PRIOR READING: Main 1.1, 2.1 Taylor 5.1, 5.2 SIMPLE HARMONIC MOTION: NEWTON’S LAW http://www.myoops.org/twocw/mit/NR/rdonlyres/Physics/8-012Fall-2005/7CCE46AC-405D-4652-A724-64F831E70388/0/chp_physi_pndul 1 not simple simple
Dec 21, 2015
PRIOR READING:Main 1.1, 2.1Taylor 5.1, 5.2
SIMPLE HARMONIC MOTION:NEWTON’S LAW
http://www.myoops.org/twocw/mit/NR/rdonlyres/Physics/8-012Fall-2005/7CCE46AC-405D-4652-A724-64F831E70388/0/chp_physi_pndulm.jpg
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not simple
simple
Newton
Known torque
This is NOT a restoring force proportional to displacement (Hooke’s law motion) in general, but IF we consider small motion, IT IS! Expand the sin series …
The simple pendulum
mg
mT
4
mg
m
L
T
The simple pendulumin the limit of small angular displacements
What is (t) such that the above equation is obeyed? is a variable that describes positiont is a parameter that describes time"dot" and "double dot" mean differentiate w.r.t. timeg, L are known constants, determined by the system. 5
C, p are unknown (for now) constants, possibly complex
Substitute:
REVIEW PENDULUM
p is now known (but C is not!). Note that 0 is NOT a new quantity! It is just a rewriting of old ones - partly shorthand, but also "" means "frequency" to physicists!
mg
m
L
T
6
TWO possibilities …. general solution is the sum of the two and it must be real (all angles are real).
If we force C' = C* (complex conjugate of C), then x(t) is real, and there are only 2 constants, Re[C], and Im[C]. A second order DEQ can determine only 2 arbitrary constants.
Simple harmonic motion
mg
m
L
T
REVIEW PENDULUM
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mg
m
LT
REVIEW PENDULUM
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Re[C], Im[C] chosen to fit initial conditions. Example:
(0) = rad and ddt(0) = rad/sec
0 000 0
1
0
0
0
0
1
0
(0) *
0.2 / * 2 Im[ ]
0.2 0.1Im[ ]
2
i ii e i e
rad s i i i C
C
C C
C C
2
0 0
0.1 0.10 ; * ?
i
cartesian polar
CiC e
Remember, all these are equivalent forms. All of them have a known 0=(g/L)1/2, and all have 2 more undetermined constants that we find … how?
Do you remember how the A, B, C, D constants are related? If not, go back and review until it becomes second nature!
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The simple pendulum("simple" here means a point mass; your lab deals with a plane pendulum)
Period does not depend on max,
simple harmonic motion( potential confusion!! A “simple” pendulum does not always execute “simple harmonic motion”; it does so
only in the limit of small amplitude.)
mg
m
L
T
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x
m
mk
k
Free, undamped oscillators – other examples
No friction
mg
mT
L
CIq
Common notation for all
• The following slides simply repeat the previous discussion, but now for a mass on a spring, and for a series LC circuit
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Newton
Particular type of force.m, k known
What is x(t) such that the above equation is obeyed?x is a variable that describes positiont is a parameter that describes time"dot" and "double dot" mean differentiate w.r.t. timem, k are known constants
x
m
mk
k
Linear, 2nd order differential equation
REVIEW MASS ON IDEAL SPRING
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x
m
mk
k
C, p are unknown (for now) constants, possibly complex
Substitute:
REVIEW MASS ON IDEAL SPRING
p is now known. Note that 0 is NOT a new quantity! It is just a rewriting of old ones - partly shorthand, but also “” means “frequency” to physicists!
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• A, are unknown constants - must be determined from initial conditions• , in principle, is known and is a characteristic of the physical system
Position:
Velocity:
Acceleration:
This type of pure sinusoidal motion with a single frequency is called SIMPLE HARMONIC MOTION
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THE LC CIRCUIT DIFFERENTIAL EQUATION
L
CIq Kirchoff’s law
(not Newton this time)
mx x
k
0q
LqC
Same differential equation as the SHO spring!
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What is inductance??
dL
dI
It is how much magnetic flux is created in the inductor coil by a given current I, in a wire.
What is the voltage change across an inductor?
A voltage change occurs WHEN there is change in magnetic flux (i.e. some of the energy is ‘converted’ to a magnetic field)
L
d d dI dIV L
dt dI dt dt