Chapter 13
Periodic Motion
Special Case:
Simple Harmonic Motion (SHM)
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
Simple Harmonic Motion (SHM)
• 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)
Simple Harmonic Motion (SHM)
• 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
The ‘Reference Circle’
P = mass on spring: x(t)
Q = point on reference circle
A = amplitude of x(t)
(motion of P)
A = radius of reference circle (motion of Q)
The ‘Reference Circle’
P = mass on spring: x(t)
Q = point on reference circle
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