0. SIMPLE HARMONIC MOTION: NEWTON’S LAW
PH421:Oscillations F09
4/17/2017
SIMPLE HARMONIC MOTION: NEWTON’S LAW
simple
not simple
PRIOR READING:
Main 1.1, 2.1
Taylor 5.1, 5.2
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1. The simple pendulum Energy approach q T m mg PH421:Oscillations F09
4/17/2017
The simple pendulum
Energy approach q mg m T
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2. The simple pendulum Energy approach q T m mg PH421:Oscillations F09
4/17/2017
The simple pendulum
Energy approach q mg m T
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3. The simple pendulum q T m Newton Known torque mg
PH421:Oscillations F09
4/17/2017
The simple pendulum q mg m T
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 …
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4. The simple pendulum in the limit of small angular displacements L q T
PH421:Oscillations F09
4/17/2017 q mg m L T
The simple pendulum
in the limit of small angular displacements
What is (t) such that the above equation is obeyed?
 is a variable that describes position
t is a parameter that describes time
"dot" and "double dot" mean differentiate w.r.t. time
g, L are known constants, determined by the system.
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5. C, p are unknown (for now) constants, possibly complex
PH421:Oscillations F09
REVIEW PENDULUM
4/17/2017 q mg m L T
C, p are unknown (for now) constants, possibly complex
Substitute:
p is now known (but C is not!). Note that w0 is NOT a new quantity! It is just a rewriting of old ones - partly shorthand, but also "w" means "frequency" to physicists!
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6. Simple harmonic motion
PH421:Oscillations F09
REVIEW PENDULUM
4/17/2017 q mg m L T
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
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7. PH421:Oscillations F09
REVIEW PENDULUM
4/17/2017 q mg m L T
Re[C], Im[C] chosen to fit initial conditions. Example: q(0) = 0 rad and dqdt(0) = 0.2 rad/sec
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8. REVIEW PENDULUM L q T m mg PH421:Oscillations F09 4/17/2017
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9. PH421:Oscillations F09
4/17/2017
Remember, all these are equivalent forms. All of them have a known w0=(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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10. The simple pendulum Period does not depend on qmax, f L
PH421:Oscillations F09
4/17/2017 q mg m L T
The simple pendulum
("simple" here means a point mass; your lab deals with a plane pendulum)
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.)
Period does not depend on qmax, f
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11. PH421: Oscillations; do not distribute
Free, undamped oscillators – other examples
11/12/07 x m k L
No friction I C q q mg m T
Common notation for all
Lecture 5/6 - damped oscllations

12. PH421:Oscillations F09
4/17/2017
• 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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13. REVIEW MASS ON IDEAL SPRING
PH421:Oscillations F09
REVIEW MASS ON IDEAL SPRING
4/17/2017
Newton x m k
Particular type of force.
m, k known
Linear, 2nd order differential equation
What is x(t) such that the above equation is obeyed?
x is a variable that describes position
t is a parameter that describes time
"dot" and "double dot" mean differentiate w.r.t. time
m, k are known constants
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14. REVIEW MASS ON IDEAL SPRING
PH421:Oscillations F09
REVIEW MASS ON IDEAL SPRING
4/17/2017 x m k
C, p are unknown (for now) constants, possibly complex
Substitute:
p is now known. Note that w0 is NOT a new quantity! It is just a rewriting of old ones - partly shorthand, but also “w” means “frequency” to physicists!
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15. A, f chosen to fit initial conditions: x(0) = x0 and v(0) = v0
PH421:Oscillations F09
4/17/2017 x m k
A, f chosen to fit initial conditions: x(0) = x0 and v(0) = v0
Square and add:
Divide:
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16. (A, f) because 2nd order linear differential equation
PH421:Oscillations F09
4/17/2017
2 arbitrary constants
(A, f) because 2nd order linear differential equation
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17. Position: Velocity: Acceleration:
PH421:Oscillations F09
4/17/2017
Position:
• A, f are unknown constants - must be determined from initial conditions
• w0, in principle, is known and is a characteristic of the physical system
Velocity:
Acceleration:
This type of pure sinusoidal motion with a single frequency is called SIMPLE HARMONIC MOTION
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18. PH421: Oscillations; do not distribute
THE LC CIRCUIT DIFFERENTIAL EQUATION
11/12/07 L C I q
Kirchoff’s law
(not Newton this time)
Same differential equation as the SHO spring!
Lecture 5/6 - damped oscllations

19. What is inductance??
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)

20. PH421: Oscillations; do not distribute
THE LC CIRCUIT
11/12/07 L C I q
Kirchoff’s law
(not Newton this time)
Lecture 5/6 - damped oscllations