FYC7_ SYSTEM OF PARTICLES AND ROTATIONAL MOTION_FORMULAE
Page 2
‘n’ number of particles of masses m, 2m, 3m, ....nm are at distances x1= 1, x2 = 4, x3 = 9, ......
xn = n2 units respectively from origin on the x-axis. The distance of their centre of mass
from origin is________
………. Position vector of centre of mass
……….
……… …….
n-particles having masses m1, m2, .....mn are with position vectors , , … … … . respectively. The position vector ( ) of their C.M is
………. ………
∑
!
Position vector of Centre of mass of continuous mass distribution.
A rigid body may be treated as a continuous distribution of matter.
The ‘particles’ then becomes differential mass elements ‘dm’ and the summation Σ
becomes integration #.
The integrals are evaluated for all mass elements in the object such that
# $%
!
& ! # &$%
' ! # '$%
In case of continuous distribution of mass to locate the position vector of centre of mass we
use the formula
# $% 1
* $% )
# $%
Few important applications for location of position of centre of mass
If two circular discs of radii ‘r1‘ and ‘r2‘ of same material, same thickness are kept in contact
then the distance of centre of mass of system from centre of a disc of radius ‘r1‘ is given by
+
, - + , - If two solid spheres of radii ‘r1‘ and ‘r2‘ of same material are kept in contact, then the
distance of centre of mass of the system from centre of a sphere of radius ‘r1’ is given by
+
, - +
, - For an equilateral triangle the centre of mass is at its centroid.
Distance of centre of mass of a uniform cone of height ‘h’ and base radius R, from the vertex
.
on the line of symmetry is
.
Formula to find shift in centre of mass when a small portion of mass is removed from a
uniform body.
Suppose a small portion of mass is removed from a uniform body as shown in the figure.
98694<6=
/012342567386429:// ; +9
869:17=68
, =
Where
= >>? = distance between original centre and centre of removed mass.
Negative sign implies that the shift of centre of mass is away from the side about which
mass is removed.
9
|Shift of Centre of mass| = +9 8694<6= , =
869:17=68
A circular portion of radius ‘r’ is removed from one edge of a circular plate of radius ‘R’
with uniform thickness, as shown in the figure. The shift in centre of mass is __________
JKLMNKO
@ABCDECFGHDGEC%I@@ ; +
JKLPOKJ
,$
Here,
%QRSQT U AV , %QWXTQ UY AV ; U AV UY ; AV
⇒ \ABCDECFGHDGEC%I@@ ; +] ^ , Y ; $ ZZ Y ; Negative sign implies that the shift of centre of mass is away from the side about which
mass is removed.
Page 6
Moments of inertia of different bodies about different axes –Table
S.NO.
1.
2.
Body
Thin rod (of mass M
and length l)
Circular ring (of
mass M and radius
R)
Axis
Moment of
Inertia
About an axis passing through the centre of
mass and perpendicular to its length
_`a
About an axis passing through one end of the
rod and perpendicular to its length.
_`a
About an axis passing through centre of ring
and perpendicular to its plane (generator axis or
transverse axis)
_ca
?a
b
About a tangent perpendicular to plane of ring.
a_ca
About a diameter of ring
_ca
a
,
b_ca
a
About a tangent in the plane of ring
3.
4.
Circular disc (of
mass M and radius
R)
Rectangular Lamina
(of mass ‘M’, Length
‘l’, Breadth ‘b’)
About an axis passing through centre of disc
and perpendicular to its plane(generator axis or
Transverse axis).
_ca
a
About a tangent perpendicular to plane of disc
b_ca
About a diameter of disc
_ca
About a tangent in the plane of disc
e_ca
About an axis through its centre and parallel to
the length in its plane
_fa
?a
About an axis through its centre and parallel to
the breadth in its plane
_`a
?a
About an axis through its centre and
perpendicular to its plane
_+
About an axis through its edge and parallel to
its length in its plane
!g About an axis through its edge and parallel to
its breadth in its plane.
_`a
About an axis through its corner and
perpendicular to its plane
_
a
d
d
`a fa
,
?a
b
b
`a - fa S.NO.
5.
Body
Square lamina (of
mass M and side a)
Axis
About an axis through its centre and parallel to
any side in its plane
About an axis through its centre and
perpendicular to its plane
Solid Sphere (of
mass M and radius
R)
8.
h
_:a
About an axis passing through diameter
(generator axis)
Hollow sphere (or)
Spherical shell (of
mass M and radius
R)
About an axis passing through diameter
(generator axis)
Solid cylinder (of
mass M , radius R
and length l)
About the axis of symmetry (an axis passing
through the centre along the length)
Hollow cylinder (of
mass M , radius R
and length l)
_:a
About an axis through its diagonal in its plane
About a tangent
About an axis perpendicular to length and
passing through the centre
9.
?a
_:a
About a tangent
7.
_:a
About an axis through its edge in its plane
About an axis through its corner and
perpendicular to its plane
6.
Moment of
Inertia
About the axis of symmetry (an axis passing
through the centre along the length)
About an axis perpendicular to length and
passing through the centre
b
?a
a_:a
b
a
_ca
i
_ca
a
_ca
e
e
b
e
_ca
b
_ca
a
`a
_ +?a -
ca
,
ca
,
d
_ca
ja
_ +?a -
a
Page 11
Comparison of translational and rotational motions
S.No.
Translational motion
Rotational motion
1.
Liner displacement = /
2.
Linear velocity = <
Angular displacement = k
3.
Linear acceleration = a
Angular acceleration = m
4.
Mass = m
Moment of inertia = I
5.
o 9<
o
Linear momentum = on
ooo
Angular momentum = op ql
6.
7.
o Force = r
Angular velocity = l
o
=s
o Torque = t
=3
o. oooo
Work u # r
=/
?
sa
o
=p
=3
Work u # t
o. ooooo
=k
?
8.
Translational KE = a 9<a a9
Rotational KE = a qla 9.
Work –energy theorem
?
?
v r/ 9<a2 ; 9<a1
a
a
o
Power = n or. <
Work – energy theorem
?
?
v tk qla2 ; qla1
a
a
ooo
Power = n t
o. l
< x - :3
l2 l1 - m3
10.
11.
12.
13.
14.
15.
Linear impulse w # r =3
y x3 -
:3a x - <3
a
a
<a ; xa a:/
?
y7 x - : z7 ; {
a
pa
aq
Angular impulse w # t =3
l1 - l2 ?
k l1 3 - m3a 2
a
a
la2 ; la1 amk
?
k7 l1 - m z7 ; {
a
Rolling motion
Rolling
If a body rotates about a fixed axis, and at the same time if the axis of rotation translates then
the motion is called combined translatory and rotatory motion (or) rolling motion .
Contd……