ROTATION
ROTATION
3.
@
ab
hi
5. Three particles each of mass m are placed at the corners of equilateral triangle side I
3
6. A thin rod of length L and mass M is bent at its midpoint into two halves so that the angle
between them is 90 degree. The moment of inertia of the bent rod about an axis passing
through the bending point and perpendicular to the plane defined by the two halves of the
rod is
a) √2 ML2
b) ML2 /24
c) ML2 /12
d) ML2 /6
pg. 1 MANISH SS
ROTATIONAL MOTION
7. The moment of inertia of a thin uniform circular disc about one of its diameter is I. its
moment of inertia about an axis tangent to it and perpendicular to its plane is
a) 2I/3
b) 2I
c) I/2
d) 6I
8. At thin uniform wire of mass m and length l is bent into a circle. Th moment of inertia of the
wire about an axis passing through its one end and perpendicular to the plane of the circle is
a) 2 mL2 / π2
b) mL2 / π2
c) mL2 / 2π2
d) mL2 / 3π2
shown. Its moment of inertia about an axis joining its free ends will
be
ab
a) ml2
b) zero
c) mL2 / π2
hi
d) mL2 / 2π2
_1
10. the moment of inertia of a uniform circular disc of radius R and mass M about an axis
86
passing from the edge of the disc and normal to the disc is
a) 1/2 MR2
3
b) MR2
c) 7/2 MR2
d) 3/2 MR2
11. Two discs of same mass and same thickness have densities as 17 g/cm3 and 51 g/cm3 . the
ratio of their moment of inertia about their central axes is
a) 1/3
b) 2/3
c) 3/1
d) 3/2
pg. 2 MANISH SS
ROTATIONAL MOTION
13.
14.
@
ab
hi
_1
86
15.
3
16.
pg. 3 MANISH SS
ROTATIONAL MOTION
17.
18.
@
ab
19. The moment if inertia of a uniform disc is maximum about an axis perpendicular to the disc
and passing through:
hi
a) D
_1
b) A
c) B
d) C
86
20. From a circular ring of mass M and radius R is an are corresponding to 90 degree sector is
3
removed. The moment of inertia of the remaining part of the ring about an axis passing
through the centre of ring and perpendicular to the plane of the ring is k times MR2. Then
the value of k is
a) 3/4
b) 7/4
c) 3/2
d) 3/5
pg. 4 MANISH SS
ROTATIONAL MOTION
22. The moment of inertia of a thin uniform rod of mass M and length L about an axis passing
through its midpoint and perpendicular to its length is l. its moment of inertia about an axis
passing through one of its ends and perpendicular to its length is
a) l0 + ML2 / 2
b) l0 + ML2 /4
c) l0 + 2ML2
d) l0 + ML2
23. a particle of mass 1 kg is kept at I 1m, 1m, 1m). the moment of inertia of this particle about
z-axis would be
a) 1 kg – m2
b) 2 kg – m2
c) 3 kg – m2
d) None of these
a) 2 mL2 / 3
b) mL2 /12
hi
c) mL2 /6
d) none of these
_1
25. four identical thin rods of mass M and length l from a square frame. Moment of inertia of
86
this frame about an axis through the centre of the square and perpendicular to its plane is
a) 1/3 Ml2
3
b) 4/3 Ml2
c) 2/3 Ml2
d) 13/3 Ml2
26. Three objects A: ( a solid sphere), B: ( a thin circular disk) and C: ( a circular ring), each have
the same mass M and radius R. they all spin with the same angular speed ω about their own
symmetry axes. The amount of works ( W ) required to bring them to resty would satisfy the
relation
pg. 5 MANISH SS
ROTATIONAL MOTION
27. sIn the rectangular lamina shown in the figure, AB =
BC/2. The moment of inertia of the lamina is minimum
along the axis passing through:
a) AB
b) BC
c) EG
d) FH
28. Find the moment of inertia of the ring shown in fig. about the axis AB.
@
29. The mass density of a rod at a distance x from its end is M0 /2L2 x. what is the MOI of the rod
of length L about an axis passing through the end of the rod & perpendicular to the rod
ab
30. Three identical spherical shells each mass m and radius r are placed as shown in fig.consider
an axis XX’ which is touching to two shells and passing through diameter of third shell.
hi
Moment of inertia of the system consisting of these three spherical shells about XX’ axis is
_1
a) 4 mr2
b) 11/5 mr2
c) 3 mr2
86
d) 16/5 mr2
3
a) Dics
b) Solid sphere
c) Rod
d) All of these
32. The figure shows a uniform block of mass M and edge lengths
a, b and c. its MOI , about an axis through one edge and
perpendicular ( as shown) to the large face of the block is
a) M/3 ( a2 + b2 )
b) M/4 ( a2 + b2 )
c) 7M/12 ( a2 + b2 )
d) M/12 ( a2 + b2 )
pg. 6 MANISH SS
ROTATIONAL MOTION
33.
34.
@
ab
hi
_1
86
35.
3
36. Two rings of same mass and radius R are placed with their planes perpendicular to each
other and centre at a common point. the radius of gyration of the system over an axis
passing through the centre and perpendicular the plane of one ring is
a) 2R
b) R/ √ 2
c) √3/√2 R
d) √3R /2
37. The two spares one of which is hollow and other solid have identically mass and moment of
inertia about their respective diameter the ratio of their radii is given by
pg. 7 MANISH SS
ROTATIONAL MOTION
a) 5:7
b) 3:5
c) √3 : √5
d) 3:7
38. Four particles each of mass m are placed at the corners of a square of side length t the
radius of gyration system about an axis perpendicular to the square and passing through the
centre is
a) l/√2
b) l/2
c) l
d) l√2
39. The ratio of radii gyration of a circular dics about tangential axis in the plane of the dics
and of a circular ring of the same radius about the tangential axis in the plane of the ring is
a) 2:1
@
b) √5: √6
c) 2:3
d) 1: √2
ab
40. The moment of inertia of uniform semi-circular wire of mass m and radius r about an axis
passing through its centre of mass and perpendicular to its plane is
hi
a) mr2 / 2
_1
b) mr2
c) mr2 ( 1 – 4/π2 )
86
d) mr2 ( 1 –+4/π2 )
41. Three lots each of Mass m and length L are joined to form an equilateral triangle as shown in
3
the figure what is the moment of inertia about an axis passing through the centre of mass of
the system and perpendicular to the plane?
a) 2 mL2
b) mL2 / 2
c) mL2 / 3
d) mL2 / 6
42. from a circular disc of radius R and mass 9M, a small dics mass M and radius R/3 is removed
concentrically. The moment of inertia of the remaining dics about an axis perpendicular to
the plane of the dics and passing through its
centre is
a) 40/9 MR2
b) MR2
c) 4 MR2
d) 4/9 MR2
pg. 8 MANISH SS
ROTATIONAL MOTION
43. From a dics of radius R and mass M, a circular hole of diameter R, whose rim passes through
the centre is cut. What is MOI of the remaining part of the dics about a perpendicular axis,
passing through centre?
a) 15 MR2 /32
b) 13 MR2 / 32
c) 11 MR2 / 32
d) 9 MR2 / 32
a) M / 2 ( R 2 2 – R1 2 )
b) M / 2 ( R 2 2 + R1 2 )
c) M / 2 ( R2 + R1 / 2)2
@
d) MR22 /2
45. (i) Center of gravity (CG) of a body is the point at which the weight of the body acts
ab
(ii) centre of mass coincides with the centre of gravity if the earth is assumed ti have
infinitely large radius.
(iii) to evaluate the gravitational field intensity due to any body at an external point, the
hi
46.
pg. 9 MANISH SS
ROTATIONAL MOTION
47.
48.
@
ab
hi
49. The moment of the force F = 4i + 5j – 6k at ( 2,0,-3), about the point (2,-2,-2) is given by
_1
86
3
50. If F is the force acting on a particle having position vector r and τ be the torque of this force
about the origin then
pg. 10 MANISH SS
ROTATIONAL MOTION
51. The value of M, as shown for which the rod will be in equilibrium is ( rod is massless)
a) 1 kg
b) 2 kg
c) 4 kg
d) 6 kg
a) O degree
b) 60 degree
c) 90 degree
ab
d) 180 degree
hi
54. Vector A points towards north and vector B points upwards then A x B point towards
_1
a) East
b) West
c) North
86
d) South
55. If A +B +C = 0 then A + B is
3
a) 9 kg
b) 15 kg
c) 21 kg
d) 1 kg
pg. 11 MANISH SS
ROTATIONAL MOTION
58. A massless beam of length 5 m is placed on two wedge A and B. if there forces 3 kN, 4kN and
5kN are applied on the beam as shown in the figure, then find normal reaction at A and B
59. A uniform rod of length 200 cm and mass 500 g is balanced on a wedge placed at 40cm
mark. A mass of 2 kg is suspended from the rod at 160 cm mark as shown in the figure. Find
@
60. Find the torque about the origin when a force of 3 j N acts on a particle whose position
vector is 2 k m
86
a) 6iNm
b) 6jNm
3
c) -6 I N m
d) 6kNm
61. Moment of a force of magnitude 20 N acting along positive x-direction at point ( 3m, 0 ,0)
about the point ( 0,2, 0) ( in N-m) is
a) 20
b) 60
c) 40
d) 30
a) 90 cm from A
pg. 12 MANISH SS
ROTATIONAL MOTION
b) 75 cm from B
c) 20 cm from B
d) 85 cm from A
63. A particle of mass m has been thrown up with initial speed u making angle θ with the
horizontal. Find the torque of its weight about the point of projection when it just reached
the highest point
64. ABC is an equilateral triangle with O as its centre F1 , F2, and F3 represent three forces acting
along the sides AB, BC, and AC respectively. If the total torque about O is zero then the
magnitude of F3 is
@
ab
65. A rod of weight W is supported by two parallel knife edges A and B and is in equilibrium in a
horizontal position. The knives are at a distance d from each other. The centre of mass of the
hi
a) W ( d – x) /d
b) Wx /d
c) Wd / x
86
d) W ( d – x)/x
3
pg. 13 MANISH SS
ROTATIONAL MOTION
67. od of mass ( M,L) as shown then find T1 & T2
68. find torque ‘o’ due to gravitational force when rod at angle theta from vertical
@
ab
hi
70. what is the angular acceleration of a particle if the angular velocity of a particle 4 times of its
initial angular velocity 1 rad/s in 2 seconds ?
71. a particle moving with an angular velocity of 200 rad /s starts de-accelerating at a rate of 2
rad/s2. Calculate the time in which it will come to rest
72. a fan is rotating with a speed of 450 rev/min. after being switched off it comes to rest in 10
s. assuming constant angular deceleration calculate the number of revolution made by it
before coming to rest.
73. A dics rotating about its axis from rest acquires the angular speed 100 reve/s in 4 second.
The angle rotated by it during these 4 second ( in radian) is
a) 100 π
b) 200 π
c) 300 π
d) 400 π
pg. 14 MANISH SS
ROTATIONAL MOTION
74.
75. The instantaneous angular position of a point on a rotating wheel is given by the equation θ
(t) = 2t3 – 6t2 . the torque on the wheel becomes zero at:-
a) t = 1s
b) t = 0.5s
c) t = 0.25s
@
d) t = 2s
76. a body rotating with uniform angular acceleration covers 100π ( radian) in the 1st 5 s after
ab
a) 40 π
b) 30 π
_1
c) 20 π
d) 10 π
86
77. A wheel starting from rest is uniformly accelerated at 2 rad/s^2 for 20 seconds. It os allowed
to rotate uniformly for the next 10 seconds and is finally brought the rest in next 20 seconds.
The total angle rotated by the wheel ( in radian ) is
3
a) 600
b) 1200
c) 1800
d) 300
78. A body rotates about fixed axis with an angular acceleration of 3 rad/s2 . the angle rotated by
it during the time when its angular velocity increases from 10 rad/s to 20 rad/s ( in radian) is
a) 50
b) 100
c) 150
d) 200
pg. 15 MANISH SS
ROTATIONAL MOTION
79. Two particles start moving from same position on a circle of radius 20 cm with speed 40 π
m/s and 36 π m/s respectively in the same direction. Find the time after which the particles
will meet again.
80. If the parallel forces acting on a lever are in the ratio 3:5 then what is the mechanical
advantage of the lever
81. The grinding stone of a flour mill is rotating at 600 rad/sec, for this power is 1.2 k watt is
used. The effective torque on stone is N-m will be
a) 1
b) 2
c) 3
d) 4
of the dics is
a) F /MR
ab
b) 2F/3MR
c) 4F/MR
d) Zero
hi
83. A wheel having moment of inertia 2 kg – m2 about its vertical axis, rotates at the rate of 60
_1
rpm about the axis.the torque which can stop the wheel’s rotation in one minute would be:
86
a) π/12 N-m
b) π/15 N-m
c) π/18 N-m
3
d) 2 π/15 N-m
84. A wheel has moment of inertia 5*10-3 kg m2 and is making 20 rev/sec. the torque needed to
stop it in 10 sec is _____________ * 10-2 N-m
a) 2π
b) 2.5 π
c) 4π
d) 4.5 π
85. A solid cylinder of mass 50 kg and radius 0.5m is free to rotate about the horizontal axis. A
massless string is wound round the cylinder with one end attached to it and other hanging
freely. Tension in the string required to produce an angular acceleration of 2 revolution S-2
is:
a) 25 N
b) 50 N
pg. 16 MANISH SS
ROTATIONAL MOTION
c) 78.5 N
d) 157 N
86. A rope is wound around a hollow cylinder of mass 3 kg and radius 40 cm. what is the angular
acceleration of the cylinder if the rope is pulled with a force of 30 N?
a) 25 m / s2
b) 0.25 rad/ s2
c) 25 rad/ s2
d) 5 m/ s2
87. A uniform circular dics of radius 50 cm at rest is free to turn about an axis which
perpendicular to its plane and passes through its centre. It is subjected to a torque which
produces a constant angular acceleration of 2.0 rad /s2. Its net acceleration in ms-2 at the
end of 2.0 is approximately:
a) 8.0
b) 7.0
@
c) 6.0
d) 3.0
ab
88. A wheel has angular acceleration of 3 rad/sec^2 and an initial angular speed of 2 rad/sec. in
a time of 2 sec it has rotated an angle ( in radian) of
hi
a) 4
b) 6
_1
c) 10
d) 12
86
3
89.
pg. 17 MANISH SS
ROTATIONAL MOTION
90.
91. A solid cylinder of mass 2 kg and radius 4 cm is rotating about its axis at the rate of 3 rpm.
The torque required to stop it after 2 π revolution is
a) 2*106 N m
2*10-6 N m
@
b)
c) 2*10-3N m
d) 12*104 N m
ab
92.
hi
_1
86
3
93.
pg. 18 MANISH SS
ROTATIONAL MOTION
94.
96. A solid cylinder of M has a string wrapped several times around its
circumference. The free end of string is attached to the ceiling and the
hi
cylinder is released from rest. Find the acceleration of the cylinder and
the tension in the string.
_1
97.
pg. 19 MANISH SS
ROTATIONAL MOTION
98.
99. A thin 1 meter scale is kept vertical by placing its lower and hinged on floor. It is allowed to
fall. Calculate the velocity of its upper end when it hits the floor.
@
ab
hi
a) 4
b) 16
86
c) 2
d) 8
3
101. When a body is under pure rolling the fraction of its total KE which is the purely
rotational is 2/5. Identify the body
102. The KE of a roiling solid cylinder of mass 4 kg about its axis of rotation is 20 kg m 2
m2 s-2. Calculate the velocity with which it is moving
103. Which of the following I if mass and radius are assumed to be same) have maximum
percentage of total KE in rotational form while pure rolling?
a) Disc
b) Sphere
c) Ring
d) Hollow sphere
104. A spherical shell rolls on a table without sloping. What is the percentage of its KE
which is rotational?
a) 50 %
b) 40 %
pg. 20 MANISH SS
ROTATIONAL MOTION
c) 70 %
d) 90%
105. When a sphere of moment of inertia I rolls down on an inclined plane the
percentage of total energy which is rotational is approximately
a) 28%
b) 72%
c) 100%
d) None of these
106. A disc is rolling on an inclined plane without slipping then what fraction of its total
energy will be in form of rotational KE.
a) 1:3
b) 1:2
c) 2:7
d) 2:5
@
a) Ring
b) Cylinder
c) Hollow sphere
hi
d) Solid sphere
_1
108. A disc of radius 2 m and mass 100 kg rolls on a horizontal floor. Its centre of mass
has speed of 20 cm/s. how much work is needed to stop it?
86
a) 3j
b) 30 kj
3
c) 2j
d) 1j
109. A solid sphere is in rolling motion. In rolling motion a body possesses translational
KE( Kt ) as well as rotational KE( Kr ) simultaneously. The ratio Kt : ( kt + kr ) for the sphere is
a) 10:7
b) 7:10
c) 2:5
d) 5:7
110. A circular disc of mass 2 kg and radius 10 cm rolls without slipping with a speed 2
m./s. the KE of disc is
111. Find angular speed of both the disc?? If speed of connecting string is 10 m/s
pg. 21 MANISH SS
ROTATIONAL MOTION
112. Find W and velocity of point A,B,,C,D if body is in pure rolling motion
113. A solid cylinder of mass 20 kg rotates about its axis with angular speed 100 rad/s.
the radius of the cylinder is 0.25m. what is the KE associated with the rotation of the
@
116.
pg. 22 MANISH SS
ROTATIONAL MOTION
117.
118. A solid is rolling without slipping on a level surface at a constant speed of 2.0 m/s.
how far can it roll up a 30 degree ramp before it stop?
119.
120.
@
121.
ab
hi
_1
86
3
122.
pg. 23 MANISH SS
ROTATIONAL MOTION
123.
124.
@
125.
ab
hi
_1
86
126.
3
pg. 24 MANISH SS
ROTATIONAL MOTION
127.
128.
@
ab
hi
_1
86
3
129.
pg. 25 MANISH SS
ROTATIONAL MOTION
130.
131.
@
ab
hi
_1
86
132.
3
133.
pg. 26 MANISH SS
ROTATIONAL MOTION
134.
135.
136.
@
ab
hi
_1
86
3
137.
pg. 27 MANISH SS
ROTATIONAL MOTION
138.
139.
@
ab
hi
_1
86
3
140.
pg. 28 MANISH SS
ROTATIONAL MOTION
141.
142. An object is rolling without slipping on a horizontal surface and its rotational KE is
two-third of translational KE. The body may be
a) Disc
b) Solid sphere
c) Hollow cylinder
d) Spherical shell
@
143. A disc of mass m and radius r rolls on a horizontal plane without slipping with the
ab
speed u. if it starts climbing up an inclined plane, the maximum height it would attain will be
hi
_1
86
3
a) 5m
b) 7m
c) 8m
d) 6m
145. A body is rolling without slipping on a moving plank as shown in figure relation of v
and v1 is
pg. 29 MANISH SS
ROTATIONAL MOTION
146. A solid sphere is rolling on a frictionless surface, shown in figure with a translational
velocity v m/s. if it is to climb the inclined surface, then v should be
@
147. A solid cylinder is rolling down on an inclined plane of angle theta. The coefficient of
static friction between the plane and cylinder is μs, then condition for the cylinder not to slip
is
ab
hi
_1
86
3
a) Forward direction
b) Backward direction
c) Normal to the surface
d) Zero
a) Forward direction
b) Backward direction
c) Normal to the surface
pg. 30 MANISH SS
ROTATIONAL MOTION
d) Zero
150. A hollow sphere placed on rough surface then find acceleration and value of friction
a) X – axis
b) Y – axis
86
c) Z – axis
d) Line at equal angles to all the three axes
3
154.
pg. 31 MANISH SS
ROTATIONAL MOTION
155. Object is projected with speed u at an angle theta then find angular momentum
after time ‘t’. when it is at maximum height.
156.
157.
@
ab
hi
_1
86
3
158.
pg. 32 MANISH SS
ROTATIONAL MOTION
159.
160.
@
ab
hi
161.
_1
86
3
162. When a mass is rotating in a plane about fixed point its angular momentum is
directed along
a) The radius
b) The tangent to orbit
c) Line at an angle 45 degree to the plane of rotation
d) The axis of rotation
163. Objected is projected with speed u at an angle theta then find angular momentum
after time ‘t’. when it is about to collides
164. Ball is projected with u at angle theta then find angular momentum at time ‘t’ about
point of projection
pg. 33 MANISH SS
ROTATIONAL MOTION
165. Object is moving with velocity 40√2 m/s on a straight line of equation y = x + 4 then
find angular momentum about origin
166. A swimmer while jumping into river from a height easily forms a loop in air if
a) 0
b) 1
c) -1
d) 2
@
168.
ab
hi
_1
86
3
169. A heavy solid sphere is thrown on a horizontal rough surface with initial velocity u
without rolling. What be its speed, when it starts pure rolling motion?
a) 3u/5
b) 2u/5
c) 5u/7
d) 2u/7
170.
pg. 34 MANISH SS
ROTATIONAL MOTION
171.
172.
@
ab
hi
_1
173. A solid sphere is rotating freely about its symmetry axis in free space . the radius of
the sphere is increased keeping its mass same. Which of the following physical quantities
86
a) Rotational KE
3
b) Angular velocity
c) Angular momentum
d) MOI
174. A circular platform is mounted on a frictionless vertical axle. Its radius R = 2m and its
moment inertia about the axie is 200 kgm^2. It is initially at rest. A 50 kg man stands on the
edge of the platform and begins to walk along the edge at the speed of 1 m/s relative to the
ground. Time taken by the man to complete one revolution is
a) 3π/2 s
b) 2πs
c) Π s /2
d) πs
175. what will be the value of maximum acceleration of the truck in the forward direction
so that the block kept on the back does not topple?
pg. 35 MANISH SS
ROTATIONAL MOTION
a) Ag /h
b) Hg/h
c) Ag/2h
@
ab
hi
_1
86
3
pg. 36 MANISH SS