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Overview
Chapter 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 theirmathematical 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 fixeddrum, 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
xy
z
1r
2r
r
O r
t
r0tlim
t
rr0tlimv
dtrd
12
zkyjxi rxyz
r-naught
Position vector in Cartesian system
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Acceleration
rdtrd
dtvda 2
2
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Kinematics in Genaral
In our prescription of r-naught, Newtons 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 earths 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 andmass 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
tsineE
x 00
meE
a 0
0
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Solution Contd.
Step 1: Do the first integration
10 Ctcosa
x
Step 2: Plug in the initial conditions to get C1
At, t = 0, 0
1
a
c0x
00 atcosa
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 theconstant 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 undergravity near earths surface taking the
frictional force of air proportional to the
first power of velocityof the particle. Theparticle 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
kt
e1k
g
dt
dx
v
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 itthro 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:
rx
y
tsinjtcosi ryjxi r
tcosj
tsini
r
vr
0r.r
as expected
r
t
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t
r
x
Coordinate to Play
yj
xi
r Pat
Position vectorin Cartesian form
where we need to handle twounknowns, x andy
P
For this case, r= const & particle position is
completely defined by just one unknown
The Key:
But also, ,rrr Pat
Position vectorin Polar form
y
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Coordinate Systems
Choice 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
cosjsini
sinjcosi r
11
cos
sin
yjxi ry,xP
rrr
How pt P (x,y) expressedin polar form??
Q. Is missing here?
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Unit Vectors
cosjsini sinj
cosi
r
2. Dissimilarity: ???????.
0.r0j.i
Polar unit vectors are NOT constant vectors; they
have an inherent dependency. (REMEMBER)
jAi AA yx
ArAA r
1. Similarity: Use of Cartesian & Polar unit vectors to expressa 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
yjxi r yjxi r rrr
???
velocity
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Velocity Expression
r r
rrr
dt
rdrr
rr
rdt
drv
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rrrv
Conclusion
Radial Part vr Tangential Part v
In general, the velocity expressed in plane polarcoordinate has a radial and a tangential part. For
a uniform circularmotion, it is purely tangential.
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Acceleration
rrrdt
d
dt
rdra
r2rrrra 2 Radial Part ar Tangential Part a
Do these steps as HW
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Example
Find the acceleration of a stone twirling on the end ofa string of fixed length using its polar form
r=l=const
r2rrrra 2
rrr
rrra
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. Findvel and accl. in plane polar coordinate
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Solution (Velocity)
;urutr
rr
rv
utruv
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Solution (Acceleration)
0
0r;ur;utr
r2rrrra 2
u2ruta 2
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Q. How Newtons Law of motion look like inpolar form?
Newtons Law (Polar)
ymFxmF yx In 2D Cartesian form (x,y) :
In 2D Polar form (r,), can we write:
mFrmFr
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Newtons Law in Polar Form
ar
a
r2rrrra
r
2
Acceleration in polar coordinate:
mr2rmF
rmrrmF 2r
Newtons eq.
in polar form
Newtons eq.s in polar form DO NOT scale
in the same manner as the Cartesian form
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Constraint
Any 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 ofa circle and tension T in the string is the
force of constraint associated with it
The rollers roll down the inclinewithout 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 slopewithout 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 Information
For a drum that is rolling without slipping downan 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)
E 2 4b
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Ex 2.4b
A pulley accelerating upward at a rateA
in aMass-Pulley system. Find how acceleration
of bodies are related (rope/pulley mass-less).
ypy1 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 rotatingby 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 force2. 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
z
WB
rEq.s of motion:
3TWzM20rr2M
1TrrM
BB
A
2
A
And, the eq. of constraint:)4(zrlzr
E 2 7 d
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Ex 2.7 contd.
Using the eq.s of motion and the constraint:
BA
2
AB
MM
rMWtz
BA
200AB
MMrMW0z
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Fictitious Force
Two observers O and O, fixed relative to two coordinatesystems Oxyz and Oxyz, respectively, are observing the
motion of a particle P in space.
O
x
y
z
O
x
yz Find the condition when
to both the observers the
particle appears to havethe same force acting on
it?
P
l
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O
x
y
z
Ox
yz 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
ttanCons
dt
Rd
Inertial Frames
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Non-inertial Frames
ttanConsdtRd
IfOxyzis accelerating
To observers O and O, particle willhave different forces acting on it
fictitioustrue
2
2
2
2
2
2
FFF
dt
Rdm
dt
rdm
dt
rdm
P bl
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Problem Exercise 2.16
A 450
wedge is pushed along a table with aconstant accelerationA. A block of mass m
slides without friction on the wedge. Find the
acceleration of the mass m.
450
A
x
ym
Fixed on Table and Ais 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 themoving wedge....
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Force Diagram
y
450
mA
x
mg
N
2
mA
2
mgN
2mA
2mgfsurface
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Accelerations
Acceleration of the block down theincline:
2
A
2
g
asurface
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
A21
2A
2gax
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: