Classical Mechanics
Lecture 22
Today’s Concept:
Simple Harmonic Mo/on: Mo#on of a Pendulum
Mechanics Lecture 8, Slide 1
Your Comments
I don't like radians. When people say pi, I say 180 degrees. Or even more pro, I say 200 gon.
Torsion is a new concept and another explana/on so I can try to understand it beIer would be helpful.
Who's the genius who decided omega should have two meanings? Did they run out of Greek leIers? Why don't they fly over there and get some more? It would probably help boost their economy at this point.
I am finding it difficult to understand how the moment of iner/a and the radius are both being used in the equa/on. Isn't the moment of iner/a dependent on the radius?
Is the period propor/onal to Rcm then?
talking about harmonic mo/ons , Lets all dance "GANGNOM style " , its a perfect prac/cal example !
Why my roommate keeps on ramming Grape Fantas into my mini fridge is way more confusing than anything that we've covered this year
I think Jason should do a rap on physics to help us learn.
Mechanics Lecture 8, Slide 2
Don’t Panic!
http://www.youtube.com/watch?v=7rOMGIbY-9s
http://www.youtube.com/watch?v=Iexp-OwOs0M
“There is a theory which states that if ever anybody discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable.” Text
“There is another theory which states that this has already happened.”
Douglas Adams Your instructors are great fans of H2G2. The original radio version is the best. I s/ll have my casseIe tapes recorded off the air in 1978.
Mechanics Lecture 8, Slide 3
I want to know why the answer to life is 42!
Drill a hole through the earth and jump in – what happens?
Just for fun – you don’t need to know this.
I want to know why the answer to life is 42!
Drill a hole through the earth and jump in – what happens?
You will oscillate like a mass on a spring with a period of 84 minutes. It takes 42 minutes to come out the other side! k = mg/RE
Mechanics Lecture 8, Slide 5
I want to know why the answer to life is 42!
Drill a hole through the earth and jump in – what happens?
You will oscillate like a mass on a spring with a period of 84 minutes. It takes 42 minutes to come out the other side! The hole doesn’t even have to go through the middle – you get the same answer anyway as long (as there is no fric/on).
Mechanics Lecture 8, Slide 6
I want to know why the answer to life is 42!
This is also the same period of an object orbi/ng the earth right at ground level. Just for fun – you don’t need to know this.
Mechanics Lecture 8, Slide 7
Torsion Pendulum
wire
τ
θ
I
Q: In the prelecture the equa/on for restoring torque is given as T=-­‐k*theta in clockwise direc/on..so if the restoring torque is in counter clockwise direc/ons then would T be posi/ve? Mechanics Lecture 8, Slide 8
CheckPoint
A torsion pendulum is used as the /ming element in a clock as shown. The speed of the clock is adjusted by changing the distance of two small disks from the rota/on axis of the pendulum. If we adjust the disks so that they are closer to the rota/on axis, the clock runs:
A) Faster B) Slower
Small disks
Mechanics Lecture 8, Slide 9
CheckPoint
If we adjust the disks so that they are closer to the rota/on axis, the clock runs
A) Faster B) Slower
A) The moment of iner/a decreases, so the angular frequency increases, which makes the period shorter and thus the clock faster.
B) T = 2pi * sqrt(I/MgRcm). If Rcm decreases, T will increase, making the clock run slower.
Mechanics Lecture 8, Slide 10
Pendulum
θ
For
small θ
RCM
XCM
Mg
θ
RCM
XCM
arc-­‐length
= RCM θ
Mechanics Lecture 8, Slide 11
The Simple Pendulum
pivot
RCM
The simple case
θ
θ
CM
L
The general case
Mechanics Lecture 8, Slide 12
CheckPoint
A simple pendulum is used as the /ming element in a clock as shown. An adjustment screw is used to make the pendulum shorter (longer) by moving the weight up (down) along the shap that connects it to the pivot. If the clock is running too fast, the weight needs to be moved A) Up B) Down
Adjustment screw
Mechanics Lecture 8, Slide 13
CheckPoint
If the clock is running too fast, the weight needs to be moved A) Up B) Down
If the clock is running too fast then we want to reduce it's period, T, and to do that we need to increase omega, the frequency it moves with and to do that we need the posi/on of the center of mass to be further from the pivot, which is achieved by moving the weight down. Mechanics Lecture 8, Slide 14
The Stick Pendulum
pivot
RCM
θ
CM
M
Same period
Mechanics Lecture 8, Slide 15
CheckPoint
Case 1
Case 2
m
m
m
In Case 1 a s/ck of mass m and length L is pivoted at one end and used as a pendulum. In Case 2 a point par/cle of mass m is aIached to the center of the same s/ck. In which case is the period of the pendulum the longest?
A) Case 1 B) Case 2 C) Same
C is not the right answer.
Lets work through it
Mechanics Lecture 8, Slide 16
Case 1
Case 2
m
In Case 1 a s/ck of mass m and length L is pivoted at one end and used as a pendulum. In Case 2 a point par/cle of mass m is aIached to a string of length L/2?
In which case is the period of the pendulum longest?
A) Case 1 B) Case 2 C) Same
T = 2π
�
2
3L
g
T = 2π
�
1
2L
g
Mechanics Lecture 8, Slide 17
T2
m
T1
m
m
Suppose you start with 2 different pendula, one having period T1 and the other having period T2. T1 > T2
Now suppose you make a new pendulum by hanging the first two from the same pivot and gluing them together.
What is the period of the new pendulum?
A) T1 B) T2 C) In between
Mechanics Lecture 8, Slide 18
Case 1
Case 2
m
m
m
In Case 1 a s/ck of mass m and length L is pivoted at one end and used as a pendulum. In Case 2 a point par/cle of mass m is aIached to the center of the same s/ck. In which case is the period of the pendulum the longest?
A) Case 1 B) Case 2 C) Same
Now lets work through it in detail
Mechanics Lecture 8, Slide 19
Case 2
Case 1
m
m
m
Lets compare for each case. Mechanics Lecture 8, Slide 20
Case 2
Case 1
m
m
m
Lets compare for each case. (A)
(B)
(C)
Mechanics Lecture 8, Slide 21
So we can work out Case 1
Case 2
m
m
m
In which case is the period longest?
A) Case 1
B) Case 2
C) They are the same
Mechanics Lecture 8, Slide 22
The Small Angle Approximation
- Exact expression
RCM
XCM
arc-­‐length
= RCM θ
% difference between θ and sinθ
θ
Angle (degrees)
Mechanics Lecture 8, Slide 23
Clicker Question
A pendulum is made by hanging a thin hoola-­‐hoop of diameter D on a small nail. What is the angular frequency of oscilla/on of the hoop for small displacements? (ICM = mR2 for a hoop)
A)
B)
pivot (nail)
D
C)
Mechanics Lecture 8, Slide 24
The angular frequency of oscilla/on of the hoop for small displacements will be given by
Use parallel axis theorem: I = ICM + mR2
= mR2 + mR2 = 2mR2
pivot (nail)
R
X
So
CM
m
Mechanics Lecture 8, Slide 25