overview chapter 1 & 2
TRANSCRIPT
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OverviewChapter 1 & 2
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“Ch 1 Must-do-problems”:
17, 20
4, 11,14, 17, 19, 35, 36
“Ch 2 Must-do-problems”:
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Learning Objectives Signatures of a particle in motion and their
mathematical representations
Velocity, Acceleration, Kinematics
Choice of coordinate systems that would
ease out a calculation.
Cartesian and (Plane) Polar coordinate
Importance of equation of constraints
Some application problems
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Skip
Vectors and free body diagram
Tension in massless string and with strings having
mass, string-pulley system.
Capstan problem
While sailing it is used
by a sailor for lowering
or raising of an anchor
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Capstan
A rope is wrapped around a fixed
drum, usually for several turns.
A large load on one end is held
with a much smaller force at the
other where friction between drum
and rope plays an important role
You may look at:
Example 2.13 & Exercise 2.24
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Velocity
Particle in translational motion
x
y
z
1r
2r
r
O r
t
r0tlim
t
rr0tlimv
dtrd
12
zkyjxir xyz
“ r-naught”
Position vector in Cartesian system
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Acceleration
rdt
rd
dt
vda
2
2
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Kinematics in Genaral
In our prescription of “r-naught”, Newton’s 2nd law:
t,r,rFrm
To know precisely the particle trajectory, We need:
Information of the force
Two-step integration of the above (diff) equation
Evaluation of constants and so initial conditions
must be supplied
A system of coordinate to express position vector
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Simple form of Forces
(Force experienced near earth’s surface)ttanconsa,amF
•
tFF
•(Effect of radio wave on Ionospheric electron)
(Any guess ???)
(Gravitational, Coulombic, SHM) rFF
•
rFF
•
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Example 1.11
Find the motion of an electron of charge e and
mass m, which is initially at rest and suddenly
subjected to oscillating electric field, E = E0sint.
The coordinate system to work on
We decide:
• Information of force.
• Initial condition.
We check:
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Solution
Translate given information to mathematical form:
m
tsinEe
m
eEreErm 0
1.
…..but the acceleration is NOT constant
As E0 is a constant vector, the motion is safely 1D
tsinam
tsineEx 0
0
m
eEa 0
0
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Solution Contd.
Step 1: Do the first integration
10 Ctcos
ax
Step 2: Plug in the initial conditions to get C1
At, t = 0,
01
ac0x
00 a
tcosa
x Hence,
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Solution Contd.
What would be the fate of the electrons?
tsina
ta
x2
00
Step 3: Carry out the 2nd. Integration and get the
constant thro’ initial conditions
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Graphical Representation
Red: “sin” term
Black: Linear term
Blue: Combination
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Viscous Drag
• Motion of a particle subjected to a resistive force
Examine a particle motion falling under
gravity near earth’s surface taking the
frictional force of air proportional to the
first power of velocity of the particle. The
particle is dropped from the rest.
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Solution
Equation of motion:
Cvmgdt
dvm
dt
xdmF
2
2
Resistive force proportional to velocity
(linear drag force)
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Final Equation
kte1k
g
dt
dxv
If the path is long enough, t
tervk
gv “Terminal
Velocity”
Solving the differential equation and plucking the
correct boundary condition (Home Exercise)
m
Ck
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Important Results & A Question
Some calculated values of terminal velocity is given:
1. Terminal velocity of spherical oil drop of diameter
1.5 m and density 840kg/m3 is 6.1x10-5 m/s.This
order of velocity was just ok for Millikan to record it
thro’ optical microscope, available at that time.
2. Terminal velocity of fine drizzle of dia 0.2mm is
approximately 1.3m/s.
How one would estimate these values?
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Circular Motion
Particle in uniform circular motion:
•r
x
y
tsinjtcosir
yjxir
tcosjtsinir
vr
0r.r
…as expected
r
t
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•
tr
x
Coordinate to Play
yjxir Pat
Position vector in Cartesian form
where we need to handle twounknowns, x and y
P
For this case, r = const & particle position is
completely defined by just one unknown
The Key:
But also, ,rrr Pat
Position vector in Polar form
y
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Coordinate SystemsChoice of coordinate system is dictated (mostly)
by symmetry of the problem
Plane polar coordinate (r, )
• Motion of earth round the sun
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• Electric Field around a point charge
Other Examples
Spherical polar coordinate P (r, , )
(r,,)
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• Mag. field around a current carrying wire
Other Examples
Cylindrical polar coordinate P (, , z)
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Plane Polar System
How to develop a plane polar coordinate system ?
x
y
i
jr
r r
To Note:
Unit vectors are drawn in the direction of
increasing coordinate….THE KEY
y,xP
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Relation with (XY)
x
y
i
j
r
r
1
cosjsiniˆ
sinjcosir
11
cos
sin
yjxiry,xP
rrr
How pt P (x,y) expressed
in polar form??
Q. Is missing here?
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Unit Vectors
cosjsiniˆ
sinjcosir
2. Dissimilarity: ???????.
0ˆ.r0j.i
Polar unit vectors are NOT constant vectors; they
have an inherent dependency…. (REMEMBER)
jAiAA yx
ˆArAA r
1. Similarity: Use of Cartesian & Polar unit vectors to express
a vector and are orthogonal to each other
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Consequence
As polar unit vectors are not constant vectors,
it is meaningful to have their time derivatives
Useful expressions of velocity & acceleration
yjxir
yjxir
rrr ???
velocity
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Velocity Expression
r r
ˆrrr
dt
rdrrrrr
dt
drv
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ˆrrrv
Conclusion
“Radial Part vr” “Tangential Part v”
In general, the velocity expressed in plane polar
coordinate has a radial and a tangential part. For
a uniform circular motion, it is purely tangential.
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Acceleration
ˆrrrdt
d
dt
rdra
ˆr2rrrra 2
“Radial Part ar” “Tangential Part a”
Do these steps as HW
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ExampleFind the acceleration of a stone twirling on the end of
a string of fixed length using its polar form
r=l=const ˆr2rrrra 2
ˆrrr
ˆrrra
2
2
“Centripetal” accln. (v2/r) Tangential accln.
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Example 1.14
A bead moves along the spoke of
a wheel at a constant speed u mtrs
per sec. The wheel rotates with
uniform angular velocity rad.
per sec. about an axis fixed in space.
At t = 0, the spoke is along x-axis
and the bead is at the origin. Find
vel and accl. in plane polar coordinate
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Solution (Velocity)
;urutr
ˆrrrv
ˆutruv
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Solution (Acceleration)
0
0r;ur;utr
ˆr2rrrra 2
ˆu2ruta 2
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Q. How Newton’s Law of motion look like in
polar form?
Newton’s Law (Polar)
ymFxmF yx
In 2D Cartesian form (x,y) :
In 2D Polar form (r,), can we write:
mFrmFr
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Newton’s Law in Polar Form
ˆara
ˆr2rrrra
r
2
Acceleration in polar coordinate:
mr2rmF
rmrrmF 2
r
Newton’s eq.in polar form
Newton’s eq.s in polar form DO NOT scalein the same manner as the Cartesian form
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ConstraintAny restriction on the motion of a body is a constraint
How these things are associated with constraints?
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More on Constraints
Bob constrained to move on an arc of
a circle and tension T in the string is the
force of constraint associated with it
The rollers roll down the incline
without flying off and the normal
reaction is the force of constraint
associated \with it
Common Both are rigid bodies where rigidity is the
constraint and force associated with it is ……………..
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Example 1
Take a simple pendulum with massless string
x
y
l
0z
.constlyx 222
Eq.s of constraint
Constraints can be viewed mathematically by one/more eq.
of constraint in the coordinate system it is used to express
(Note: Constraints may involve inequations too)
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Example 2
If a drum of radius R has to roll down a slope
without slipping, what is the corresponding eq.
of constraint ?
R
x
Q. How eq. of constraints help us get useful information?
xRxR
Eq. of constraint:
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Useful InformationFor a drum that is rolling without slipping down
an incline from rest, its angular acceleration ()
is readily found using the equation of constraint.
R
a
at2
1xR 2
R
a
x
Home Study: Ex. 2.4a(Prob 1.14)
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Ex 2.4bA pulley accelerating upward at a rate A in a
“Mass-Pulley” system. Find how acceleration
of bodies are related (rope/pulley mass-less).
yp
y1 y2
1
2
R
2p1p yyyyRl
Constrain Equation:
21p yy2
1y
A
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Ex 2.7
B
A
Initial conditions:
• B stationary and A rotating
by a mass-less string with
radius r0 and constant angular
velocity 0. String length = lz
Q. If B is released at t=0, what is its
acceleration immediately afterward?
Steps: 1. Get the information of force
2. Write the eq. of motions
3. Get the constraint eq. if there is any and use it effectively
r0
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Ex 2.7 contd.
MA
MB
T
T
zWB
r
Eq.s of motion:
3TWzM
20rr2M
1TrrM
BB
A
2
A
And, the eq. of constraint:
)4(zrlzr
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Ex 2.7 contd.
Using the eq.s of motion and the constraint:
BA
2
AB
MM
rMWtz
BA
2
00AB
MM
rMW0z
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Fictitious Force
Two observers O and O’, fixed relative to two coordinate
systems Oxyz and O’x’y’z’, respectively, are observing the
motion of a particle P in space.
O
x
y
z
O’x’
y’z’ Find the condition when
to both the observers the
particle appears to have
the same force acting on
it?
P
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O
x
y
z
O’x’
y’
z’ P
Formulation
rr’
R
rrR
We look for:
0
dt
Rd
dt
rrd
dt
rdm
dt
rdm
FF
2
2
2
2
2
2
2
2
ttanConsdt
Rd
“Inertial Frames”
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Non-inertial Frames
ttanConsdt
Rd
If O’x’y’z’ is accelerating
To observers O and O’, particle will
have different forces acting on it
fictitioustrue
2
2
2
2
2
2
FFF
dt
Rdm
dt
rdm
dt
rdm
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Problem Exercise 2.16
A 450 wedge is pushed along a table with a
constant acceleration A. A block of mass mslides without friction on the wedge. Find the
acceleration of the mass m.
450
A
x
ym
Fixed on Table and “A”
is measured here
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Solution
We would like to use the conceptof fictitious force.....
450
A
x
y Attach the ref. frame to the
moving wedge....
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Force Diagram
y
450
mA
x
mg
N
2
mA
2
mgN
2
mA
2
mgfsurface
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Accelerations
Acceleration of the block down theincline:
2
A
2
gasurface
2
1
2
A
2
gax
2
1
2
A
2
gay
These are wrt the wedge!
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“Real” Accelerations
2
A
2
g
A2
1
2
A
2
gax
2
g
2
Aay
fictitioustrue
2
2
2
2
2
2
aaa
dt
Rdm
dt
rdm
dt
rdm
To find the acceleration of m, we must recollect: