Rotationl Motion Notes and MCQs

Notes 40 Pages

Joint Entrance Examination (Main)

Physics

Contributed by

Yash Kuruvilla

107
Unit - 5
Rotational Motion
Page 1
108
SUMMARY
* Important Formula, Facts and Terms
1. Centre of mass of system of particles






n
1n
n
n
n
n21
n
n
2
1
1
1
cm
m
rm
mmm
rmrmrm
R
for rigid body


n
n
cm
rmR.M
for Two body System

2
2
1
1
rmrm 
OR
1 2 1 2 2 1
1 1
2 2
r m r +r m +m
+1= +1 OR =
m m
r r
 
 
1 2 1 2
2 1
1 1 2 1 2
2
r m +m rm rm
= r = OR r =
m m +m m +m
r
 

 

2.
1 2 n
1 2 n
cm
m v + m v + ......... + m v
V =
M
  

1 2
.........
cm
n
P M v p p p
    
    
similarly
M
amamam
a
n
n
2
2
1
1
cm


n21
cm
FFFaMF 
3. Torque =
T
=
F
r

=
I

sinrF
 


= product of force and perpendicular distance between point of rotation
and line of action.
Angular momentum =
L r P I   
   
|
L
| = rpsin

= product of linear momentum and perpendicular distance between point of
rotation and line of action.
Moment of inertia = I =
2
nn
2
22
2
11
rmrmrm 
=


n
1n
2
nn
rm
Page 2
109
4. Low of conservation of angular momentum
As
L I 
 

p
d L d
I I
dt dt

   
 
 

d L
dt
  


when 

= then
L
remains constant
Its geometrical representation in planetary motion
Let
dt
dA
is an areal velocity
Then m
dA L dA L
= OR =
dt 2 dt 2m
5. Radius of gyration {K}
As I =
2
nn
2
22
2
11
rmrmrm 
If all particles have same mass then
= m


2
n
2
2
2
1
rrr 


n
rrr
nmI
2
n
2
2
2
1


Here (nm = M)
2
mk

n
)rrr(
k
2
n
2
2
2
1


6. Some relations between linear and rotational motion.
v = rw
v w r 
  
Here w =
d
dt

2
2
d w d
dt
dt

  


r T
a a a 
  
where
T
wv r
r
a a  
2 2
2 2 2 2
r t
a a a v r     


2 2 2 2 2 4 2
r r r        
7. Equilibrium of a rigid body.
When
0FFFF
n2
1

it is in linear equilibrium
When
2
1
0
n
P P P P      
   
it is in rotational equilibrium
8. Two theorm for moment of inertia
YXZ
III 
Theorm of perpendicular axis.
2
cm
MdII 
Theorm of Parallel axis.
Page 3
110
9. Rolling down of body on an inclined plane.
V
2
2
2
2
2 2 sin
,
1
1
gh
a
K
K
R
R
 

 
 
 
 


 
 
 
 
Condition for rolling without sliding

 
,
K
R
1
tan
2
2
s



For ring
 tan
2
1
s
(K = R)
disc







2
R
Ktan
3
1
s
solid sphere








 R
5
2
Ktan
7
2
s
10. Rotational Kinetic Energy.
R.K.E =
2
2 2 2
2
1 1
&
2 2
V V
I MK I MK
R
R
 
    
 
 

I2
L
2

Total kinetic Energy = Rotational K.E. + Linear K.E.

2 2
2 2 2
2 2
1 1 1
1
2 2 2
V K
MK MV MV
R R
 
   
 
 
Now (1)
2
2
R
K
E.
LinearK
E.K.R

(2)
2
2
2
2 2 2
2
.
.
1
K
R
K
R
Rotational K E K
Total K E
R K
 


Percentage rotational K.E. =
%100
1
2
2
2
2
R
K
R
K










Page 4
111
(3)
2
2
R
K
22
2
1
1
KR
R
E.KTotal
E.KnalTranslatio




Comparison between physical quantities of linear motion and rotational motion
Translational motion Rotational motion
Linear displacement,
d
Angular displacement,

Linear velocity,
V
Angular velocity,
w
Linear acceleration,
dt
vd
a 
Angular acceleration,
dt
wd

Mass, m Moment of inertia, I
Linear Momentum,
vmP 
Angular momentum,
wIL 
Force,
amF 
Torque,
dt
Ld

Newton's Second Law of Motion, A result similar to newtown's Second Law,
dt
Pd
F 
dt
Ld

Translational kinetic energy K =
2
mv
2
1
Rotational kinetic energy K =
2
1
I
2

m
2
p
2

I2
L
2

Work, W =
dF
Work, W =

Power, P = Fv Power, P =
w
Equations of linear motion taking place Equations of rotational motion taking place
with constant linear acceleration with constant angular acceleration :
v = V
O
+ at w = tW
O

d =
2
O
1
v t+ at
2
2
0
at
2
1
tW 
2ad =
2
2
0
v -v
2a =
2
0
2
WW 
Law of conservation of linear momentum Law of conservation of angular momentum
when
0F 
then
P
is constant Impulse when
0P 

then
L
is constant Impulse
linear
12
PPtF 
rotational
2 1
t L L   
  
Page 5
Value of V, a, t for some Rolling Bodies
Shape of Body velocity velocity acceleration Time
2
2
R
K
Ring/Hollow cylinder
gh
 
2
1
singl 
sing
2
1
sing
2

1
Disc/Solid cylinder
gh
3
4
2
1
singl
3
4







sing
3
2
sing
3
2
1
Solid Sphere
gh
7
10
2
1
singl
7
10







sing
7
5
sing
14
5
2
Shell/Hollow spher
gh
5
6
2
1
singl
5
6







sing
5
3
sing
10
3
2
Moment of inertia an radius of gyration for some symmetric bodies
Body Axis Figure I K For rolling
body
2
2
R
K
Thin rod of Passing through its
2
ML
12
1
32
L
-
Length L centre and per pendicular
to its length
Ring of Any diameter
2
MR
2
1
2
R
-
radius R
Ring of Passing through its
2
MR
R 1
radius R centre and perpen-
dicular to its plane
Circular disc Passing through its
2
MR
2
1
2
R
2
1
radius R centre and perpen-
dicular to its plane of
Circular disc Any diameter
2
MR
4
1
2
R
-
of radius R
Page 6
113
Body Axis Figure I K For rolling
body
2
2
R
K
Hollow Geometrical
2
MR
R 1
cylinder of axis of the
radius R cylinder
Solid cylinder Geometrical axis
2
MR
2
1
2
R
2
1
of radius R of the cylinder
Solid sphere Any diameter
2
MR
5
2
R
5
2
5
2
of radius R
Hollow Any diameter
2
MR
3
2
R
3
2
2
3
sphere
of radius R
MCQ
For the answer of the following questions choose the correct alternative from among the given ones.
1. The centre of mass of a systems of two particles is
(A) on the line joining them and midway between them
(B) on the line joining them at a point whose distance from each particle is proportional
to the square of the mass of that particle.
(C) on the line joining them at a point whose distance from each particle inversely
propotional to the mass of that particle.
(D) On the line joining them at a point whose distance from each particle is proportional
to the mass of that particle.
2. Particles of 1 gm, 1 gm, 2 gm, 2 gm are placed at the corners A, B, C, D, respectively
of a square of side 6 cm as shown in figure. Find the distance of centre of mass of the
system from geometrical centre of square.
{A} 1 cm
{B} 2 cm
{C} 3 cm
{D} 4 cm
3. Three particles of the same mass lie in the (X, Y) plane, The (X, Y) coordinates of their
positions are (1, 1), (2, 2) and (3, 3) respectively. The (X,Y) coordinates of the centre of
mass are
{A} (1, 2) {B} (2, 2) {C} (1.5, 2) {D} (2, 1.5)
Page 7
114
4. Consider a two-particle system with the particles having masses
1
M
, and
2
M
. If the first
particle is pushed towards the centre of mass through a distance d, by what distance should
the second particle be moved so as to keep the centre of mass at the same position?
{A}
21
1
MM
dM

{B}
21
2
MM
dM

{C}
2
1
M
dM
{D}
1
2
M
dM
5. Four particles A, B, C and D of masses m, 2m, 3m and 5m
respectively are placed at corners of a square of side x as
shown in figure find the coordinate of centre of mass take A
at origine of x-y plane.
{A}






10
x7
,x2
{B}






7
x10
,x2
{C}






7
x10
,
2
x
{D}






10
x7
,
2
x
6. From a uniform circular disc of radius R, a circular disc of radius
6
R
and having centre at a
distance +
2
R
from the centre of the disc is removed. Determine the centre of mass of remaining
portion of the disc.
{A}
70
R
{B}
70
R
{C}
7
R
{D}
7
R

7. A circular plate of uniform thickness has a diameter of 56 cm. A circular portion of diameter
42 cm. is removed from +ve x edge of the plate. Find the position of centre of mass of the
remaining portion with respect to centre of mass of whole plate.
{A} - 7 cm {B} + 9 cm {C} - 9 cm {D} + 7 cm
8. Two blocks of masses 10 kg an 4 kg are connected by a spring of negligible mass and placed
on a frictionless horizontal surface. An impulse gives velocity of 14 m/s to the heavier block in
the direction of the lighter block. The velocity of the centre of mass is :
{A} 30 m/s {B} 20 m/s {C} 10 m./s {D} 5 m/s
9. A particle performing uniform circular motion has angular momentum L., its angular frequency is
doubled and its K.E. halved, then the new angular momentum is :
{A} ½ {B} ¼ {C} 2L {D} 4L
10. A circular disc of radius R is removed from a bigger disc of radius 2R. such that the circumferences
of the disc coincide. The centre of mass of the remaining portion is R from the centre of mass
of the bigger disc. The value of

is.
{A} ½ {B} 1/6 {C} ¼ {D} 1/3
11. Three point masses M1, M2 and M3 are located at the vertices of an equilateral triangle of
side 'a'. what is the moment of inertia of the system about an axis along the attitude of the triangle
passing through M1, ?
{A}
 
4
a
MM
2
21

{B}
 
4
a
MM
2
32

{C}
 
4
a
MM
2
31

{D}
 
4
a
MMM
2
321

Page 8
115
12. A body of mass m is tied to one end of spring and whirled round in a horizontal plane
with a anstant angular velocity. The elongation in the spring is one centimeter. If the angular
velocity is doubted, the elongation in the spring is 5 cm. The original length of spring is…
{A} 16 cm {B} 15 cm {C} 14 cm {D} 13 cm
13. A cylinder of mass 5 kg and radius 30 cm, and free to rotate about its axis, receives an angular
impulse of 3 kg M
2
S
-1
initially followed by a similar impulse after every 4 sec. what is the angular
speed of the cylinder 30 sec after initial imulse ? The cylinder is at rest initially.
{A}
1
Srad7.106

{B}
1
Srad7.206

{C} 107.6 rad S
-1
{D} 207.6 rad S
-1
14.
Two circular loop A & B of radi r
a
and r
b
respectively are made from a uniform wire. The ratio
of their moment of inertia about axes passing through their centres and perpendicular to their planes
is
8
I
I
A
B

then
ra
rb
Ra is equal to…
{A} 2 {B} 4 {C} 6 {D} 8
15. If the earth were to suddenly contract so that its radius become half of it present radius, without
any change in its mass, the duration of the new day will be…
{A} 6 hr {B} 12 hr {C} 18 hr {D} 30 hr
16. In HC1 molecule the separation between the nuclei of the two atoms is about


m10A1A27.1
10


.
The approximate location of the centre of mass of the molecule is
i
ˆ
A


with respect of Hydrogen
atom ( mass of CL is 35.5 times of mass of Hydrogen)
{A} 1 {B} 2.5 {C} 1.24 {D} 1.5
17. Two bodies of mass 1kg and 3 kg have position vector


k
ˆ
j
ˆ
2i
ˆ

and (-3i-2j+k) respectively
the center of mass of this system has a position vector……
{A}
k
ˆ
2
i
ˆ
2 
{B}
k
ˆ
j
ˆ
i
ˆ
2 
{C}
k
ˆ
j
ˆ
i
ˆ
2 
{D}
k
ˆ
j
ˆ
i
ˆ

18. Identify the correct statement for the rotational motion of a rigid body
{A} Individual particles of the body do not undergo accelerated motion
{B} The center of mass of the body remains unchanged.
{C} The center of mass of the body moves uniformly in a circular path
{D} Individual particle and centre of mass of the body undergo an accelerated motion.
19. A car is moving at a speed of 72 km/hr the radius of its wheel is 0.25m. If the wheels are
stopped in 20 rotations after applying breaks then angular retardation produced by the breaks
is ……
{A} -25.5
2
s
rad
{B} -29.5
2
s
rad
{C} -33.5
2
s
rad
{D} -45.5
2
s
rad
20. A wheel rotates with a constant acceleration of 2.0
2
secrad
If the wheel start from rest. The
number of revolution it makes in the first ten seconds will be approximately.
{A} 8 {B} 16 {C} 24 {D} 32
21. Two discs of the same material and thickness have radii 0.2 m and 0.6 m their moment of inertia
about their axes will be in the ratio
{A} 1 : 81 {B} 1 : 27 {C} 1 : 9 {D} 1 : 3
Page 9
116
22. A wheel of mass 10 kg has a moment of inertia of 160 kg
2
m
about its own axis. The
radius of gyration will be _______ m.
{A} 10 {B} 8 {C} 6 {D} 4
23. One circular rig and one circular disc both are having the same mass and radius. The ratio of
their moment of inertia about the axes passing through their centres and perpendicular to their
planes, will be……
{A} 1 : 1 {B} 2 : 1 {C} 1 : 2 {D} 4 : 1
24. One solid sphere A and another hollow sphere B are of the same mass and same outer radii.
The moment of inertia about their diameters are respectively
A
I
and
B
I
such that…
{A}
BA
II 
{B}
BA
II 
{C}
BA
II 
{D}
dB
dA
I
I
B
A

(radio of their densities)
25. A ring of mass M and radius r is melted and then molded in to a sphere then the moment of
inertia of the sphere will be…..
{A} more than that of the ring {B} Less than that of the ring
{C} Equal to that of the ring {D} None of these
26. A circular disc of radius R and thickness R/6 has moment of inertia I about an axis passing through
its centre and perpendicular to its plane. It is melted and recasted in to a solid sphere. The moment
of inertia of the sphere about its diameter as axis of rotation is …
{A} I {B}
8
I2
{C}
5
I
{D}
10
I
27. One quater sector is cut from a uniform circular disc of radius R. This
sector has mass M. It is made to rotate about a line perpendicular to
its plane and passing through the centre of the original disc. Its moment
of inertia about the axis of rotation is…
{A}
2
MR
2
1
{B}
2
MR
4
1
{C}
2
MR
8
1
{D}
2
MR2
28. A thin wire of length L and uniform linear mass density  is bent
in to a circular loop with centre at O as shown in figure. The moment
of inertia of the loop about the axis xx' is ….
{A}
2
2
L
8


{B}
3
2
L
16


{C}
3
2
5 L
16


{D}
3
2
3 L
8


29. Two disc of same thickness but of different radii are made of two different materials such that
their masses are same. The densities of the materials are in the ratio 1:3. The moment of inertia
of these disc about the respective axes passing through their centres and perpendicular to their
planes will be in the ratio.
{A} 1 : 3 {B} 3 : 1 {C} 1 : 9 {D} 9 : 1
Page 10

Rotationl Motion Notes and MCQs

Rotationl Motion Notes and MCQs

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