momentum
Hence,
conscrvcd.
systcm (Mv,
is Constant.
remains
mass In
Momentum MOTION
The 1.e., point where then
of concentrated.
where CENTRE
MASS OF
Centre
the
If motion moment
mass
MAm is the respectively
MApm absence
have, We called M
of is (m centre the of
the the
conservation of OF of
Man)-0 =Ft the particles +m,
moment entire mass
of CENTRE mass total of
external for
Ycm centre t mass from .be Chapter
Fext 0 where = m, mass mass
cm
Constant. = m×r the system a
=F, ofand t.....)
of the
= ext forces,
of of OF mass /m|I,4 position
a Aem massMASS its theof the origin of
constant) system position
= system system m, the of
O, particles
the d'(Ycm) is vectors system
d1' governed M
velocity of vector i=1 is
of particles: offparticles.
the
is
of defined
Rotational
Motionand is
by w.r.t.the m,
centre of massesimagined
the the
equation
The m,, as
of centre reference that
product
mass My,
to
point
Particles
System of
of 3 be Shantau
Dutta
10. 9. 8 7. 6. 4. 3. 2
5 1.rotational
motion. axis RotatoryA
bodyMotion rigid.
body
undergoes if Bodies
force, Rigid
afa FLWP
such
(a) and The can coincide,body.
Intheof The mass If the origin In the If ofThemotion The
rotatory iocated so, Ifm,=mlies The moves
of
body
kinematics externalcentre pure
we the rotation. rotating
can Thenature
location be centre ofa moments position position for in
centre
body itdoes
assumed takechoice the is
rotatory }m, or particles at between in said
be
position if body motion.
rolling a undergoes
same not
the cm
found of the of force of i, the of circleabout to
distribution of and mass. centre of
mean The undergo
of of of displacement. be
the to vaiue mass remains is c0-ordinate
motion, zero. is the centre centre +m2Tz)(r +r,) (m them mass
MemoryKeep in
using of dynamics, be motion. But of
tm,)
(m, motion and a rigid.
ceníre ofg zero positionequal fixed some
cenire
concentrated and masses of it of onofa the any
integration constant.
is then the mass of
changes the
system
mass mass centres
of axis dispiacement, change
of of of samecentre axis masses line
mass mass whoie the system.at vector the No
mass at remains then
throughout velocity of ofthe of with joining of boty real
of of rotation origin, a the two of in
depends
of at of the 2 every
all shape
as the body of body
continuous the the gravity
system time
unchanged the
centre identical is
these every
the said
body. centre mass ofpasses then is particies. particle can by
on the the independent in particles. circleslion e
of to particie
be th e
of about the translatory
ofa
the ofdimension centrethrough mass particles possess perfectly action
bodies shapemass. a sum in
body body pure the of
the in
of of is it of
,
analogy In
The acceleration to The
velocity. The linear The TheAverage ANGULAR
radius
angular
The
174
unit lim a= instantaneous
(Q=lim unit (C)
avg.average angular velocity) vector
symmetricalFor(d) (b)
of It angular
angular is Abo dt
A of the It Centrepoint. The where, dm
is denoted
angular
acceleration
angular VELOcITY
per does Xem
definedlinear veiocity C. C.M. and
A0
d is
defined velocity M. not of
acceleration acceleration angular unit MX,
dt do by
velocity
rad/sec. is lies where , mass of is y
as acceleration time. is depend
a. at athe and
A defined AND xdm,
as
velocity o= tbodies
he of R uniform total z
is It is ofa are
the At is geometrical on the uniform Yem
is thea, denoted
ANGULARACCELERATION the
rad/sec2 rate as mass. radius rod mass
a.vg o distribution,
uniform
mass co-ordinates 1
(similar the of of
instantaneous of by semicircular
of length the
Ofa angle centre the ydm,
change .
rotating semicircular system.
to L
instantaneous covered is of Zem
of at small
body angular wire its Mzm
angular middle
ty mass
is wire. is
the at
Therefore,
the Also, distributioninertia
Radius mass.of the gyration: beRadius
The ofIts where the byThe rotation...n,MOMENT A where
SIaxis rigid
Relationsthip
distances Or
entire
definedradius
M=Zm;
where, unit r moment andr,, y=rwsin þ
ofabout of is then body is
radius gyration
MK2I= mass as rotation.
is the
of kgm. I=m,r?+ m, I, OFthe
of X the gvration the perpendicular of moment having angle between
of inertia ..
particles of givendistance
the ,
INERTIA
gyration k It be between
is body is of
constituent
of a of their anguiar
from n
given axis contiuous inertia
Em; were
from bodv atensor.
distance r+..respective AND o
the (k) by, would & veiocity
equals the is
axis 2 concentrated. about particlesRADIUS r.
m, }m be axis mass
m,given
same
of distances
ofthe
of i s the and
rotation. m4 r, by,
root rotation axis small distribution of OF
m as =m; i=l n
inear
mean with its of masses
from
rotation mass GYRATION
moment at veiocity:
PHYSICS
square its dm the
which, is m,
actual given axis
may from m,,
of of if of
Hence,
conscrvcd.
systcm (Mv,
is Constant.
remains
mass In
Momentum MOTION
The 1.e., point where then
of concentrated.
where CENTRE
MASS OF
Centre
the
If motion moment
mass
MAm is the respectively
MApm absence
have, We called M
of is (m centre the of
the the
conservation of OF of
Man)-0 =Ft the particles +m,
moment entire mass
of CENTRE mass total of
external for
Ycm centre t mass from .be Chapter
Fext 0 where = m, mass mass
cm
Constant. = m×r the system a
=F, ofand t.....)
of the
= ext forces,
of of OF mass /m|I,4 position
a Aem massMASS its theof the origin of
constant) system position
= system system m, the of
O, particles
the d'(Ycm) is vectors system
d1' governed M
velocity of vector i=1 is
of particles: offparticles.
the
is
of defined
Rotational
Motionand is
by w.r.t.the m,
centre of massesimagined
the the
equation
The m,, as
of centre reference that
product
mass My,
to
point
Particles
System of
of 3 be Shantau
Dutta
10. 9. 8 7. 6. 4. 3. 2
5 1.rotational
motion. axis RotatoryA
bodyMotion rigid.
body
undergoes if Bodies
force, Rigid
afa FLWP
such
(a) and The can coincide,body.
Intheof The mass If the origin In the If ofThemotion The
rotatory iocated so, Ifm,=mlies The moves
of
body
kinematics externalcentre pure
we the rotation. rotating
can Thenature
location be centre ofa moments position position for in
centre
body itdoes
assumed takechoice the is
rotatory }m, or particles at between in said
be
position if body motion.
rolling a undergoes
same not
the cm
found of the of force of i, the of circleabout to
distribution of and mass. centre of
mean The undergo
of of of displacement. be
the to vaiue mass remains is c0-ordinate
motion, zero. is the centre centre +m2Tz)(r +r,) (m them mass
MemoryKeep in
using of dynamics, be motion. But of
tm,)
(m, motion and a rigid.
ceníre ofg zero positionequal fixed some
cenire
concentrated and masses of it of onofa the any
integration constant.
is then the mass of
changes the
system
mass mass centres
of axis dispiacement, change
of of of samecentre axis masses line
mass mass whoie the system.at vector the No
mass at remains then
throughout velocity of ofthe of with joining of boty real
of of rotation origin, a the two of in
depends
of at of the 2 every
all shape
as the body of body
continuous the the gravity
system time
unchanged the
centre identical is
these every
the said
body. centre mass ofpasses then is particies. particle can by
on the the independent in particles. circleslion e
of to particie
be th e
of about the translatory
ofa
the ofdimension centrethrough mass particles possess perfectly action
bodies shapemass. a sum in
body body pure the of
the in
of of is it of
,
analogy In
The acceleration to The
velocity. The linear The TheAverage ANGULAR
radius
angular
The
174
unit lim a= instantaneous
(Q=lim unit (C)
avg.average angular velocity) vector
symmetricalFor(d) (b)
of It angular
angular is Abo dt
A of the It Centrepoint. The where, dm
is denoted
angular
acceleration
angular VELOcITY
per does Xem
definedlinear veiocity C. C.M. and
A0
d is
defined velocity M. not of
acceleration acceleration angular unit MX,
dt do by
velocity
rad/sec. is lies where , mass of is y
as acceleration time. is depend
a. at athe and
A defined AND xdm,
as
velocity o= tbodies
he of R uniform total z
is It is ofa are
the At is geometrical on the uniform Yem
is thea, denoted
ANGULARACCELERATION the
rad/sec2 rate as mass. radius rod mass
a.vg o distribution,
uniform
mass co-ordinates 1
(similar the of of
instantaneous of by semicircular
of length the
Ofa angle centre the ydm,
change .
rotating semicircular system.
to L
instantaneous covered is of Zem
of at small
body angular wire its Mzm
angular middle
ty mass
is wire. is
the at
Therefore,
the Also, distributioninertia
Radius mass.of the gyration: beRadius
The ofIts where the byThe rotation...n,MOMENT A where
SIaxis rigid
Relationsthip
distances Or
entire
definedradius
M=Zm;
where, unit r moment andr,, y=rwsin þ
ofabout of is then body is
radius gyration
MK2I= mass as rotation.
is the
of kgm. I=m,r?+ m, I, OFthe
of X the gvration the perpendicular of moment having angle between
of inertia ..
particles of givendistance
the ,
INERTIA
gyration k It be between
is body is of
constituent
of a of their anguiar
from n
given axis contiuous inertia
Em; were
from bodv atensor.
distance r+..respective AND o
the (k) by, would & veiocity
equals the is
axis 2 concentrated. about particlesRADIUS r.
m, }m be axis mass
m,given
same
of distances
ofthe
of i s the and
rotation. m4 r, by,
root rotation axis small distribution of OF
m as =m; i=l n
inear
mean with its of masses
from
rotation mass GYRATION
moment at veiocity:
PHYSICS
square its dm the
which, is m,
actual given axis
may from m,,
of of if of
