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Space shuttle
Part 1 Mechanics
MECHANICS
Particle Motion Rotation Oscillation
1. Kinematics: description of motion
1.1 Frame of reference and coordinate system
1.2 Physical quantities
1.3 Ideal model and motion
How does the matter move?
Why does the matter move?
Kinematics Dynamics
2. Dynamics: relation of motion to its causes
2.1 Newton’s laws of motion
2.2 Work and energy
2.3 Momentum and impulse
MECHANICS
Kinematics Dynamics
How does the matter move?
Why does the matter move?
Reference of frameReference of frame
quantities to describe motion quantities to describe motion method to describe motion method to describe motion
linear quantities linear quantities Angular quantitiesAngular quantities calculate method calculate method
Project motion
Project motion
circular motion
circular motion
curve motion
curve motion
Particle motionParticle motion
Chapter1-3 Particle Motion
scalarvectorunit vectormagnitudedirectionlengthcoordinate axisdisplacementdistance
vector additioncomponent vectorscomponents
positivenegativescalar productvector product time intervalinstantcurved lineline-segmentarroworigin point
parallelperpendicular
Key word:
particleframe of referencepositiondisplacementaverage (/instantaneous) velocity average (/instantaneous ) accelerationspeed free fallacceleration due to gravity projectiletrajectoryderivativenormal componenttangential component
Key word:
1.1 Ideal Model
Particle: It is the body that has only the mass, but not its shape and size.
Which one is a particle?
a pingpong the earth
Ideal Models: Simple pendulum, rigid body, point charge, harmonical oscillator…
1. Basic Concepts
1.2 Frame of Reference and Coordinate Axis
Frame of Reference: relative, usually refer to earth
o x
The Coordinate System: math conception
attached to the real-word bodies
Cartesian natural
Other coordinates: polar, spherical, cylindrical, elliptical…
1.3 Scalars and Vectors
Scalar: described by a single number with a unit, such as 1kg(mass), 103kg/m3(density), 1A(electrical current).
Vector: has both magnitude and a direction, such as Eavr
,,,
Represent by: A
AAAAA ˆˆ
M
NA
A
B
C
vectorUnit:A
ACAB
,
A
jAiAA yxˆˆ
22yx AAAA
O x
y
xy AAtg /
ji ˆˆ, :represent unit vectors in direction of +x-axis or +y-axis
xA
yA
yx AAA
1) Components of a vector:
jAAiAA yyxxˆ,ˆ
2) Vector Addition
(1) adding with components;(2) adding by geometrical way.
? BA
B
A
BAC
B
A
BAC
1P2P
3P
4P
...21 PPPP
jBiBBjAiAA yxyxˆˆ,ˆˆ
j)BA(i)BA(BAC yyxx
3) Scalar Product (Dot Product)
cosAB
cosBABAC
B
A
BA
//Suppose:
BABA
Then: 2AAA
BA
//Suppose:
BABA
Then:
BA
A
B
Example:
rdFdWSFW
;
4) Vector Product (Cross Product)
sinABBAC BAC
B
A
BAC
BAorBA
////Suppose:
0BA
Then: 0AA
Example:
vmrLFrM
;
Direction: determined by right-hand rule
c
2. Physics quantities to describe the particle motion2.1 Position Vector , Displacement and Motional Equation2.1 Position Vector , Displacement and Motional Equation
1) position vectors
kzjyixr ˆˆˆ
1coscoscos
/cos;/cos;/cos222
rzryrx
Direction is determined by:
222 zyxrr
Magnitude is determined by:
OPr
BPABOA
o
x
y
z
P(x,y,z)
xy
z
A B
C
r
2) displacement vectors
r
Br
BA
Ar
O x
y Displacement Vector: r
AB rrr
ABr
Caution!
ABr
ABs Displacement is different from distance.
Discussion: A very small displacement during a small time interval
O
y
Ar
Br
BA
x
rd
dsABds
dsrd When time interval approaches to 0:
ABrd A very small displacement:
A very small distance:
Cdr
rdAC
rdrˆrdCBACrd
OCOA Let:
CBACAB
d
rd
drCB
Caution!
drrd
Example: a radar station detects an airplane approaching directly from the east. At first observation, the range to plane is 360m at 400 above the horizon. The plane is tracked for another 1230 in the vertical east-west plane, the range at final contact being 790m. Find displacement of the airplanes during the period of observation.
Solution:
3) Motional equation
Motional equation
)(),(),( tzztyytxx
ktzjtyitxtrr ˆ)(ˆ)(ˆ)()(
x2+y2=62
x
y
Pathequation
Path graph
kzjyixr ˆˆˆ
ty
tx
2sin6
2cos6Example:
2.2 Velocity and Speed2.2 Velocity and Speed
S
r
x
Az
yo
C
B)t(r
)tt(r
1) Average Velocity
t
trttr
t
rvav
)()(
kt
zj
t
yi
t
xvav
ˆˆˆ
t
svav
2) Average Speed
sr
avav vv
r
)()( trttrr
S
r
x
Az
yo
C
B)t(r
)tt(r
3) Instantaneous Velocity
dt
rd
t
rv
t
0
lim
kdt
dzj
dt
dyi
dt
dxv ˆˆˆ
dt
ds
t
slimv
0t
4) Instantaneous Speed
vv
Caution!
dsrd sr
Example: Chose the correct equation
dt
rdv
)1(
dt
rdv
)2(
dt
rdv
)3(
dt
drv )4(
Example: How to determine the direction of V in the curved-line motion?
x
y
Example: How to find Velocity on an x-t graph?
t
x
P
O
Slope of tangent = instantaneous speed
t1
02 vQ=tg2=01 vp= tg1
tangentVp=?
A
BVA=?
VB=?
Q
t2
VQ=?
)( ttv
)(tv v
)(tv
Instantaneous acceleration
t
vlima
0t
kdt
vdj
dt
ydi
dt
xd
kdt
dvj
dt
dvi
dt
dva zyx
ˆˆˆ
ˆˆˆ
2
2
2
2
2
2
A
)( ttv
B
)t(v)tt(vv
t
)t(v)tt(v
t
vaav
Average acceleration
2
2
dt
rd
dt
vd
2.3 Acceleration2.3 Acceleration1) acceleration in Cartesian coordinates1) acceleration in Cartesian coordinates
kdt
zdj
dt
ydi
dt
xdk
dt
dvj
dt
dvi
dt
dv
dt
vda zyx ˆˆˆˆˆˆ
2
2
2
2
2
2
2z
2y
2x
2
2z
z2
2y
y2
2x
x
aaaaa
dt
zd
dt
dva,
dt
yd
dt
dva,
dt
xd
dt
dva
kzjyixr ˆˆˆ
Components of velocity and acceleration
kdt
dzj
dt
dyi
dt
dx
dt
rdv ˆˆˆ
2z
2y
2x
zyx
vvvvv
dt
dzv,
dt
dyv,
dt
dxv
Example : direction of acceleration
Concave side of the path
aa
a t
x
OA
B
C EF
D
Inflection point
G
Example : Chose the correct equation:
dt
dva )1(
dt
vda
)2(
2
2
)3(dt
rda
dt
vda
)4(
Example: The position of a particle is given by
(1) calculate: when t=2s.
(2)when is the velocity perpendicular to acceleration.
jtitr ˆ)21(ˆ2
av
,
jtitr
)21(2
idt
vda
jitdt
rdv
2
22
(1)
0t(2)
Solution:
Example: The motion of a particle is described by the function
What kind of motion does it undergo?
j)1t2(i)t23(r 22
Tow kinds of problems in kinematics
dt
dr
dt
da ktzjtyitxr ˆ)(ˆ)(ˆ)(
derivative
integral
Calculus-based-physics!
asvv
attvs
atvv
2
2
1
20
2
20
0
Example: deduce the following equation if particle move in straight line with a=c, and t=0, v=v0, x=x0 .
Example: Suppose the position of an object is given by x = t3-9t2+15t+1(SI).a) Find the initial velocity. When does the object turn
around?b) Find the displacement and the distance traveled for
the time interval t=0 to t=2s.
Solution: )5t)(1t(315t18t3dt
dxv)a 2
s/m15v,0t
v=0, the object turns around at t=1,t=5
Because condition for turning around is:
m2m)13()0(x)2(xx)b
m12)1(x)2(x)0(x)1(xS
Example: The position of a particle is given by .a) What kind of motion does it undergo?b) Find the displacement and the distance traveled for
the time interval t=/ to t= 2/.
jtsinRitcosRr
Solution: tsinRy,tcosRx)a 222 Ryx
The particle moves along a circle with constant speed
,jtcosRitsinRdt
rdv
jtsinRitcosRdt
vda 22
r2
Ra,Rvvv 22y
2x
iR2)/(r)/2(rr)b RS
Example: A radio-controlled model car is moving on a plane (xy-plane). The car has x- and y-coordinates that vary with time according to
x=2t, y=19-2t2(SI).a) Find the car’s coordinates at time t=1s and t=2s, the
n find the displacement and average velocity during the time interval.
b) Find the instantaneous velocity and acceleration at t=1s.
c) Find the path equation of the car.d) When the car is nearest to the origin point of xy-pla
ne?e) What is the distance for t=0s to t=1s.
Solution:
j)t219(it2r)a 2
m)j11i4(r,s2t,m)j17i2(r,s1t 21
s/m)j6i2(t
rv,m)j6i2(rrr av12
j4dt
vda,jt4i2
dt
rdv)b
2s/mj4a,s/m)j4i2(v,s1t
Solution:
2/x19y)c 2 This is a parabola
22222 )219(4) ttyxrd
minrr,0dt
drwhen
22 )t4(2dt
dsvjt4i2v)e
m08.6rr,s3t min
vdtds
mttt
tdtvdtst
t
t
t
34.1)])2(12ln()2(1[2
1
)2()2(1
10
22
22
1
2
1
20 at
2
1tvs
Example:The motion of an object falling from rest in aresisting medium is described by the equation dv/dt=A-Bv,Where A and B are constants. In terms of A and B, finda) The initial acceleration.b) The velocity at which the acceleration becomes zero (th
e terminal velocity).c) Show that the velocity at any given t is given by
)e1(B
Av Bt
Solution: 0v,BvAdt
dva)a 0
t
0
v
0dt
BvA
dv)e1(
B
Av Bt
Aa0
B
Av,0BvA
dt
dva)b
dt)BvA(dv)c
h
x
Example: The man on the bank drag the boat with constant velocity. Try to find the velocity and acceleration of the boat, When the distance between the boat and the bank is
x.Solution: Set up coordinate axis in the picture, then draw
the position vector of the boat.
22 hrx,jhixr
idt
dx
dt
rdv
x
hxvi
22
3
22ˆ
x
hvi
dt
da
ox
y
r
2) Tangential and Normal Acceleration2) Tangential and Normal Acceleration
P
R
kdt
zdj
dt
ydi
dt
xdk
dt
dvj
dt
dvi
dt
dv
dt
vda
2
2
2
2
2
2yyx
Can we represent the acceleration of a particle moving in a curved path in terms of components parallel and perpendicular to the velocity at each point?
v
tevv
dt
)ev(d
dt
vda t
tt e
dt
dv
dt
edv
:e
:e
n
t tangential direction
normal direction
ne
te
O
te)t(et
)tt(et
tt e
dt
dv
dt
edva
P
R)t(v
One
)t(et
for a very small time interval
0,0 t
,ee tt nt eded
tn edt
dve
dt
dva
v
dt
ds
ds
d
dt
d
tn
2
edt
dve
va
Q
)( ttv
)tt(et
ta
tn
2
edt
dve
va
tn aa
n
2
n ev
a
tt e
dt
dva
Direction: normal direction
Describe the change rate of the magnitude of velocity with time
2
n
va Magnitude:
Describe the change rate of direction of velocity with time
dt
dvat Magnitude:
Direction: tangential direction
t
n
222
2t
2n
a
atg
)dt
dv()
v(aaa
tn
2
tn edt
dve
vaaa
na
ta
v
a
dtdva )1
dtvdat
)2
dt
vdat
)3
Example: correct the following formula
1) In straight line motion
2) In free fall motion
3) In projectile motion
4) In uniform circular motion
5)In nonuniform circular motion
tn aaa 0
0 nt agaa
sincos gagaga tn
02
tn aR
vaa
222
tntn aaadt
dva
R
va
Example: what is the character of aaa nt
Example: A particle moves in a circle of radius R. The distance is described by the equation
(b,c are constants, b2>Rc)
a) When an= at?b) When a= c?
2ct2
1bts
Solution:
ctbdt
ds)a
cdt
dat
R
)ctb(
Ran
22
For an = | at | c
R
c
bt
2t
2n aaa)b
c
bt
Example8: Find an , at and of projectile motion at any time. Suppose t=0, v=v0 , and makes an angle with +x.
Solution:
Set up x,y coordinate axis
x
y
g
0v
Projectile motion can be considered as a combination of horizontal motion and vertical motion
cosox
gtoy sin 22
yx
ga
xvyv
nata
x
n gga cos
yt gga sin
x
n gga cos
yt gga sin
22 sincos
sin
)gt()(
)gt(g
oo
o
22 sincos
cos
)gt()(
g
oo
o
cos
sincos
o
2
32
o2
o
n
2
g
)gt()(
a
Another solution: ,ox cos ,gtoy sin 22
yx
dt
dat
2o
2o gt)-sin()cos(
sin
)gt(g o
22
tn aag
na
2 2t
2n aga
x
y
g
0v
xvyv
nata
y
xo
A
R
Suppose a particle moves in a circle of radius R. We can use the single quantity as a coordinate, Suppose a particle moves in a circle of radius R. We can use the single quantity as a coordinate, is called angular coordinate, and usually measured in radians.
1) Angular Displacement, Velocity and Acceleration
(2) Angular velocity:
3. Angular quantities to describe the particle motion
(1) Angular Displacement
)()( ttt
Average angular velocitytav
Instantaneous angular velocitydt
d
Caution :
Angular velocity is a vector!
Right Hand Rule: the vector is represented by an arrow drawn so that if curled fingers of the right hand point in the direction of the rotation, the direction of the vector coincides with the direction of the extended thumb.
dtd
If rotation is counter clockwise increasing positive
If rotation is clockwise decreasing negative
dtd
(3) Angular Acceleration:
dt
d
Average Angular Acceleration
Instantaneous Angular Acceleration
tav
find the angular velocity and angular acceleration
812
68 2
tdt
d
ttdt
d
Solution:
Example: The angular position
32 242 tt
Rdt
da,R
Ra,R t
22
n
2) Relationship between Linear Quantities and Angular Quantities
r
in circular motion
prove:
General rule:
rvrdt
d
dt
dsrdds
Compare circular motion with straight-line motion
Straight-line motion Circular motion
x
v = dx/dt = d/dt
a = dv/dt = d/dt
a=const. =const.atvv 0
200 at
2
1tvxx
)(2 020
2 xxavv
t0 2
00 t2
1t
)(2 020
2
Example: An early method of measuring speed of light makes use of a rotating slotted wheel. A beam of light pass through a slot at the outside edge of the wheel, travel to a distant mirror, and return to the wheel just in time to pass through the next slot in the wheel. r=5cm and there are 500slots at wheel’s edge, L=500m, c=3*108m/s. What is the constant angular speed of the wheel and linear speed of a point on the edge of wheel.
Example: A pulsar is a rapidly rotating neutron star that emits a radio beam like a lighthouse emit a light beam. We receive a radio pulse for each rotation of the star. The period T of rotation is found by measuring the time between pulses. The pulsar in the crab nebula has a period of rotation of T=0.033s that is increasing at the rate of 1.26*10-5s/y.
1) What is the average pulsar’s angular acceleration?
2) If its acceleration is constant, how many years from now will the pulsar stop rotating
t
T
TtT
t
2
2)2
(
137
5
104101.3
1026.1
t
T
29 /103.2 srad
sT
twhen
t
10
0
0
103.82
/0
Example: A particle move around a circle of radius R=1.0m. The angular coordinate vary with time according to
=2 + 12 t - t3 (SI). a) Find the normal acceleration and tangential acceleration
in rim of wheal at time t=1s.b) When will the particle stop? How many circles does the
particle turn?
Solution:
,t312dt
d)a 2 t
dt
d 6
)SI(6a,81a,s1t
t6Ra,)t312(R
t2
n
t222
n
0t312,stopwhen)b 2 s2t
16
2,0,18,2 12
tst
4. Relative Velocity
o x
?v,h/km30v,h/km20v
)earth(E),dog(D),car(C:let
CEDECD
4. Relative velocity
yA
zA
xA
OA
yB
zB
xBOB
PAr
P
PBr
BAPBPA rrr
BAv
Suppose here are two reference frame A and B. xA//xB, yA//yB , zA//zB . B moves with constant velocity along x axis relative to A.
for point P:
BAPBPA vvv
BAPBPA aaa
APPA vvv
Caution :
BAPBPA vvv
BAPBPA aaa
(1) They are vector addition.
(2) You can not change the sequence of the subscripts.
APPA vv
vPA :velocity of P relative to A
B B
ECCE vvv
?v
,h/km30v,h/km20v
)earth(E),dog(D),car(C:let
CE
DECD
D D
Example: The compass of an airplane indicates that it is headed due north, and its airspeed indicator shows that it is moving through the air at 240km/h. If there is a wind of 100km/h from west to east , what is the velocity of the airplane relative to the earth?
Solution: N
E
S
W
)earth(E),air(A),plane(P:let
PAv
AEv
?vPE
EPPE vvv A A
,ˆ/100 ihkmvAE
?PEv
,ˆ/240 jhkmvPA
Example: When a train’s speed is 10m.s-1 eastward, raindrops that are falling vertically with respect to the earth make traces that are inclined 30o to the vertical on the windows of the train.
a) What is the horizontal component of a drop’s velocity with respect to the earth? With respect to train?
b) What is the magnitude of the velocity of the raindrop with respect to the earth? With respect to train?
ism10vTEˆ/
030?RTv
ERRE vvv T T
)earth(E),rain(R),train(T:letSolution:
?REv
sm10v0v RTRE /, ////
s/m20v
s/m310v
RT
RE
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