom)

60. Em (top) = Em (bottom)

61. Em (top) = Em (bottom)

64. = m/s

65. Practice Problems
Try these problems in the Physics 20 Workbook:
Unit 6 p. 6 #11