..

. •,

..
''
..

: U~\1': S
~ ' '
I··•

'
ROTATIONAL DYAMIC·~
~ ' ' '
I

~
-~ 1. A wheel is rotating at a speed of 1000 rpm and its KE is 106.J . Wha_t is moment of inertia of .
the wheel about its axis of rotation? .
[Ans. 182.56 kg m- 1 2 ·• .,
~ 2. A wheel of mass 5 kg and radius 0.4 m is rolling .on a road wjthout sliding with angular
. .
~~ velocity . . .: . · 2
10 rad s-1 • The moment of inertia of the wheel about its a~Is ¢>f (0tatIon Is 0.65 kg m . What
is the percentage of KE of translatibn in the total K.E of the-'lf1eel ?
[Ans. 55.2 %) ·
3. A thin metal hoop of radius 0.25 ~ and mass 2 kg starts from rest and rolls down an inclined
plane. If its linear velocity on reacnlng the foot of. the Riane is 2 mis, what is its rotation KE at
that instant?[Ans. 4 J] · ·
4 . A circular disc of mass M and radius r is. set rolling on·· a, table. I
If OJ be its angular velocity,
show that its total K.E. is given by (3/4) Mv ., where ·v is~~ linear velocity . M.I. of circular
2

disc= (1/2) mass)( (radius)2. . :. . . •.' . 1 •

~ 5. A s9lid cylincJ~r rolls down an inclined plane. Its mass i·s i. kg and. radius 0.1 m. If the higher
of the inclined ~cine is 4 m, what 1s its rotational kinelic energy when it r~aches the foot of
-~ the plane? Assume that the surfaces are sn:iooth.·
(Ans. 26 .13 J) · ·
~ ' .
6. The earth h~s a mass of 6x 10 kg ~nd a radius of 6 .4 x 10 m . Calculate the amount work
24 6

~ in joules that must be done if its rotation were to be slowed down so that tt\El duration the of
day becomes 30 hours instead of 24 hours. Moment of inertia of -e·arth =- (2/5) mass x
~ =- (radius)2. . ·. . . · :. " · ·
' ,
lI
=-ic5 7. A buck~t of rnass 8 kg. fs supporte'd by a light rope ~ound aro~!"d a solid wooden cylinder of I
mass · ·- · _ ,
\·!
12 kg and radius 20 cm, free to rotate about.its- axis. A man holding the free er\d of .the rope
~ In a well 50m
with b_ucket _and the.cyllnder at rest inltlally, lets g~ the bucket freely cf ownward
deep. Neglecting fnctlon, obtain the sped of b~cket and angular speed of the cylinder just
~ Take g = 10 m/s2
before the bucket enters water. . .

~ [Ans. 23.9 m/s ; 119.5 rad/s}

2
8. If a constant torque ~f 500 N-m turns a wheel of moment of inertia 100 kg m about .an axis
passing through its centre, find the gain in angular velocity in 2s.
[Ans. 10 rad/sl
is rotating about its central· axis at an angular
9. A cylinder of length 20 cm and ra~us 10 cm _
speed of 100 rad~s. Y:Jhat ta1Jge~t1al force! ~111 s~~P the cylinder at a uniform rate in 1Os?
Given moment of inertia of the cylinder about tis axis of r.otation is 0.8 kg m2.
lAns. 80 N1 '
Fig. 5 (b) 32. A string
1o. A wheel of radius 10 ~m can rotate freely about its centre as shown in

. . -
r ,
wrapper over its _rim is pulled2 ~Y a force, of 5 N. It is found that the torque produces an
angular acceleration of 2 rad/s in th e whe~I. Calculate the moment of inertia of the wheel.
I
! 5l
-
/
 vr
~
1
~
~
1
~
!1N
11. A flywheel of moment of inertia 5.0 kg m2·is rotate
d at a speed of 60 rad/s. Because of
friction on the axle, it comes to rests in 5 minutes. Find
total work done by the friction (c) Angular momentum
(a) average torque. of the friction (b)
of the wheel one minutes before it .9
';: , stops rotating.. _ .,,, . . . ·
• ,"f > .\ ' > ! e ti : ., •
,. i . • ,.,-: ~ ''

.
[A11s. (a)--.1 Nrtr(b)·9 kJ (o} 60.~g m s l
A • • , ,•
· ·. ~
2 -1
... :y.~~
12. A boy is .seat~d ·In a_revolving .chair, revol~ir,g at
~r.l 'ar;igular.:~p-eed .of ·1~~:.c rpm. ·By ·some 5
~
ar~ang~me~t, the b_o~ decreases th~ moment of inerti
a·of the syst~m from ~ ~9- m2 to 2 kg
m . What wdl-.be the new angular speed?
. ·- - : [Ans ..360 rpm] C;;z
.
1a:-u the · eartfi: expands siiddenly to twice its . diam
:, [AAS :·91,1,} ·. < , ete'r, ~hat w'ou.ld .b~ le~gth ~f the day? C::
; . . I . ·· .• .
C::
o!
14. A circular ~!s~ ~ome~t Qf:_!ne~ia (1 is rationing
·in .a horizontal plane about its symmetry C:::
-axis _with_a-constant a!lgH(ar 'lelocity ·w ·.: Another disc
1 1 of moment of inertia I b . is dropped co- ~
- axially onto the-rofatlng-~_dis~·: lniti~lly, the s·econd· disc
has zero angular speed . .Eventually ,
initially rotating disc due·to f[ictio'ri.
'*
both the discs· rotate with conJ:;tanr an_gular' speed
WI • Calculate th-e· energy lost by the C::
.· 1 ~
[Ans. ;,..-Jbco co ]
·. 2 ' 1 ~
15. Three particles, each of mass m, '3re situated at
the -~ertices of .an equilateral triangle ABC
of side L Find the moment bf inertia of the system abou
t the line AX-perpendicular to AB in ~
the plane of ABC. · · ·
:: X.'
~
I •
~
:•
i
,': C

~
~
~

'~

~
52
~
~
 ~

~
, .
..... I

16. F®r par°"'i of masses 4 kg , 2 kg , 3 kg and 5 kg are respectively lo¢ated at,the 1oµr
~ comers P-., a, c, o o1 a g~uare oi s\de '\ m . .Ca\cu\a\e \he momen\ oi \nen,a o1 ·t he S'1tt\.em

~
about
(i) The axis passing through poin\ of in\erseq'tion of \he diagona\s and perpendicu\ar \o the
~ plane of the square (ii) diagonal BD.
{Ans. 7 kg m2: 8 kg m2; 3.5 kg m2]
1
!
~ 17. A rod of length L, whose lower end is resting along the horizontal plane, starts to topple from
~ the vertical position. What will be the velocity of the upper end when it hitl the ground?
·[Ans . .J3gD] . ·· '
~ ·: I .
18. Two small balls A and B each of mass ni~ fJre attached. rigidly to the ends of a light rod of
~ length d. The structure rotates about the- perpendicular .bjsector of the rod at an angular
, speed m Calculate the angular mqmentum of the individual balls and of the system about
~
1
the axis of rQtation . • •

~ [Ans·. .!.. ~m d 2 • .!..m (J) d 2 ]

·. 4 ' 2
~ 19. An equilateral tricfngle is formed qY, thre·e identical uniform rods ea ch of length 40 cm and
mass 2 kg . Find the moment of inertia of the system about an axis pass!ng through one
~ corner. ....
~ SoJution : Tl)e situation is shown:.in figure. Suppose L be the.length of each rod and M the
mass. We have to find the moryieni·of inertia of the equilate.ral .triangle a~!4 an ~xis passing
~ through the corner C and perpendi9ular to plane ABC. -If I 1, '2..h .be the moment of inertia of
rods AC , CB and AB respe ctively about C, then the moment of inertia of the system is
:::;.; • ' ~ t ..

I = 11 +12 + I 3 •• • ; '
~

~
Now

_ _ ML ML _
2 2
ML2 ·J
Il /C /

.,r;.
<) [ JI - Ii - -12- +4- - ,/ /
3 ·. I
~
"·-/_. !--=--
- - - - L :::::====:5
~ -
ML2 ( l ✓3J 2
• • : ··
I , =-+M _..,_ (according to theorem of parallel axes)
~ 12 2
. '
~ Ml} M .3l2 5ML2
=-+ =--
~ 12 4 6
2 '
!!) ML ML2 5 2 9 3 2 ,· I
:. 1 = - + - + - ML =-M 2 =-x 2x{0.4) ;: Q.48~g xm~
3 3 6 6 2 : ,
.
!' •
' ''
i
20. A_solid ~ oo_de~ sphere :oils down the differ~nt incline~ planes of the same height but
!) different mclmal1ons . W~II 1t_ rea~h the bottom with the same speed in each case? Will it take
longer to roll down one tnchne~.,t:>lan e than the other? Ifs~ which one and why?
;, •. '

~
fl 53·. ,.
 , . :·. . . . .
cl
Solution : The ve . ~ I

c
locity ac~uired PY n in cl in ed pl an e
·g .body of m a of
he ig ht hi sg iv en by s ~ 1olhng d o
~ __ _ C
0- ~~
v= (1/:ig;R 0
~

As for a solid sphe

2)
\ _ _ _ il,_!_!Z/ ~
re of radi~~ R, I~ 2 . i
~
(2/5~ MR

:. v =
( 2Mgh \
(M +(2/ S)M) =
1
~cl g\
1 f :-.. ~
l , l/ ~ 'L
~

e
·· · ·
Because this
j_p__.f : \v
1s independent of : · · ·
the bottom with th the mchnat1on 0 df h
o t e p Iane. Hen e th e sp he re ~
e same velocity in c will re ac h
each case .
Now the time of de e-=
t= J ~ (M +1-
lgM\
scend of a.rolling
body over an incli
•.· . ~ - - - - - ned plane is gi ve
Yl _l_·,~ - J ~ ( M +: :~- ;,l ~.
n by ~
J_ 'J- :. ~ \
t
71
~
R2 JJ sm0 ·. l_ gM \ R )J sin0 Jot ·n ~ ~
This is turn implie
s \that smaller the
sphere will take lo
nr.er time to reach
inclination, longer
~i ll be th e tim e of ~
the bottom of le~~_ de sc en d. H en ce
21. A carpet of m inclined plane. th e
ass M made of in
cylinder of radius ex te ns ible material is ro ~
fi and is kept on a ro lled along its le ng
ugh floor. Th e :_ca th in th e from of
on the floor when
the axis of the cylin
a
negligibly small pu
sh is given io it.
rpet starts un ro llin a
g w ith ou t sl id in g C:
drical part of the ca Calculate th e ho
rpet when its radi riz on ta l ve lo ci ty
So lu tio n : If L be
the length of the
us reduces to R/
2 .
of C:
carpe~is given by cy lin drical carp~t and
p its de ns ity , th en
· th e m as s of ~
M = ~ R lp2

When the carpet •, .

..
e
unrolls to half of its
initial radius i.e ·~_R/
M '= 1r (R li )2 Lp 2, th en its m as s
is 0
= M l4
When th~ carpet
• ~
unrolls , i~s p~tent
and kinetic energy ial energy decrea
of rotation l'ncre~ ses while its ki ne
ses. Hen,ce ·.
tic en er gy of tra ns
la tio n QI
MgR-(Mg 14) (R /2 ) • •
I
.
= .!__(M / 4~•2 +.!__
Io} ~
2 · 2
l
Here . I= -x mas - l
s x (r ad iu s) 2 = -x
I •

·. ·. 0
(M / 4)x (&/ 2)2 and
2,
2 = _v _ = 2: w ~
. R /2 R
:. M gR - Mg .R =.!..
xM v2 +.!..I.!._ M ( R"\l' x 4v2 C
4 2 2 '4 2 L2 4 \ 2) j , R2
.
7 I
1 Mv2 3
or M gR = 2 ...---.1,_ _
8 8 Mv + --
16
= - Mv2
16
:. v.J(l4g,Rt31
~
j •:· . . th to
22. A small Sphere rolls down witho~t shpp,.ng f; ~m art T~e hbrizontal part is 1m above the
of la track in a vertical plane. Toe

?
~
,track has an elevated section and h?nzon a p . ·e the i ound Find the distance on the
1
ground level end the top of t~e t~.ac~I ~
grou')ds with resp~ct !o the polnt.w c j st'I does
f'\;rt~~~;
pelo.:\he e~d of the track, where ,t~e
the '.sphere continue to rotate about its
I
j.

sphere lands. During its flight as a pro ec I IJ! , . • . . •

~ centre of mass? . . ·. . 1 i .
~ So\ution :Suppose M be the mass ot sph~re and R. be it.s ·radius. let v A be the ~e\~c1\y o~
sphere when it raches at the point A. Since _in ,g~ing .fro_m .th~.top t~f 1~~ ~::~~e:~~~~· ~~
~ falls by a vertlcal height equal to (2A -1 .0) m. ~1ence a~for ing i.
energy, we get ·
~
~ Mg(2.4-1.0)=½Mv~ +½Ia? 1
. .. .. .. . ... .-.. :-. ( 1)

~ Where mis.. its angular velocity at point A

'

~ · Putting 1 - (2/5) MR~ ·and OJ= V0 .IR , we get

..
, V. 2.' ;
~
.; • ·~] ' 7.
, V V ' ,
Mgxl.4=M -:i,+d =M,x-..2.
' . . [ 2 5· · 10 2.4 -
~

~ v; = 2x..9.&=}~.6orvA= ./(f9JJ ml s ~

~
~
At the point A, the sphere leaves the horizontal section as . a. projectile with a ho.rizontal
=4 velocity vA and vertical velocity zero. If it reaches the ground in time 1 and aJ a distance R
from ~then
4 1 2
-x9.8xt =1.0
~ 2 .. .. .. .... .. .. (2)

=' And R=vAt=.J(f9:6) xt ... : .. .. . .... . ' (3)

~ Solving (2) and (3), R = ~ x0.4518= ,2rn
I
~ During its flight as projectile, according to j law of conservation of angular momentum, the
sphere wm continue to rotate about its axis. 1
~ I

23. A uniform disc of radius R is spinned to the angular velocity OJ and then carefully placed on
~ a horlz.ontal surface. How long will the pisc be rotating on the surface if the friction
coefficient Is eq~al to µ? The pressure exerted by the disc on the surface can be regarded
.,!}!, ·as uniform. · . . ·: I · .
'•
_,?, Solution : When the spinning disc is car~fully placed on the horizontal surface, a frictional
force acts on it, due to which a retarding torque is develpped·about the centre of disc.
:, , The disc, can be supposed 'to be· made bf various· c6ncentric elementary strips Let
/' consider one such elementary strip uf rad!us r and width d~. Mass of the elementary ~trip. us

?
•
? 55
r --
.-~
dm = -
M
1i R
2
(2,r rdr}

Weight of elementa~ strip =(M /ml.2 Xmdr )g

~,
~]

The weight will be equal to normal reaction and hence frictional force on the strip.
i!
F =µV =(pM I 1IR2 ) (2trdr)g .
Hence, torque due to frictional force ~'
dr =Fr= µM (2,r rdr) gr
,rR2

Integrating on both sides, net tocqu.e ~ct;ng_on .the dis~

~~
~
µM JI. . ·. .-µM· 21lRl } ,nR2
r = -211' gJr 2dr = Ia a · r=.--2 x-.- g = - a
1tR2 ·. 1CR 3 2
0

4µg (JJ 3(J) R .

a = - andt= - = - - ·
3R a 4~ ' .. c::l
24. A uniform bar of length 61 ~nd ma11 8m hes on a ~mooth horizontal table. Two point
masses m and .2 m..moving in same horizontal piano with'JSpeeds 2v and v respectively strike
e8
the bar as shown in figure and stick to the bar after collision. The angular speed of the bar ~
after collision is. And the total ~ergy' after collision is ... ... .
'.
~
2m
~
~
-r-;-- 3a
\I ~
t •
I
~
-~ a >~< 2a --?- 2 v ~
I
I
I .: m ~
I
cE-
25. A s~ooth uniform rid of length L and mass M has t;·o identical beads of negligible size
each of mass m, w ich can slide freely along the rod~. Initially the two bead, are at the
centre of the rod a d the system is rotating with an : angular speed % about an axis
~
1
perpendicular to the rcid and passing through mid-po!r,t ~f the rod as shown in figure, there
are no external forces. When the beads reach the e~s of the rod, the angular speed of the
system is? ·
I

26. A stone of mass 0.3 kg tied to the end of a string in a:_horizontal plane is whirled round in a
circle of radius 1 m with a speed of 40 rev/min. (a) -What is the tension in the string? (b)
What is the maximum speed with which the stone ~ be whirled around if the string can
withstand a maximum tension of 200 N? -; :
(Ans. 5.27 N, 25.8 Mis) ··
•
56

' ·...
 '1 :: .
~ . -·
~ 27. A long playing record revolves with the speed of 33 1/3 rev/min and has a radius ofi1,t5 cm.
.l
~
Two coins are placed at 4 cm and 14 cm away from the center of the record 0 (1f the
coefficient of friction between the coins and the record Is 0.15, which of the two coins ,· I!
~
revolve with the record? [Ans. A coin at 4 om] · · 1•'

28. A 70 kg man stands against the wall of a cylindrical drum of radius 3 m rotating ~~put its
~ vertical axis with 200 rev / min. The coefficient of friction between the walls and his clothes is
0.15. Wt\at is the minimum rotational speed of the cylinder to enable the man to temain
~ stuck to the wall without slipping when ttie floor.I~ su9de~ly removed?

~ [Ans . 4.7 rad/sec]

' I
. · ·
29. A car is speeding on a horizontal curved road of radius 60 m. The coefficient of friction
~ between the wheels and the road is 0.5, The height of center of gravity of the car from the
road level is 0.3 m and the distance between the wheels is 0.8m. Calculate the maximum
~ safe velocity for the vehicle to negotia\e the curve. Will the vehicle skid or topple if this
velocity is exceede.d? , [Ans. 28 ms-1]
~ 30. What will be. the duration of the day, i} the earth suddenly shrinks to 1/64 of its original
~ volume, mass remaining unchanged. '
[Ans. 1.5 hrs] ·
~ ~1 . A cylinder of mass ·5 kg · and radius 30' cm and free to rotate about its axis, receives an
angular impulse ·of 3 mg m2?_, initially..'fdllowed by a similar i,:npulse afte,every 4 sec. What
~ is the angular speed of the cylinder after 30 sec aft~r the initial' impulse? The cylinder is at
~ rest iriiti~lly. [Ans . 106.7 rad/sec]

•
~
1
32. A ring, a disc, and a sphere all •of the ilmeraidus. and mass rolls down -the inclined plane
from the same height h. Which pf the. three reaches t~e botto·m (a) earliest and (b) latest?
[Ans. (a) sphere, (b) ring] ·

•
33. Two discs of moment of inertia 11 and h about their respective axes (normal to the disc and
passing· thru' tj,e center)° and rQtating with angular speed.s iv1 and cv2 .are brought into

•• contact' face to face with their a~i's

:, , ·of rotatioh coincident.
(a) what is the angular speed of ~Metwo disc system .
·

(b) S~ow that the K.E. of the Qombi ned system is less than the sum of the initial kinetic
energies of the two discs. ·
wlJ .., )2]
~ [Ans. o, Iif 2 , A_(V , - (1)2'
= J, oJ1 + J2w2 +·- -· (
1,+12:· 2 1,+12
)

~ 34. A uniform disc rotating freely ~out a vertical axis makes 90 rev/mjn. A small piece of wax of
mass m gm s falls vertically on·tpe disc and sticks to it at a distan¢e of r cm from the axis. If
~ the number of rotations per minute reduces to 60, find the moment of inertia of the disc.
[An s. 1 = 2 mr2] 1
~ •

35. A metre stick is held vertically.;with one end on the fldnr and is then allowed to fall. Find the
~ velocity of the other end when: it hits the floor, assuming that the end on the floor does not
I slip. [Ans. 5,4 ms- 1J , ;i ·
tJf 36. A constant to,rque of 20 nm is exerted on , a piv~ted b~am
for 1o sec during which the
~I angular velocity of the wheel Increases from 0 to 100! rps . The external torque is then·
removed and wheel comes to rest in 100 1 sec due: to fription at its bearings. Calculate {a)
~
~
57
/
L I. ,
 - -- ~

. ~
! .
.
'
moment of Inertia of the wheel (b) friction~I·torque (c) total number of revolutions made by
the wheels. · . · [An~. (a) I =V,r _kgm 2
(b12 Nm (c-) 5.500]
~
:.,JIC
"

• tI
f
I 37. A chord is wound round the. cir~vmferenc"'e ·of a wheel of diameter of 0.3 m axis of the wheel Q fir"
is horizontal. A 0.5 kg mass Is ~ttacheo at the end of the chord and is allowed to fall from
I . rest. If the weight falls 1.s•ni in 4 sec,',what i, the angular acceleration of the wheel? Also
,,. Jll..,,
., -at"'
find the moment of inertia of the .~he~I. . ~
2 2
[~l'JS.:1.~5 re~/sec , 0.588 k~m ] ~IC
38. A flywheel of mass 100 kg ~nd radius of gyration 20 Gill mounted on a weightless horizontal "fr 11""
axle of radius 2 c·m, free to rotatein frictionless bearings, has a weightless string wound on ~ Ill...
the axle carrying at its free ~nd a mas of 5 kg. Prove that the acceleration of this mass when ~.I I)'
the system is released fro~ rest is g/2001 ms-2 , and:that if the string slips off the axle after ~
the mass has descended 2 m, a couple of 0.318 kg wt m would bring the flywheel to rest in ,.. II.,_,
5 revolutions. . · . c=-::
~r
39. A body .weighing 30 kg is slanding on a flat boat so that his distance from the shore is 20m.
He walk a distance of 8 rn tpwards. the shore on the boat and then stops. If the boat weight a
120 kg, then calculate the boy's djstance from the shore.
[Ans. 98/5 m] ;tt!:':""lb

MCQs
o_;:-8.
t;.-1
1.

2.
(a) 54 radian
I

f. fly wheel orlgin~lly at rest ls to reach an angular velociti of 36 radlan/ s in 6 second. The total angle
11
it turns through the 6 second ls
· (b) 108 radian
:
(c) 6 radian (d) 216 radian
In the ~djoinlng figure along which axis the momen't '.of inertia of the triangular lamina will be
maximum- [Give!) that AB < ,. BC < AC] -
:I
' ~_ft
(a) AB ~ ..
.. .. iJI
~
(b) BC I

(b) CA
(d) For all axis k t

3. Three particles, each of mass m are situated at the ve~~es of an equilateral triangle ABC of side e cm
(as shown in the figure). The moment of ine1t ia of the ~Y,Stem about a line AA perpendicular to ABand
2
~
~
in the plane of ABC, in gram cm units will be :- ·: ;.
(a) 2 mt
2
•
(b)
5 2
-me
X .
C ~
4
3 2
~- [
(c) -mt
2 ~~J
A
(d) Im/ ' ... ~J
4. A ci~cular disc is to be made by using iron ahd 'aluminium so that it acquir~ maximum moment ol
inertia al:iout geometrical axis. It is possible wi\h :- ·
~1
(a) aluminiurrf at interior and iron surrounded to it.
'
l5 g
I
 (b} Iron at Interior and aluminium surrounded to It. .j
(c) using Iron and alumlnlum layers In alternate ord~r. ··:
(d) sheet of Iron Is used at both external surface and alurnmlum s~eet as internal layer.
5 Off two eggs which have identical.'sizes, shapes and we{ghts, orie is raw and the other is half-boiled.
The ratio between the moment of ~ertia of lhe raw egg and tha~ of the half-boiled egg about a central
axis is :-
(a) one (b) greater ~an one (c) less than one (d) incomparable
1
\ . . I 1
6. "(he moment of . Inertia of a rod'· about an axis · ~~rough , Its, ~entre and perpendlcu ar to It Is
12
2
ML (where M Is the mass and Lis the length ~f the ·rod). Toe rod Is bent in the middle so that the two
half make an angle of 60 . The moment o( Inertia of the b~t r~ about the same axis would be :-
. I J
I 2 I 2 , ( .2. ·1 ML
(a) -mL (b) - mL .(cl~ mt.. •r (d) r;;
48 12 24 · . 8v3
2
7. If the mass of hydrogen atom is 1. 7 ; 0- \ and. intei'atomic d~tance in a molecule of hydrogen is4
8
10- cm, then the mornent of inertia 1in kg:n/ J of a molecul~ of hydrogen about the axis passing
through the centre of mass and perpendicular to the line joining the atoms will be:- •
32 24
(a).6.8 . ~0~ · . (b) 1.7 · 10~ · (c:) 13.6 10- 27 (d) 13.6 10-4 7
8. If a body completes one revolution in 7t sec then the moment of inertia would qe:-
(a) Equal to rotational kinetic energy (c) Double of rotational kinetic energy
(c) Half of rotatiqnal kinetic energy : (d) F~ur times of the rotational kinetic energy ,. -
1 ·•

9. I
A rigiq body can be hinged about ~!.I point on lhe x•axis. when it Is hinged such that the hinge is at x, , I"

is
the mome~t ~rinertia given °by I~ / ~zx
+ 99. Th1t x-coord~~te of centre.of mass Is :- {,:. . .
(a) x=2 · · (b) x=0 (c) x=l - · (d) x=3 - , .j '. li

~
10. A rod of mass M and length L is placed in a horizont~l plane with..one end hinged -about the vertical
axis. A horizontal force of F= Mg Is applied at a dist~~ce SL from th; :hklged end. The
·~:l -\
H
1 '
I _,
2 6
::4 angularacceleration of the rod wtll be :-
l'
I •
IL
I
~ (a) 4g (b) Sg (c) 3g (d) 4g
I

SL 4L 4L 4L
~
!1. A person supports a book between finger and thumb as shown (the point of grip ls assumed to
~ be at the corner of the book). If the .book has a weight of W then the person is
producing atorque on the book of
.A (a) W ~ anticlockwise
2
~
(b) W!!_ anticlockwise
~ 2
(cl Wa anticlockwise ~ .
b I
~ (d) Wa clockwise

:+~ 12. A string Is wrapped around the rim of a wh~el of moment of inertia 0.20 kg-m 2and radius 20 crn.
The wheel Is free t? rotate about its axis and 4nltially the wheel Is rest. The string Is now pulled by a
force of 20N. The angular velocity of the string after 5 seconds will be :-
(a) 90 rad/s

(b) 70 rad/ s.
~
A) (cl 95 rad/s
(d) 100 rad/s
}'
''iO
 ..•'
13. In the figure (A) half of the meter scaliz Is made, of wood:•• . f I
whllla the other half O •tee · The woode n
part ls pivoted at 0 . A force F is applied at the end of st~! part. In figure . ted
at d and the same force Is applied at the wooden end:- .:'.
(BJ the steel part is pivo
..1 (b) -
--
(a) More angular acceler ation will be produced In (A ) :: wood &1HI
More angular acceleration will be produced In (B) ·: : \ -:0 ~:~\ p
(c) Same angular acceleration will be produced In both .o·
- - ~ L- _ _1..---;F'!'T"""' QI
conditions IA! 181
(d) lnformation is Incomplete

~
(a) 10
2
2
rad/sec
1 2
.·

is 5100 kg·m then its angular acceleration will ~e :·

• y
· ·
14. In the following figure r and r ~re 5 cm and 30 ~ res~ct lvely.
If the moment of lnerlla of the
wh l
ee
-
QI

QI

3
(b) 10- racl/sec2
2
(c) 10- rad/sec2
1
.
(d) 10- ~/se /
(!;3
15. A non uniform rod OA of liner mass den~ity A.= ~ x (~
= const.) is suspended from ceiling
withhlnge joint O & light string as shOYJl"I In figure. And the ~lar
acceleration of rod )I.1st after the ~
string is cut
(a) 2g
~
L
~
(b) !
L ~
(c) 4g
3L ~
(d) None of these
~
16. A particle ol mass m moves with a con'stant velocity. Which
about Its angular momentwn :
(a) it is zero when it is at a and moving along oa
(b) the same at all points along the line de
t
0
v0 _· ·
of the following sLatements is not corr~
~
1l A c ~
(c) of the same magnitude but oppositely directed at b and d
I
(d)increases as it moves along the line BC
•o•- Jm- x
8
~
17. If the 4Wth were to. suddenly contract to.!.. th of Its present radius
.. . n without any change In Its mass then ~
the duration of the new day will be nearly :-
(a)
24
hour (b) 24n hour (cl
24
hour 2
(d) 24n hour
~
n
~
l • ,
n . .
18. The angular velocity of a body chang~ from ro to ro without applyin
l 2 g , torque. The ratio of Initial
radius of gyration to the final radius of gyration is :- ~
(a) .ra;; :~ . (b)~ :fl; (c) roe: rol {d) (ill : 00 2
19. A circular tum table has a block of lee pla4ed at Its centre. The system
rotates with an angular speed
ro about an axis passing .through the centre: of the table. If
the ice melts on its own without any
evaporation, the speed of rotation of the system :- .
(a) becomes zero .. · : (b) r~in s constant at tl-ie same value of Cil
(c) increases to value greater thar) 'Q) • · (d) decreases to a value less than Cil
.1.
I 60
 20. A thin circular ring of mass M and radius 'r' i~ rotating about its:axls
with a constant angular velocity
Four objects each of mass m, are kept gently to the oppt>slte ends of two
ffi .
perpendicular diameters
of the ring. The new angular ve'9clty of fhe ring will be :- ··
(a) Mro • ( b ) ~ ._-: . · (c) (M+4 m)c.o · (d) (M+4m)co
4m M+4m : :. M M+4m
21. A person Is standin g on the edge of a tlrcular p\atfonn , whlch Is
mo\J\ng w\th constan t angulat t . ~
aoou\ an u~ ?M\ \n~ \ntoog'n \\, c.~\te. and ?et-pend\cu\at \o \'ne.
?\ane. o\ ?\atlotm . \\ \?enc>n \,
mov\ng a\ong am; radius toward s axis 6,1 rotation then the angular ve\od \)I
w\\\ :-
{a) decreaw , lb) remain unchan ged
(c) Increase (d) data ls Insufficient
22. An ant Is sitting at the edge of a rolatifig disc If the ant reaches the other
end, after moving along the
diameter, the angular velocity of the disc will ··
(a) remain constant (b) first decreases and then' increases
(c) first increases, then decrease :.
a
(d) Increase continuously
23. A boy stands over the centre of ~1orlzonlal platf 01m which Is
j
rotating freely with a speed of 2
revolutlons/s about a vertlcal axis thtough the cen tre of the platfonn and
st alght up through ,the boy.
He holds 2 kg masses in each of his hands close lo his body. The combin
2
ed moment of inertia of the
system is 1 kg-m. The boy now st~etches his anns so as to hold the masses
far from his body. In this
situation the moment of inertia of t~e system increases to 2' k~-m.2. The
kin et it energy of the system

~ in the latter case as compa red with that in the nrevlous case wilt-.·
(a) Remain unchanged (b) Decn=lase
-
'I
~ (c) Increase · · (d) Re~~ln uncertain
24. A horizontal platform is rotating with uniform angular velocity arq.md the vertical
rl
~ its centre. At some instant of time a viscous fluid of mass ".'m" ls.i:1ropp
axis p~ssing through I
ed at the centre and Is allowea
to spread out and finally fall. The angular velocity during this perl~d :-
~
(a) Decreases continuously (b) Decreases ini\ialty and Increases again
~ (c) Remains unaltered
• I
(d)' lncreases cor-rtinuously

~ 25. A particle starts from the point (Om;.Sm) and moves with ~nif orm!velocity
of 3I mis . After 5 seconds,
the angular velocity of the particle a~ut the origin will be I
I
~

--
y
8
(a) -rad /s
289
(b) I r.~d/ s 3ml,
~ 8 -:•,
f
24 g·

--~--·
8m
(c)- rad/s
~ 289
(d) -
17
rad/s ·1 .
0
26. Two rotating bodies have same angular momentum . but their \Jnome
~ respectively (1 >1 ). Which body will have higher~i11~ic ~n~~- o~ ~otation
nts of inertia are 1 and 1
1 2
1 2 :-
~ (a) First · (b):Second· .. .
(c) Both will have same klnet~c·en~m i
.,!!I) . (d)'Notpossible to.predict
·21 . A thin rod of length l is susp.ended from on_e . end· and rotated with
n rotations per second . Tne
_!' rotational kinetic energy o( the rpd will be:- · · ,
2.22
~ (a) 2ml 1t n
1 . · 222
· · ·. (b) - ml n
2 222 l 222
1t (c) - ml 1t n (cl) - ml 1t n
2 3 6
fJ 28. A rigid body of mass ·m r_otates with arygular velocity c.o about an axis at a distanc
e d from the centre of
p mass G. The radjus ¢f .gyriition about a parallel axis through G is K. The
the body Is :- · ,.. . • •.: ·
kln~tic energy of rotation of
. ·

~ I

IJ., 61 .
 1 1 . 1 2 2 2 l (d+k 2 )ui
(a)-mk 2ci (b) - md 2ci - m(d .+k )ro
(c) _ (d) m 2
. 2_ . . . 2_ 2 . ds A and B respectively.
29.· A weightless rod is acted ·on by upw~rd ,parallel forces of 2N and 4N_at~ Of' 6N should act in the
The total length of the rod Is AB .. 3 m. To keep the rod in e.qullib~um. a force · ·
following manner:- · · · · · · ·
!' ,

· (a) Downwards at any point between A and B

(b) Downwards at.mid point of AB
(c) Downwards at a point C such that AC =lm
(d) Downwards at a point D such that BO = 1m . . . · · · ·· ·
" · f t The power
30. A rod Is hinged at its centre and rotated ~y applying a consta'rlt torque starting ram res ·
developed by the external toi;que as a functioh of time is :-
(a) (b) • · •(c) . (d)

,j. :
:r·, ; ·-_
·.p., .
P.,, :r,..

,.
•''.

31. If ' . . ~ - time .... time

• lime . , . lime th · f' t-
a nng, a disc, a solid sphere and a cylinder of same radius rolls down on inclined plane, · e irs one
to reach the bottom will be ~ \
(a) disc · (b) riJ g · (c) solid ~phere (d) cylinder
32. A body .i~ roUin_g without slipping on a horizontal surfac~· and its rotational kinetic energy is equal to
the transla!ional kinetic enerm,.
The body is :- ·.
(a) disc , . - · (b) sphere : (c) cylinder ·: (d) ring ·
33. A solid5 ylinder of mass M ~nd rad\us R r.olls·without slipping down an inclined. plane of length Land
helgh't h. What Is th_e 5.peed )df its centre of mass when the cylinder reaches its bottom :-

(a) J2gi, (b) ~ · ' ·.. (ci ✓¾ gh (d) J4i{, .

34. A disc of ·mass M and. radi~s R r9lls on a horizontal surface and th~n rolls up. an inclined plane as
shown in the figure. If the v~locity of the disc is v, the height to which the disc will rise will be :-
(a) 3v2 (b) 3y2
2g 4,k
h
v2 . 2 '.
·
(c)- · (d) y_·
4g 2g
35. A solid sphere is/rolling on a frictionless surface, sho~n i,tl figure with a translational velocity v m/s. If
• it is to climb the inclined surface then v should be:- ·
i
j o\ -v/ -
t~·
- --

(a) c..jl0/7gh _ (b) >Jigh (c) 2gh ·:· (d) 10/7gh

36. A rod hinged at one end is released from the horizontal position as shown in the figure. When
itbecomes vertical its -lower half separates wjthout e~~rting any reaction at the breaking point, Then
the maximum angl~ ' 0 '· made by the hinged upper h~f with the vertical is
(a) 30° .c ·

(9)45°
(t ) 60°
B .
'•
'

(d) 90° C
•
62
 ~7 . _Thire. \s tod o\ \e.ng\h £. The. \le.\oc\\\a o\ \\s \Wo e.nds ~e. " \ and "i \n 0'9'90'>\\e d\te.d.\o~ t\otma\K~
\he. too. ToQ 0\9\ance. o\ \he. \ns\an\ane.oo\ ax\1 o\ to\8\\on \tom" 1 ~ ·.-
" 2 V l
(a) zero (b) - - C le) ___.:_i::__ (d) 2f.
v1+v 2 v1+v 2
38. A unlfonn rod AB of mass m and length e at rest on s smooth horizontal surface. An Impulse P Is
applied to the end B. The time taken by the rod to tum through a right angle is :."
.(a) 2,rn,I
p ' A
(b)7tml
3P t'
(c) 7tml
~ 12P
p I I

~ (d) ~1tml
3P
-----8
I

~ 39. An equilateral prism of mass m rests on a rough horizontal surface with coefficient of friction µ. A
horizontal force F is applied .on the prism a~ shown in the figure. If the c~efficient of friction is
~ sufficiently high_so that the prism does not slide before toppling, then the rnlnimUJ'll force required to
topple the prism Is-· ·,: /
,,
~ (a) mg- (b) mg •• I

✓ 3 4 a a
~ ,/ . ~- - l ~•

(c) µmg (dl,umg ;.~ .,,

~ ✓3 • 4 . a

40. A uniform rod of mass M and length L lies radially on· a disc rotating with angular speed ro in a
~ 0

horizontal plane about its axis. The rod d~es not slip 'on the disc and -the centre of the rod is at a
distance R from tllle centre of the disc,.- The·n the kinetic energy of the rod is-
~

~ (a) !.. mcn2 (R2+ L2) (b)!..~~2~ 2 r:~I'"lt-::c-.. · .

2l . 12- 2 ·: l\. _iR
~
(,;) -molL2 (d) Nori_~ _of these "
24 I ,'

~ 41. A particle of mass m Is projected with a velocity v making an angle of 45 with the .horizontal. The
magnitude of the angular momentun; ;of the p'rojecllle about the point of proJ.ection when the particle
;;::) ls at its maximum height h Is :· : !

:;;:;, (a) zero (b) mv

1
·;: (c)
111
~~ (d) m~2gh 1
( 4✓2g) _:· J2g j

~ ' i
! 42. Two point masses of 0.3 kg and' 0.7 kg are fixed at the ends of a rep of length 1.4 rn and of
q negligible mass . The rod Is set fotatlng about an axis perpendlcular toi its length with a uniform
angul.ar speed, The point on tnl rod through which the .~Is. should pass in order that the work

1~ I
required for rotation o_f the rod is ij,1nimum, is located at a c:11s.tati'Ce of :- ·
(a) 0.42 rn from mass of 0.3 kB
(c) 0.98 m from mass of 0.'.'i kg
(b) 0.70 m frq~ mass of 0.7 kg
(d) 0.98 m froh1 mass of 0.7 kg
·

~
~
63

p_
 43. Two spheres each of mass , M and radius R/2 are connected with a massless rod of length 2R as
shown In the-figure. What VJ!U be the moment of Inertia of the system about an axis pass\ng through·
the centre of one of the sp~~res and perpendicular to the v rod
(a)~MR 2 (b)IMk 2
5 5 ·
' 5 5
(c)-MR 2 (d) - MR 2
2 21 2R
I • '(
44. A cord is wound 1 over a cylinder of radius rand momen~•of inertia I. A mass m is attached to
the free end of the cord. The cylinder is free to rotate ·.about its own horizontal axis. If mass m is
released from rest, then the velocity of the mass after i1 '.had falien through a distance h will be-
(i2
(a) (2gh/12 (b) ( 2mlghr )112 : (c) ( ~ghr )112
l+mr
(d) ( mghr
l+2mr
)112 c;:
45. A .solid sphere of radius R is placed on smooth horizo'1\al surface. A horizontal force 'F' Is applied at
height 'h' from the lowest point. For the maximum a~~leration of centre of mass, which is correct-
~
(a) h =- R .. ~
• (b) h = 2R
(c) h = 0 • ~
(d)No relation between h and R ~
Answers '-'"-' p""''
~
ij 1.(b) 2.(a) 3.(b) 4.(a) 5.(b) e
6.(b) 7.(d)
I •

8.(c) ·....
I ..
9.(c) 10.(b) c
11 .(b)' 12.(d) 13.(b) 14.(b) 15.(c) C
16.(d) 17.(c) 18.(a) 19.(d) 20.(b) ·
c
I
21 .(c)

2~.(b)
22.(c)

27.(c)
23.(d.)

28.(c)
I
24.(a)

29.(d)
25.(b)

30.(b)
'e
~

31 .(c) 32.(d) 33.(c) 34.(b) 35.(b)

36.(c) 37 .(c). . . 38.-(c)

C
39.(a) 40.(a)
C
41.(b) 42.(c) 43:(a) 44.(c) 45.(c)
4
4

64