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

1

not simple

simple

The simple pendulumEnergy approach

mg

mT

2

The simple pendulumEnergy approach

mg

mT

3

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

7

mg

m

LT

REVIEW PENDULUM

8

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

mg

m

LT

REVIEW PENDULUM

9

0 0

0

2* 0.1( ) ;

ii t i tt e eC C C 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

11

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

13

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

14

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!

15

A, chosen to fit initial conditions:

x(0) = x0 and v(0) = v0

x

m

mk

k

Square and add:

Divide:

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2 arbitrary constants(A, because 2nd order linear differential equation

17

• 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

18

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!

20

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

THE LC CIRCUIT

L

CIq

Kirchoff’s law (not Newton this time)