Chapter 13 Periodic Motion. Special Case: Simple Harmonic Motion (SHM)

Chapter 13

Periodic Motion

Special Case:

Simple Harmonic Motion (SHM)


• Only valid for small oscillation amplitude

• But SHM approximates a wide class of periodic motion, from vibrating atoms to vibrating tuning forks...

Starting Model for SHM:

mass m attached to a spring

DemonstrationDemonstration


• x = displacement of mass m from equilibrium

• Choose coordinate x so that x = 0 is the equilibrium position

• If we displace the mass m, a restoring force F acts on m to return it to equilibrium (x=0)


• By ‘SHM’ we mean Hooke’s Law holds:for small displacement x (from equilibrium),

F = – k x ma = – k x

• negative sign: F is a ‘restoring’ force(a and x have opposite directions)

Demonstration: spring with force meterDemonstration: spring with force meter

What is x(t) for SHM?

• We’ll explore this using two methods

• The ‘reference circle’:x(t) = projection of certain circular motion

• A little math:Solve Hooke’s Law kx

dt

xdmF

2

2

The ‘Reference Circle’

P = mass on spring: x(t)

Q = point on reference circle

P = projection of Q onto the screen




A = amplitude of x(t)

(motion of P)

A = radius of reference circle (motion of Q)




f = oscillation frequency of P = 1/T (cycles/sec)

= angular speed of Q = 2/T (radians/sec)

= 2f

What is x(t) for SHM?

P = projection of Q onto screen.

We conclude the motion of P is:

See additional notes or Fig. 13-4 for See additional notes or Fig. 13-4 for

tt

Atx

)(

cos)(

Alternative: A Little Math

• Solve Hooke’s Law:

• Find a basic solution:

kxdt

xdmF

2

2

m

k

tAtx

)cos()(

Solve for x(t)Solve for x(t)

• v = dx/dt v = 0 at x = A |v| = max at x = 0

• a = dv/dt|a| = max at x = A a = 0 at x = 0

Tmk

tAtx

/2/

)3/cos()(

See notes on x(t), v(t), a(t)See notes on x(t), v(t), a(t)

0

0

0

arctan

cos)cos()(

x

v

AxtAtx

Show expression for Show expression for

• going from 1 to 3,increase one of A, m, k

• (a) change A : same T

• (b) larger m : larger T

• (c) larger k : shorter T

Tmk

tAtx

/2/

)0cos()(

Do demonstrations illustrating (a), (b), (c)Do demonstrations illustrating (a), (b), (c)

Summary of SHMfor an oscillator of mass m

• A = amplitude of motion, = ‘phase angle’

• A, can be found from the values of x and dx/dt at (say) t = 0

m

ktAtx

kxF

)cos()(

Energy in SHM

• As the body oscillates, E is continuously transformed from K to U and back again

22

2

1

2

1kxmv

UKE

See notes on vmaxSee notes on vmax

E = K + U = constant

Do Exercise 13-17Do Exercise 13-17

Summary of SHM

• x = displacement from equilibrium (x = 0)

• T = period of oscillation

• definitions of x and depend on the SHM

Tf

xdt

xd

tAtx

22

)cos()(

22

2

Different Types of SHM

• horizontal (have been discussing so far)

• vertical (will see: acts like horizontal)

• swinging (pendulum)

• twisting (torsion pendulum)

• radial (example: atomic vibrations)

Horizontal SHM

Horizontal SHM

• Now show: a vertical spring acts the same,if we define x properly.

spring-block

22

2

m

k

xdt

xd

Vertical SHM

Show SHM occurs with x defined as shownShow SHM occurs with x defined as shown Do Exercise 13-25Do Exercise 13-25

‘Swinging’ SHM: Simple Pendulum

Derive for small xDerive for small x

pendulum simple

22

2

L

g

xdt

xd

Do Pendulum DemonstrationsDo Pendulum Demonstrations

‘Swinging’ SHM: Physical Pendulum

Derive for small Derive for small Do Exercises 13-39, 13-38Do Exercises 13-39, 13-38

pendulum physical

22

2

I

mgd

dt

d

Angular SHM:Torsion Pendulum (fiber-disk)

Application: Cavendish experiment (measures gravitational constant G). The fiber twists when blue masses gravitate toward red masses

Angular SHM:Torsion Pendulum (coil-wheel)

Derive for small Derive for small

pendulumtorsion

22

2

I

dt

d

Radial SHM:Atomic Vibrations

Show SHM results for small x (where r = R0+x)Show SHM results for small x (where r = R0+x)

Announcements

• Homework Sets 1 and 2 (Ch. 10 and 11): returned at front

• Homework Set 5 (Ch. 14):available at front, or on course webpages

• Recent changes to classweb access:see HW 5 sheet at front, or course webpages

Damped Simple Harmonic Motion

See transparency on damped block-springSee transparency on damped block-spring

SHM: Ideal vs. Damped

• Ideal SHM:

• We have only treated the restoring force:

• Frestoring = – kx

• More realistic SHM:

• We should add some ‘damping’ force:

• Fdamping = – bv

Demonstration of damped block-springDemonstration of damped block-spring

Damping Force

• this is the simplest model:

• damping force proportional to velocity

• b = ‘damping constant’ (characterizes strength of damping)

dt

dxbbvF damping

SHM: Ideal vs. Damped

• In ideal SHM, oscillator energy is constant:

E = K + U , dE/dt = 0

• In damped SHM, the oscillator’s energy decreases with time:

E(t) = K + U , dE/dt < 0

Energy Dissipation in Damped SHM

• Rate of energy loss due to damping:

0

)(2

damping

bv

vbv

vFdt

dE

What is x(t) for damped SHM?

• We get a new equation of motion for x(t):

• We won’t solve it, just present the solutions.

dt

dxbkx

dt

xdm

FFma

2

2

dampingrestoring

Three Classes of Damping, b

• small (‘underdamping’)

• intermediate (‘critical’ damping)

• large (‘overdamping’) mkb

mkb

mkb

2

2

2

dt

dxbkx

dt

xdm

2

2

‘underdamped’ SHM

‘underdamped’ SHM:damped oscillation, frequency ´

2

2

)2/(

4

)cos()(

2

m

b

m

k

tAetx

mkb

tmb

‘underdamping’ vs. no damping

• underdamping:

• no damping (b=0):

m

k

m

b

m

k2

2

4

‘critical damping’:decay to x = 0, no oscillation

• can also view this ‘critical’ value of b as resulting from oscillation ‘disappearing’:

tmbeBtAtx

mkb

)2/()()(

2

See sketch of x(t) for critical dampingSee sketch of x(t) for critical damping

2

2

40

m

b

m

k

‘overdamping’: slower decay to x = 0, no oscillation

)frequency! a( 4

)cosh()(

2

2

2

over

over)2/(

m

k

m

b

tAetx

mkb

tmb

See sketch of x(t) for overdampingSee sketch of x(t) for overdamping

Application

• Shock absorbers:

• want critically damped (no oscillations)

• not overdamped(would have aslow response time)

Forced Oscillations

(Forced SHM)

Forced SHM• We have considered the presence of a

‘damping’ force acting on an oscillator:

Fdamping = – bv

• Now consider applying an external force:

Fdriving = Fmax cosdt

Forced SHM

• Every simple harmonic oscillator has a natural oscillation frequency

• ( if undamped, ´ if underdamped)

• By appling Fdriving = Fmax cosdt we force the oscillator to oscillate at the frequency d

(can be anything, not necessarily or ´)

What is x(t) for forced SHM?

• We get a new equation of motion for x(t):

• We won’t solve it, just present the solution.

tFdt

dxbkx

dt

xdm

FFFma

dmax2

2

drivingdampingrestoring

cos

x(t) for Forced SHM

• If you solve the differential equation, you find the solution (at late times, t >> 2m/b)

2d

22d

max

d

)()(

)cos()(

bkm

FA

tAtx

Amplitude A(d)

• Shown (for = 0):A(d) for different b

• larger b: smaller Amax

• Resonance:Amax occurs at R, near the natural frequency,= (k/m)1/2

Do Resonance Demonstrations Do Resonance Demonstrations

Resonance Frequency (R)

• Amax occurs at d=R (where dA/dd=0):

2d

22d

max

)()( bkm

FA

2

2

R 2m

b

m

k

natural, underdamped, forced: > ´ > R

• natural frequency:

• underdamped frequency:

• resonance frequency: 2

2

R

2

2

2

4

m

b

m

k

m

b

m

k

m

k

Introduction toLRC Circuits

(Electromagnetic Oscillations)

See transparency on LRC circuitSee transparency on LRC circuit

Electric Quantity Counterpart

• charge Q(t) x(t)

• current I = dQ/dt v = dx/dt(moving charge)(generates a magnetic field, B)

Electrical Concepts

• electric charge: Q

• current (moving charge): I = dQ/dt

• resistance (Q collides with atoms): R

• voltage (pushes Q through wire): V = RI

Voltage (moves charges)

• resistance R causes charge Q to lose energy:

V = RI

• (voltage = potential energy per unit charge)

• C and L also cause energy (voltage) changes

Circuit Element (Voltage)

• R = resistance VR = RI(Q collides with atoms)

• C = capacitance VC = Q/C(capacity to store Q on plate)

• L = inductance VL = L(dI/dt)(inertia towards changes in I)

0

0)(

RCL VVV

PE

Change in Voltage = Change in Energy

• voltage = potential energy per unit charge

• recall, around a closed loop:

• Which looks like:

01

01

0

2

2

dt

dQRQ

Cdt

QdL

RIQCdt

dIL

VVV RCL

02

2

dt

dxbkx

dt

xdm

Circuit Element Counterpart

• 1/C = 1/capacitance k• L = inductance m• R = resistance b

• (Extra Credit: Exercise 31-35)• Use this table to write our damped SHM as

damped electromagnetic oscillations

In the LRC circuit, Q(t) acts just like x(t)!

underdamped, critically damped, overdamped

Driven (and resonance): Vdriving = Vmax cosdt

Documents

Chapter 13 Periodic Motion. Special Case: Simple Harmonic Motion (SHM)