PRIOR READING: Main 1.1, 2.1 Taylor 5.1, 5.2

SIMPLE HARMONIC MOTION:���NEWTON’S LAW

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0

t '0

t

! = d" '2IE #mgLcm 1# cos" '( )( )"0

"

!

The simple pendulum Energy approach θ

mg

m T

!r; !r = L

1

t = d! '2mgLcmmLcm

2 1" cos!max " 1" cos! '( )( )!0

!

#

t = d! '2gLcm

cos! '" cos!max( )!0

!

#

The simple pendulum Energy approach θ

mg

m T

Lcm = L

2

t = d! '2gLcm

cos! '" cos!max( )!0

!

#

t = d! 'gL

!max2 "! '2( )!0

!

# = 1$ 0

arcsin ! '!max !0

!

!t = arcsin ""max

# arcsin "0"max

$! "# $#

!" t( ) = "max sin #t +$( )

! 0 "gL

angular frequency

Newton !! = I

!"""

Known torque !! =!r "

!F# = ?

!! = Lmgsin" #̂ I

!""! = mL2 ""! •̂

!!! = "

gLsin!

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 …

sin! = ! " !3

3!+! 5

5!+ ...

The simple pendulum θ

mg

m T

!r; !r = L

3

!!! = "gL

sin! #

!!! = "gL! or !!! + g

L! = 0

θ

mg

m

L

T

The simple pendulum in the limit of small angular displacements

Compare with !!x + k

mx = 0

What is θ(t) such that the above equation is obeyed? θ 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. 4

C, p are unknown (for now) constants, possibly complex

!(t) = Cept

Substitute: p2! + gL! = 0 p = ±i g

L= ±i!0

REVIEW PENDULUM

!!! +

gL! = 0

!!!(t) = p2Cept = p2!(t)

p is now known (but C is not!). Note that ω0 is NOT a new quantity! It is just a rewriting of old ones - partly shorthand, but also "ω" means "frequency" to physicists!

θ

mg

m

L

T

5

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.

!(t) = Cei"0t +C 'e# i"0t

!(t) = Cei"0t +C*e# i"0t Simple harmonic motion

θ

mg

m

L

T

REVIEW PENDULUM

6

!(0) =Cei"0 01! +C *e!i"0 0

1!

0 =C +C*= 2Re[C]" Re[C]= 0

!(t) = Cei"0t +C*e# i"0tθ

mg

m

L T

REVIEW PENDULUM

7

Re[C], Im[C] chosen to fit initial conditions. Example: ���

θ(0) = 0 rad and dθdt(0) = 0.2 rad/sec

!!(0) = i"0Cei"0 0

1! ! i"0C *e

!i"0 0

1!

0.2rad / s = i"0 C !C *( ) = i"0 2i Im[C]

" Im[C]= 0.2!2"0

=0.1"0

C = 0+ i 0.1!0

cartesian!"# $#

=0.1!0

ei"2

polar!"#

; C*= ?

!(t) = 0.1!0

ei!2ei!0t + 0.1

!0

e!i!2e!i!0t

!(t) = 0.1!0

ei !0t+! /2( ) + e!i !0t+! /2( )( )

!(t) = 0.1!0

2cos !0t +! / 2( )( )

!(t) = 0.2!0A!

cos !0t +! / 2"!

"

#$

%

&'

θ

mg

m

L T

REVIEW PENDULUM

8

!(t) =Cei"0t +C*e!i"0t; C = 0.1"0

ei#2

!(t) = Acos "0t +#( )

!(t) = Bp cos"0t + Bq sin"0t

!(t) = C exp i"0t( ) +C *exp #i"0t( )

!(t) = Re Dexp i"0t( )#$ %&

Remember, all these are equivalent forms. All of them have a known ω0=(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!

9

The simple pendulum ("simple" here means a point mass; your lab deals with a plane pendulum)

!

"(t) = "max cos #0t + $( ) !!! = "

gL!

!

"0 =gL

!

T = 2" Lg

Period does not depend on θmax, φ

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.)

θ

mg

m

L

T

10

x

m

m k

k

Free, undamped oscillators – other examples

m!!x = !kx

No friction

θ

mg

m T

!r; !r = L

!!! " #

gL!

L

C I q

!!q = ! 1

LCq

!!! +"02! = 0

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

12

Newton F(x) = m!!x

Particular type of force. m, k known

F(x) = !kx

!kx = m!!x

!!x + kmx = 0

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

x

m

m k

k

Linear, 2nd order differential equation

REVIEW MASS ON IDEAL SPRING

13

x

m

m k

k C, p are unknown (for now) constants, possibly complex

x(t) = Cept

Substitute: p2x + kmx = 0 p = ±i k

m= ±i!0

REVIEW MASS ON IDEAL SPRING

!!x + k

mx = 0

!!x(t) = p2Cept = p2x(t)

p is now known. Note that ω0 is NOT a new quantity! It is just a rewriting of old ones - partly shorthand, but also “ω” means “frequency” to physicists!

14

A, φ chosen to fit initial conditions: ���x(0) = x0 and v(0) = v0

x

m

m k

k x0 = Acos!v0 = "#0Asin!

x(t) = Acos !0t +"( )

x02 +

v02

!02 = A

2 cos2" + sin2"( ) = A2Square and add:

!v0"0x0

= tan#Divide:

15

x(t) = Acos !0t +"( )

x(t) = x02 +

v02

!02 cos

kmt + arctan "v0

!0x0

#

$%&

'(#

$%&

'(

x(t) = Acos! cos"0t # Asin! sin"0t

x(t) = Aei!

2ei"0t + Ae

# i!

2e# i"0t

x(t) = Re Aei!ei"0t#$ %&

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

16

• A, φ are unknown constants - must be determined from initial conditions • ω0, in principle, is known and is a characteristic of the physical system

x(t) = Acos !0t +"( )

dxdt

! !x(t) = "#0Asin #0t +$( )

d 2xdt 2

! !!x(t) = "#02Acos #0t +$( )

= "#02x(t)

Position:

Velocity:

Acceleration:

This type of pure sinusoidal motion with a single frequency is called SIMPLE HARMONIC MOTION

17

THE LC CIRCUIT

!!q = ! 1

LCq

L

C I q

VL +VC = 0 Kirchoff’s law (not Newton this time)

VL = L

dIdt= L d

2qdt 2

= L!!q

VC =qC