r.r. mishra · pulley-mass system h x 1 x 2 constraint eq. (h x 1) (h x 2) " a 1 a 2 0. 2....
TRANSCRIPT
R.R. Mishra
Department Of Physics
BITS Pilani
Mechanics, Oscillations
and Waves (MEOW)
1. Mechanics (R.R. Mishra)
20 ~ 22 Lectures
2. Oscillations & Waves (D.D. Pant)
20 ~ 22 Lectures
Major Division
Textbooks :
1. An Introduction to Mechanics :
Daniel Kleppner & Robert
Kolenkow
2. The Physics of Vibrations &
Waves : A. P. French
Mechanics
Chapter No. 2 : Review of Newton’s
Equations
Chapter No. 3 : Linear Momentum
Chapter No. 4 : Work, Energy & Power
Chapter No. 6 : Angular Momentum
Chapter No. 8 : Non-inertial systems and
Fictitious Forces
A world simple enough to be
understood, would be too simple
to produce a mind that can
understand it. J.D. Barrow
Chapter 2
• Constrained Motion
• Newton’s Equations in Polar
Coordinates
Constrained Motion
Examples
1. Pulley-mass system
h
1x
2x
Constraint eq.
)xh()xh( 21
0aa 21
2. Block on a fixed wedge
x
y
Constraint equation :
h
1h
yx
xy a
ha
Prob. 2.16
450
A Angle of wedge : 450
Acceleration of
wedge : A, to right
No friction between block and
wedge
Q : What is the acceleration of the
block w.r.t. the ground
Block on accelerated wedge
450
A
N
mg
2tA2
1hyx Aaa yx
mAmg2
N
2
N
)gA(
2
mN
x
y
2At2
1 h
h
Constraint Equation :
)gA(2
1a;)gA(
2
1a yx
The block will climb up the wedge iff A > g
450
N
mg
x
y
h
h
X
Suppose the wedge is left to itself and
is free to move on a frictionless surface.
Xhyx xyx Aaa
2
Nmg
2
N
2
N
3
mg
2
N
3
gA;
3
g2a;
3
ga xyx
(Mass of wedge : m)
1M
2M
3M
Prob. 2.20
Consider the “pedagogic machine”. All
surfaces are frictionless. Find the
acceleration of block when the system
is released. 1M
The Pedagogic Machine
Constraint Equation :
)yy()xx( 021
0aaa 321
1M
2M
3M
1x
0y
2x
y
x
y
22aMT
333 aMgMT
131 a)MM(T
0aaa 321
Four unknowns, , and four
equations ! 321 a&a,a,T
1M
2M
3M
T
T
T
All four unknowns can be solved for.
In particular,
gMMM2MMMM
MMa
2
3323121
321
Newton’s Equations in Polar Co-ordinates
Review of Newton’s Eq. in Cartesian
Co-ordinates
The Cartesian System
i
j F
i
j
jFiFF yx
jvivv yx
Equations of Motion
j)t(yi)t(x)t(r
jyixdt
rd)t(v
jyixdt
vda
m
x
y
r F
Fam
jFiF)jyix(m yx
yx Fym;Fxm
Polar Co-ordinates
P r
,r
The Polar Grid and Unit Vectors
r
r
1ˆˆrr
0ˆr
Resolving Vectors in Polar Coordinates
r
A
ˆArAA r
Unit vectors in the polar co-ordinates
vary from point to point, unlike unit
vectors in the Cartesian co-ordinates.
Expressing in terms of .
)ˆ,r(
)j,i(
)ˆ,r( )j,i(
jsinicosr θ+θ=
jcosisin-ˆ θ+θ=θ
r
x
y
θ
θ
r
Newton’s Equations of Motion in Polar
Coordinates
r
x
y
rθ
θ
jsinicosr θ+θ=
jcosisin-ˆ θ+θ=θ
r-θ&θr
rrr =
We have,
ˆrrr
Another differentiation leads to
rr-ˆrˆrˆrrrva 2
ˆ)r2r(r)r-r( 2
ˆaraawithComparing r
rrrrrv
rv&rvr
r2ra&r-ra 2
r
r
2 F)r-r(m∴
F)r2r(m } Newton’s Eqs. in
polar coordinates
Prob. 2.29
A car moving radially
outward on revolving
platform.
Coeff. of friction = μ
Ang. Velocity of
platform = ω (constant)
Velocity of car w.r.t.
platform = v0
Car starts from centre of platform
v0
ω
Find :
a) Acceleration of car as a function of
time using polar coordinates. Show by
vector diagram
b) The time at which the car starts to
skid
c) Direction of frictional force at the
time of skidding
a) We have :
tvr 0
t
22
r rrra
0v2r2ra
v0
x
y
x
y
ra
aa
2
r ra
0v2a
b) Net force on car :
22
r FFF 22
r aam
2
0
242 v4rm
22
0
2
0
2242
0 t4vmv4tvm
The car will start to skid when
gmF 2
0
2
0
222
v
v4gt
c) Direction of frictional force at the
time of skidding
rF
F
F
2
r rmF
0vm2F
Direction of
skidding
Prob. 2.33
x
r
θ
y ω
m
Rod with a mass m
on it. Rod rotates
with constant
angular velocity
ωon a horizontal
plane and mass free
to slide.
i) Show that motion is given by tt- eBeA)t(r
Find β.
ii) Show that for a particular choice of
initial conditions, it is possible to obtain
a solution such that r continually
decreases and that for all other choice, r
will eventually increase.
x
r
θ
y ω
m
a) The Polar equations
are :
0)rr( 2
NFrm2
N
rdt
rd 2
2
2
The coordinate Should be such a
function of that twice differentiation of
w.r.t. will be proportional to itself.
rrt
t
An intelligent guess :
te)t(r
Substituting this into the LHS
The radial equation is
rr 22
tt e&eBoth are solutions for r
The given equation being a linear
equation, the most general solution is
tt eBeA)t(r
where, A & B are constants to be determined
ii) Let the initial conditions be :
00 v)0(r&r)0(r
The complete solution is then :
t
00
t
00 evr2
1evr
2
1)t(r
Now,
t00
t
00 evrevr2
1
dt
dr
Additional Part
rm2N
What is the normal force of the rod on
the bead?
t0
0
t00
2 ev
rev
rm
For to be negative, the coeff. of
must be negative.
dtdrte
Prob. 2.35
v0
A block of mass m slides
on the inside of a ring
fixed to a frictionless
table. It is given an initial
velocity of v0. Coefficient
of friction between the ring and the block
is μ. Find the velocity and position at a
later time t.
N
μN
N-Fr =
N-F μ=θ
ANS :
NmR)rr(m 22
NmR)r2r(m
The polar equations are :
Substituting N from the first equation in
the second
2
Putting
2
dt
d
t
0
2td
d
0
t11
0
t1 0
0
tvR
vRv
0
0
t1dt
d
0
0
0
t
0 0t1
tdd
t1n1
0
0
Prob. 2.37
A bowl shaped racing
track, on which a
racing vehicle can
move in horizontal
circles without
friction.
Q : What should be the equation of the
vertical cross section of the track, so
that it takes the same time T to circle the
track, whatever be its elevation
x
y
θ
x
y
θ
N
mg
mgcosN
x
mvsinN
2
xg
v
dx
dytan
2
,T
x2v,Since
xkTg
x4
dx
dy2
2
2
xky
2
(Parabola)