PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.
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Transcript of PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.
![Page 1: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/1.jpg)
PHYS 20 LESSONS
Unit 6: Simple Harmonic
Motion
Mechanical Waves
Lesson 5: Pendulum Motion as SHM
![Page 2: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/2.jpg)
Reading Segment #1:
Kinematics and Dynamics ofPendulum Motion
To prepare for this section, please read:
Unit 6: p. 13
![Page 3: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/3.jpg)
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.
![Page 4: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/4.jpg)
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
http://canu.ucalgary.ca/map/content/torque/aboutanaxis/apply/physicalpendulum/applet.html
![Page 5: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/5.jpg)
Consider moving a pendulum to a deviation angle and then releasing it.
rest
![Page 6: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/6.jpg)
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 towardsthe pivot (i.e. away from the mass).
![Page 7: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/7.jpg)
FT
Fg = mg
rest
y
x
Next, we establish the y-axis along the string.
Notice that Fg is the diagonal force here.
![Page 8: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/8.jpg)
FT
Fg = mg
rest
Fg y
Fg x
Next, we draw the x- and y-components of Fg.
![Page 9: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/9.jpg)
FT
Fg = mg
rest
Fg y
Fg x
mg
Fxgsin
sinmgFxg
mg
Fyg
cos
cosmgFyg
We can determine each component as follows:
![Page 10: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/10.jpg)
FT
Fg = mg
mg cos
mg sin
rest
The two “vertical” forces are equal but opposite in this position, and so there is no acceleration along the y-axis.
![Page 11: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/11.jpg)
FT
Fg = mg
mg cos
mg sin
rest
a
The net force is mg sin , and so the pendulum’s acceleration is entirely along the x-axis.
That is, the acceleration is tangential.
![Page 12: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/12.jpg)
FT
Fg = mg
mg cos
mg sin (Fs)
rest
a
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).
x
![Page 13: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/13.jpg)
FT
Fg = mg
mg cos
mg sin (Fs)
xrest
av
The pendulum speeds up until it reaches equilibrium position, where it has achieved maximum speed.
Equilibrium: Max speed
![Page 14: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/14.jpg)
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 towardsthe pivot (i.e. away from the mass).
FT
Fg = mg
mg cos
mg sin (Fs)
xrest
a
![Page 15: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/15.jpg)
FT
v
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.
ac
FT
Fg = mg
mg cos
mg sin (Fs)
xrest
a
![Page 16: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/16.jpg)
FT
Fg = mg
mg cos
x rest
aac
FT
v
Fg
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.
FT
Fg = mg
mg cos
mg sin (Fs)
xrest
a
![Page 17: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/17.jpg)
Reading Segment #2:
Pendulum Motion as SHM
To prepare for this section, please read:
Unit 6: pp. 13 - 14
![Page 18: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/18.jpg)
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.
![Page 19: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/19.jpg)
x
Fs = mg sin
LL
xsin
For the triangle shown:
![Page 20: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/20.jpg)
x
Fs = mg sin
L
L
xsin
Combining this with
sinmgFs
we discover that
L
xmgFs
or
xL
mgFs
![Page 21: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/21.jpg)
x
Fs = mg sin
L
xL
mgFs
Since m, g, and L are constant for the pendulum, it follows that
xFs
Thus, the “spring force” has a direct relationship with the displacement x.
![Page 22: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/22.jpg)
x
Fs = mg sin
L
However, technically speaking, for the pendulum to be in SHM, the restoring force must be directly proportional to the displacement along the arc.
This is not actually true, so a pendulum does not undergo SHM.
a
![Page 23: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/23.jpg)
< 15
x a
Fs = mg sin
L
However, if the angle of deflection is less than 15, then we can safely assume that the displacement is equal to the arclength.
Under this condition, Fs x and the pendulum undergoes SHM.
a
![Page 24: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/24.jpg)
Period of Pendulum Motion
As we have seen, for small amplitudes ( < 15), we can consider mg sin as a restoring force, much like Fs .
x
Fs = mg sin
L
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.
![Page 25: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/25.jpg)
x
Fs = mg sin
L
Starting with the formula we derived earlier:
xL
mgFs
We now substitute Hooke’s Law Fs = k x. This leaves us with
xL
mgxk
![Page 26: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/26.jpg)
x
Fs = mg sin
L
Thus,
xL
mgxk
L
mgk
or,
mgLk
![Page 27: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/27.jpg)
If we substitute this into the period formula for a mass-spring system, we discover the following:
k
mT 2
but,
g
L
k
mmgLk or
So,
g
LT 2
![Page 28: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/28.jpg)
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)
g
LT 2
![Page 29: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/29.jpg)
Note:
L is measured from the axis of rotation to the centre of the mass.
g
LT 2
For this formula to be used, < 15
L
axis
![Page 30: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/30.jpg)
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.
![Page 31: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/31.jpg)
= 2.38 s
Find period:
Tf
1
fT
1
42.0
1
![Page 32: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/32.jpg)
Find length:
g
LT 2
g
LT
2
2
2
g
LT
2
![Page 33: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/33.jpg)
g
LT
2
22
2
g
LT
g
LT
2
2
4
![Page 34: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/34.jpg)
g
LT
2
2
4
LTg 22 4
2
2
4Tg
L
![Page 35: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/35.jpg)
2
2
4Tg
L
2
2
4
38.281.9
= 1.4 m
![Page 36: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/36.jpg)
Practice Problems
Try these problems in the Physics 20 Workbook:
Unit 6 p. 66 #1 - 4
![Page 37: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/37.jpg)
Reading Segment #3:
Energetics of Pendulum Motion
To prepare for this section, please read:
Unit 6: p. 15
![Page 38: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/38.jpg)
Energetics of Pendulum Motion
Again, consider moving a pendulum to a deviation angle and then releasing it.
rest
![Page 39: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/39.jpg)
Pendulum Motion Applet
Analysis of Pendulum Motion
To analyze the energy of pendulum motion, click on the following link:
Note: Click on “show” if you wish to see theacceleration vector and its components.
![Page 40: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/40.jpg)
rest
= A
At the position of maximum displacement, the angle of deviation represents the amplitude of oscillation.
i.e. max = A
![Page 41: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/41.jpg)
rest
A
hmax
Equilibriumposition
At maximum displacement, the mass is at a maximum height (relative to the equilibrium position).
Thus, it has maximum gravitational potential energy.
Max Epg
Ref h: h = 0
![Page 42: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/42.jpg)
rest
A
hmax
Equilibriumposition
The mass is also at rest, and so it has no kinetic energy.
Max Epg
Ek = 0
![Page 43: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/43.jpg)
rest
A
hmax
Equilibriumposition
The total mechanical energy at this position is
Max Epg
Ek = 0
kpm EEEg
(max)gpE The mass only has gravitational potential energy
![Page 44: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/44.jpg)
rest
vmax
A
hmax
Equilibriumposition
Max Epg
Ek = 0
As the pendulum falls, it speeds up.
When it reaches the equilibrium position, the mass has attained its maximum speed.
![Page 45: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/45.jpg)
rest
vmax
A
hmax
Equilibriumposition
Max Epg
Ek = 0
Since the speed is at a maximum, the mass has maximum kinetic energy.
Max Ek
![Page 46: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/46.jpg)
rest
vmax
A
hmax
Equilibriumposition
Max Epg
Ek = 0
However, since the mass has reached the equilibrium position, it has no gravitational potential energy.
Ref h: h = 0
Max Ek
Epg = 0
![Page 47: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/47.jpg)
rest
vmax
A
hmax
Equilibriumposition
Max Epg
Ek = 0
Max Ek
Epg = 0
The total mechanical energy at the equilibrium position is
kpm EEEg
(max)kE The mass only has kinetic energy
![Page 48: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/48.jpg)
rest
vmax
A
hmax
A
Equilibriumposition
Max Epg
Ek = 0
Max Ek
Epg = 0
Max Epg
Ek = 0rest
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.
Maximum displacement
![Page 49: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/49.jpg)
Energy Formulas of Pendulum Motion
To derive an energy formula for pendulum motion, we compare maximum displacement with equilibrium position.
rest
vmax
A
hmax
Equilibriumposition
Max Epg
Ek = 0
Max Ek
Epg = 0
![Page 50: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/50.jpg)
If this is an ideal pendulum, then there is no energy lost due to friction or air resistance. rest
vmax
A
hmax
Equilibriumposition
Max Epg
Ek = 0
Max Ek
Epg = 0Thus, the total mechanical energy must remain constant.
i.e.
Em (top) = Em (bottom)
![Page 51: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/51.jpg)
rest
vmax
A
hmax
Equilibriumposition
Max Epg
Ek = 0
Max Ek
Epg = 0
Em (top) = Em (bottom)
max(max) kp EEg
![Page 52: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/52.jpg)
rest
vmax
A
hmax
Equilibriumposition
Max Epg
Ek = 0
Max Ek
Epg = 0
Em (top) = Em (bottom)
max(max) kp EEg
2maxmax 2
1mvmgh
![Page 53: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/53.jpg)
rest
vmax
A
hmax
Equilibriumposition
Max Epg
Ek = 0
Max Ek
Epg = 0
Em (top) = Em (bottom)
max(max) kp EEg
2maxmax 2
1mvmgh
2maxmax 2
1vgh
This is not on the formula sheet, and so you must derive this each time.
![Page 54: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/54.jpg)
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.
![Page 55: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/55.jpg)
Find the acceleration due to gravity:
g
LT 2
g
LT
2
2
2
g
LT
2
![Page 56: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/56.jpg)
g
LT
2
22
2
g
LT
g
LT
2
2
4
![Page 57: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/57.jpg)
g
LT
2
2
4
LTg 22 4
2
24
T
Lg
![Page 58: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/58.jpg)
2
24
T
Lg
2
2
9.2
80.04
= 3.755 m/s2
![Page 59: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/59.jpg)
Find the maximum speed:
Em (top) = Em (bottom)
max(max) kp EEg
rest
vmax
A
hmax
Equilibriumposition
Max Epg
Ek = 0
Max Ek
Epg = 0
![Page 60: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/60.jpg)
Em (top) = Em (bottom)
max(max) kp EEg
2maxmax 2
1mvmgh
![Page 61: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/61.jpg)
Em (top) = Em (bottom)
max(max) kp EEg
2maxmax 2
1mvmgh
2maxmax 2
1vgh
![Page 62: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/62.jpg)
2maxmax 2
1vgh
2maxmax2 vgh
![Page 63: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/63.jpg)
2maxmax 2
1vgh
2maxmax2 vgh
maxmax 2ghv
![Page 64: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/64.jpg)
maxmax 2ghv
015.0755.32
= 0.34 m/s
![Page 65: PHYS 20 LESSONS Unit 6: Simple Harmonic Motion Mechanical Waves Lesson 5: Pendulum Motion as SHM.](https://reader036.fdocuments.in/reader036/viewer/2022062515/56649cb05503460f949754e4/html5/thumbnails/65.jpg)
Practice Problems
Try these problems in the Physics 20 Workbook:
Unit 6 p. 6 #11