Download - Chapter 13 periodic motion
![Page 1: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/1.jpg)
Chapter 13 periodic motion
Collapse of the Tacoma Narrowssuspension bridge in America in 1940(p 415)
![Page 2: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/2.jpg)
SHMSHM
dynamicsdynamics kinematicskinematics
Dynamic equation
Dynamic equation
Circle of reference
Circle of reference
Kinematicsequation
Kinematicsequation
oscillationoscillation
EnergyEnergy Superposition of shm
Superposition of shm
Damped oscillation
Damped oscillation
Forcedoscillation
Forcedoscillation
![Page 3: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/3.jpg)
periodic motion / oscillationrestoring forceamplitude
cycleperiod
frequencyangular frequencysimple harmonic motionharmonic oscillator
circle of referencephasor
phase anglesimple pendulum
Key terms:
![Page 4: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/4.jpg)
physical pendulumDampingDamped oscillationCritical dampingoverdampingunderdampingdriving forceforced oscillationnatural angular frequencyresonancechaotic motionchaos
![Page 5: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/5.jpg)
Ideal model:
A) spring mass system
1) dynamic equation
xkdt
xdm
xkamF
2
2
02
2
xm
k
dt
xd
)tcos(Ax
022
2
xdt
xd
m
k
2
T
§1 Dynamic equation§1 Dynamic equation
![Page 6: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/6.jpg)
Small angle approximation sin
02
2
L
g
dt
d
B) The Simple Pendulum
2
2
2
2
sin
sin
dt
dLg
Lsdt
sdmmg
maF tt
)cos(0 t
l
g2
![Page 7: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/7.jpg)
mg
d
O
I
mgdI
mgd
dt
d
dt
dImgd
I
2
2
2
2
2
sin
C) physical pendulum
![Page 8: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/8.jpg)
Example: Tyrannosaurus rex and physical pendulum
the walking speed of tyrannosaurus rex can be estimated from its leg length L and its stride length s
![Page 9: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/9.jpg)
Conclusion:Equation of SHM
0xdt
xd 22
2
Solution: )tcos(Ax
![Page 10: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/10.jpg)
Solution:
2
'
g r
GmMF ,Rr M
R
rr
3
4
R34
MM
3
33
3
'
RO
gF r
M
rR
GmMF
3g
2
2
3 dt
rdmr
R
GmM 0r
R
GM
dt
rd32
2
2
Example:A particle dropped down a hole that extends from one side of the earth, through its center, to the other side. Prove that the motion is SHM and find the period.
![Page 11: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/11.jpg)
mTkM 22 )
4(
Example:An astronaut on a body mass measuring device (BMMD),designed for use on orbiting space vehicles,its purpose is to allow astronauts to measure their mass in the ‘weight-less’ condition in earth orbit.
The BMMD is a spring mounted chair,if M is mass of astronaut and m effective mass of the BMMD,which also oscillate, show that
![Page 12: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/12.jpg)
![Page 13: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/13.jpg)
02
2
2
RIm
kydt
ydWe have
Example:the system is as follow,prove the block
will oscillate in SHM
)4(
)3(
)2()(
)1(
2
21
1
Ra
kxT
IRTT
maTmg
Solution:
o
y
![Page 14: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/14.jpg)
Alternative solution
222
2
1
2
1
2
1mvIkymgy
Rv
(1)
(2)
Take a derivative of y with respect to x
![Page 15: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/15.jpg)
0xdt
xd 22
2
Solution: )tcos(Ax
)tsin(Adt
dxv
)tcos(Adt
xda 2
2
2
2.1 Equation
§ 2 kinematic equation§ 2 kinematic equation
![Page 16: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/16.jpg)
)tcos(Ax )tsin(Av
xtAa 22 )cos(
![Page 17: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/17.jpg)
1) Amplitude (A): the maximum magnitude of displacement from equilibrium.
2) Angular frequency(): T
2
Spring-mass:m
k
Simple pendulum:l
g
is not angular frequency rather than velocity .it depends on the system
Caution:
A) Basic quantity:
2.2) the basic quantity——amplitude、 period,phase
![Page 18: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/18.jpg)
In phase
2,1,0k
k212
2,1,0k
)1k2(12
Out of phase0
012 012
Lag in phaseAhead in phase
3) Phase angle ( = t+ ): the status of the object.
![Page 19: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/19.jpg)
)tcos(Ax
sinAv,cosAx,then 00
2020 )
v(xA
0
0
x
vtg
0xdt
xd 22
2
Caution:Caution:
B) The formula to solve: A, ,
1) is predetermined by the system.
2) A and are determined by initial condition:
if t=0, x=x0, v=v0 ,
Is fixed by initial condition
![Page 20: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/20.jpg)
Solution:
m
k ,5
4
100 2
202
0
vxA
m4.1
10
0 x
vtg
00 vand
4
)4
5cos(4.1 tx
An object of mass 4kg is attached to a spring of k=100N.m-1. The object is given an initial velocity of v0=-5m.s-1 and an initial displacement of x0=1. Find the kinematics equation
)cos( 0 tAx
![Page 21: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/21.jpg)
Circle of reference method
)tcos(Ax
)tsin(Av
)tcos(Aa 2
![Page 22: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/22.jpg)
Compare SHM with UCM
x(+), v(-), a(-)In first quadrant
Angle between OQ and axis-x
Phase
Angular VelocityAngular Frequency
ProjectionDisplacement
x
Radius
UCM
Amplitude
SHM
A
Ox
AA
Q
P
![Page 23: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/23.jpg)
20
30
o xo x /3
6
3
1
x(m)
o t(s)
0.8
1
x(cm)
o t(s)
63
Example:Find the initial phase of the two oscillation
1
2
3
4
1
2
![Page 24: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/24.jpg)
a b
10
-2
x (m)
t (s)
2
2
,40
22 2
att
btt
2
3
2
ba
t
tfor
0
4
3
1
)4(20
tt
a 0t
SHM: x-t graph,find 0 , a , b , and the angular frequency
Solution:
From circle of reference
![Page 25: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/25.jpg)
§ 3 Energy in SHM§ 3 Energy in SHM
)(sin2
1)(sin
2
1
2
1 222222 tkAtAmmvKE
)(cos2
1
2
1 222 tkAkxPE
2222
2
1)(cos)(sin
2
1kAttkAPEKE
Kinetic energy:
Potential energy:
Total energy of the system:
![Page 26: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/26.jpg)
![Page 27: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/27.jpg)
Mk
X
O
Solution:
1v
,kA2
1Mv
2
1 21
21 ,v)mM(Mv 21
,kA2
1v)mM(
2
1 22
22
mM
MAA 12
Mm
k
Example:Spring mass system.particle move from left to right, amplitude A1. when the block passes through its equilibrium position, a lump of putty dropped vertically on to the block and stick to it. Find the kinetic equation suppose t=0 when putty dropped on to the block
)cos( 02 tAx
20
![Page 28: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/28.jpg)
Example:A wheel is free to rotate about its fixed axle,a spring is attached to one of its spokes a distance r from axle.assuming that the wheel is a hoop of mass m and radius R,spring constant k. a) obtain the angular frequency of small oscillations of this system b) find angular frequency and how about r=R and r=0
![Page 29: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/29.jpg)
![Page 30: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/30.jpg)
)cos( 111 tAx )cos( 222 tAx
)cos(21 tAxxx
)cos(2 1221
2
2
2
1 AAAAA2211
2211
coscossinsin
AAAA
arctg
k212 max2121
2
2
2
1 2 AAAAAAAA
)12(12 kmin2121
2
2
2
1 2 AAAAAAAA
4.1 mathematics method
§ 4. Superposition of SHM§ 4. Superposition of SHM
![Page 31: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/31.jpg)
2211
2211
coscos
sinsintg
AA
AA
(2-1)M1
A1
1
M
A2
xo
xx1 x2
ω
A
A2
2
M2
)cos(2 12212
22
1 AAAAA
B) circle of reference
x= x1+x2= Acos( t+ )
x1 =A1cos( t+1 )
x2 =A2cos( t+2 )
![Page 32: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/32.jpg)
Solution:
x3 3O1A
2AA
Draw a circle of reference,
)tcos(Axxx 21
cmt )4
32cos(23
Example:x1=3cos(2t+)cm, x2=3cos(2t+/2)cm, find the superposition displacement of x1 and x2.
![Page 33: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/33.jpg)
![Page 34: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/34.jpg)
dt
dxbkx
dt
xdm
bvkxmaF
2
2
5.1 Phenomena
)cos()( 2
tAetx mbt
5.2 equation
If damping force is relative small
2
2
4m
b
m
k
2
2
4m
b
m
k
)(0
0
0
22
11
tata ececxgoverdampin
dampingcritical
ngunderdampi
§ 5 Damped Oscillations§ 5 Damped Oscillations
![Page 35: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/35.jpg)
0.0 0.5 1.0 1.5 2.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
underdamping
overdamping
No oscillation
Amplitude decrease
0
)cos()( 2
tAetx mbt
Critical damping
![Page 36: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/36.jpg)
tFdt
dxbkx
dt
xdm sin02
2
The steady-state solution is
2
220
2
0
)cos()(
mb
mF
A
tAtx
where 0=(k/m)½ is the natural frequency of the system.
The amplitude has a large increase near 0 - resonance
:§6 Forced Oscillations:§6 Forced Oscillations
drive an oscillator with a sinusoidally varying force:
![Page 37: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/37.jpg)
![Page 38: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/38.jpg)
![Page 39: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/39.jpg)
Projectile motion with air resistance
(case study:p147)
1. Projectile motion with air resistance
![Page 40: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/40.jpg)
A plane moves in constant velocity due eastward,a missile trace it,suppose at anytime the missile direct to plane,speed is u,u>v,draw the path of missile
h
x
(x,y)
(X,Y)(X0,h) v
u
2. Tracing problem
![Page 41: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/41.jpg)
vy
O
u
y(0)=0, x(0)=0
Y=h,X(0)=0
h
x
(x,y)
(X,Y)
22 )()( xXyY
yY
udt
dy
22 )()( xXyY
xX
udt
dx
tnvnyny
tnvnxnx
y
x
)()()1(
)()()1(
22 )()( xXyY
xXu
dt
dxvx
22 )()( xXyY
yYu
dt
dyv y
tVnXnX )()1(
0YY
![Page 42: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/42.jpg)
Solution:
Example:the orbits of satellites in the gravitational field
d
dr
r
v
d
dr
dt
drv
Er
GmM
mr
Lmv
vvv
Er
GmMmv
mrLvLmrv
Lvmr
r
er
r
e
2
22
222
2
22
1
2
1
v
rvv
r
me
ms
3. Planets trajectory
![Page 43: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/43.jpg)
We get:222 22 LrmGmEmrr
Ldrd
e
cos1 e
Rr
reference : 《大学物理》吴锡珑 p 149
![Page 44: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/44.jpg)
2r/r̂GMmUF
Solution1: Newton’s laws
)rr(mma)r(F 2r
0)r2r(mma)(F
GM/hp,
hyperbola1e
parabola1e
ellipse1e
cose1
pr 2
3. Planets trajectory
r/GMmU
![Page 45: Chapter 13 periodic motion](https://reader035.vdocuments.site/reader035/viewer/2022081503/56814dfd550346895dbb6a82/html5/thumbnails/45.jpg)
Solution2: conservation of mechanical energy
E)r(Umv2
1 2 0)r2r(m
GM/hp
hyperbola0E
parabola0E
ellipse0E
cos))GM(
h(
mE2
11
pr
2
2
2