Summary Mind map Physics : Textbook For Class 11 - NEET, JEE chapter rottational motion

Centre of Mass Centre of mass for various shapes Motion centre of mass Moment of Inertia Theorm of Perpend
velocity of centre of mass
•Single point where the total Uniformly distributed mass centre of mass → → → i) For discrete system of particles Iz=Ix + Iy
mass is concentrated → M1V1 + M2V2 + M3V3 +--
Vcm=
•For homogeneous object, M1+M2+M3+------ m
2
centre of mass lie at 1) Rod m
L/2 L/2 Acceleration of centre of mass
1
Note:-
their geometric centres r1
r2 m __ For n particles We cannot apply perpen

_
L → → →
3
__
__
,O M1a1 + M2a2 + M3a3 +-- r3 __
m theorm in case of diago
•Centre of mass, may or may 2 → __
__
__

acm= __
_ n
diagonals are not mutua
not lies inside the object (0,0)
M1+M2+M3+M4------ __
___
__
rn
_
__
•Centre of mass, may __
__


change it‛s location x
2) Square Lamina
Isolated System
L
M1r12 + M2r22 + M3r32+Mnrn2
•No external forces acting I =
L L

C.M
on the system M1+M2+M3+------Mn
L L ,
2 2
Centre of Mass For •They can have mutual force
L of attraction
System of n Particles Disc
M1 M2
3) Semicircular Ring < ii) For non-point mass
r2 <
x
Σmr r1
rcm= y
Σm m2 M1r1=M2r2
(x1y1z1) I = ∫dI = ∫r2dm
(x2y2z2)
O, 2R
m1
= ∫dm r2 m
C.M m3
(x3y3z3)
r1
r2 (xcmycmzcm)
for n particles (O,O) Moment of Inertia
r3 mn
4) Hemispherical shell
rcm
(xnynzn) r1
r dm It1= 5 MR2
rn I=Mr2 M 4
MR2
x I=
2
O, R  m= Mass of body

2 r= Perpendicular distance from Hollow cylinder
General Equation (O,O)
the axis of rotation
m1r1 + m2r2 + m3r3 + -----mnrn Moment of Inertia I=MR
rcm= 5) Solid circular cone
m1 + m2 + m3 + -----+ mn
Tensor Quantity I=MR2 Rotational
Parallel Axis Theorm
analogous
In terms of Cartesian co-ordinates of mass Conditions:-
h O, h
1) Axis where considered to be parallel
m1x1 + m2x2 + m3x3 + -----mnxn 4 to each other
xcm=
m1 + m2 + m3 + -----+ mn
(O,O)
Two Point Mass 2) One of the axis must pass through
centre of mass
ICOM= M1r21 + M2r22
m1y1 + m2y2 + m3y3 + -----mnyn Iparallel ICOM
6) Solid hemispherical
ycm= a
m1 + m2 + m3 + -----+ mn r M
O, 3R
COM It= 2MR2
m1z1 + m2z2 + m3z3 + -----mnzn 8
com
zcm= (O,O)
M1 r1 r2 M2
m1 + m2 + m3 + -----+ mn
m2r m1r
r 1= r 2=
m1+ m1 m1+ m2

m1m2 Iparallel=ICOM+Ma2 I=Moment of Inerti
Cavity in object Icom= mreal mreal=
Centre Of Mass For m1+m2
Non Point Mass If some mass is removed from a
body C.M will shift towards

the side with more mass