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Angular Displacement, Velocity and
Acceleration
1. The angular speed of truck wheel is increased from 900 rpm to 2460 rpm in 26 seconds. The number of revolutions by the truck engine during this time is _______ (Assuming the acceleration to be uniform). [17 March, 2021 (Shift-I)] 2. A clock has a continuously moving second’s hand of 0.1 m length. The average acceleration of the tip of the hand (in units of ms–2^ ) is of the order of [6 Sep, 2020 (Shift-I)] ( a ) 10 –1^ ( b ) 10 –2^ ( c ) 10 –4^ ( d ) 10 – 3. A bead of mass m stays at point P ( a , b ) on a wire bent in the shape of a parabola y = 4 Cx^2 and rotating with angular speed ω (see figure). The value of ω is (neglect friction) [2 Sep, 2020 (Shift-I)]
( a ) 2 2 gC ( b ) 2 gC^ ( c ) 2 g C
( d ) 2 gC ab
Moment of Inertia, Radius of
Gyration
4. Four particles each of mass 1 kg are placed at four corners of a square
of side 2 m. Moment of inertia of system about an axis perpendicular to its plane and passing through one of its vertex is _____ kgm^2. [27 Jan, 2024 (Shift-I)]
5. Two identical spheres each of mass 2 kg and radius 50 cm are fixed
at the ends of a light rod so that the separation between the centers is 150 cm. Then, moment of inertia of the system about an axis perpendicular to the rod and passing through its middle point is
x kg m^2 , where the value of x is [31 Jan, 2024 (Shift-II)]
Rotational Motion
6. Ratio of radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for moment of Inertia about their diameter
axis AB as shown in figure is
x
. The value of x is:
[05 April, 2024 (Shift-I)]
4R
2R
A
B
A
B
R
M
Diameter
( a ) 34 ( b ) 17 ( c ) 67 ( d ) 51
7. Three balls of masses 2 kg, 4 kg and 6 kg respectively are arranged at centre of the edges of an equilateral triangle of side 2 m. The moment of inertia of the system about an axis through the centroid and perpendicular to the plane of triangle, will be ______ kg m^2. [06 April, 2024 (Shift-II)] 8. A uniform solid cylinder with radius R and length L has moment of inertia I 1 , about the axis of cylinder. A concentric solid cylinder
of radius 2
R
R ′ = and length^ 2
L
L ′ = is carved out of the original
cylinder. If I 2 is the moment of inertia of the carved out portion of
the cylinder then 1 2
I
I
(Both I 1 and I 2 are about the axis of the cylinder) [24 Jan, 2023 (Shift-II)]
9. Moment of inertia of a disc of mass M and radius ' R ' about any of
its diameter is
2
4
MR (^). The moment of inertia of this disc about an
axis normal to the disc and passing through a point on its edge will
be, 2 2
x MR. The value of x is ________. [1 Feb, 2023 (Shift-II)]
10. A ring and a solid sphere rotating about an axis passing through their centers have same radii of gyration. The axis of rotation is perpendicular to plane of ring. The ratio of radius of ring to that of
sphere is 2 x
. The value of x is [6 April, 2023 (Shift-2)]
JEE-Main
158 JEE Main and Advanced Physics PYQs
11. The moment of inertia of semicircular ring about an axis, passing through the center and perpendicular to the plane of ring, is (^1) MR 2 x
, where R is the radius and M is the mass of semicircular ring.
The value of x will be [8 April, 2023 (Shift-I)]
12. Two discs of same mass and different radii are made of different materials such that their thicknesses are 1 cm and 0.5 cm respectively. The densities of materials are in the ratio 3:5. The moment of inertia of these discs respectively about their diameters will be in the ratio of 6
x (^). The value of x is [31 Jan, 2023 (Shift-II)]
13. A solid sphere and a solid cylinder of same mass and radius are rolling on a horizontal surface without slipping The ratio of their radius of gyrations respectively (Ksph : K (^) cyl) is (^) 2 : x , then value of x is_______. [15 April, 2023 (Shift-I)] 14. ICM is the moment of inertia of a circular disc about an axis ( CM ) passing through its center and perpendicular to the plane of the disc. IAB is it’s moment of inertia about an axis AB perpendicular to plane and parallel to the axis CM at a distance 2 3
R from the center.
Where R is the radius of the disc. The ratio of IAB and ICM is x : 9. The value of x is ___________ [25 Jan, 2023 (Shift-I)] C
R
M
A
B
R
15. Solid sphere A is rotating about an axis PQ. If the radius of the sphere is 5 cm then its radius of gyration about PQ will be x cm. The value of x is _________. [24 Jan, 2023 (Shift-I)]
10 cm Q
A
P
16. Two identical solid spheres each of mass 2 kg and radii 10 cm are fixed at the ends of a light rod. The separation between the centres of the spheres is 40 cm. The moment of inertia of the system about an axis perpendicular to the rod passing through its middle point is _______ × 10–3^ kg - m^2 [6 April, 2023 (Shift-I)] 17. If a solid sphere of mass 5kg and a disc of mass 4 kg have the same radius. Then the ratio of moment of inertia of the disc about a tangent in its plane to the moment of inertia of the sphere about its tangent will be 7
x (^). The value of x is ___________. [25 Jan, 2023 (Shift-II)]
18. A light rope is wound around a hollow cylinder of mass 5 kg and radius 70 cm. The rope is pulled with a force of 52.5 N. The angular acceleration of the cylinder will be ________ rad s–^. [13 April, 2023 (Shift-II)] 19. Match List–I with List–II [28 June, 2022 (Shift-I)] List-1 List - II A. Moment of inertia of solid sphere of radius R about any tangent.
I. 52
MR
B. Moment of inertia of hollow sphere of radius (R) about any tangent
II. 72
MR
C. Moment of inertia of circular ring of radius (R) about its di- ameter.
III. 12
MR
D. Moment of inertia of circular disc of radius (R) about any di- ameter.
IV. 12
MR
Choose the correct answer from the options given below: ( a ) A-II, B-I, C-IV, D-III ( b ) A-I, B-II, C-IV, D-III ( c ) A-II, B-I, C-III, D-IV ( d ) A-I, B-II, C-III, D-IV
20. The radius of gyration of a cylindrical rod about an axis of rotation perpendicular to its length and passing through the centre will be ________ m. Given, the length of the rod is 10 3m. [26 July, 2022 (Shift-II)] 21. Moment of Inertia (M.I.) of four bodies having same mass ‘M’ and radius ‘2R’ are as follows: [25 June, 2022 (Shift-II)] I 1 = M.I. of solid sphere about its diameter I 2 = M.I. of solid cylinder about its axis I 3 = M.I. of solid circular disc about its diameter I 4 = M.I. of thin circular ring about its diameter If 2 ( I 2 + I 3 ) + I 4 = xI 1 then the value of x will be ________. 22. The momentum of inertia of a uniform thin rod about a perpendicular axis passing through on end is I 1. The same rod is into a ring and
its moment of inertia about a diameter is I 2. If
2 1 2
is 3
I x I
π , then the
value of x will be _________. [29 June, 2022 (Shift-II)]
23. The radius of gyration of a cylindrical rod about an axis of rotation perpendicular to its length and passing through the centre will be ________ m. Given, the length of the rod is 10 3m. [26 July, 2022 (Shift-II)] 24. Four identical discs each of mass ‘ M ’ and diameter ‘ a ’ are arranged in a small plane as shown in figure. If the moment of inertia of the system about ‘OO’ is 2 4
x Ma. Then, the value of^ x^ will
be [28 July, 2022 (Shift-I)] O
O′
25. The radius of gyration of a cylindrical rod about an axis of rotation perpendicular to its length and passing through the centre will be ________ m. Given, the length of the rod is 10 3m. [26 July, 2022 (Shift-II)]
160 JEE Main and Advanced Physics PYQs
36. A uniform thin bar of mass 6 kg and length 2.4 meter is bent to make an equilateral hexagon. The moment of inertia about an axis passing through the centre of mass and perpendicular to the plane of hexagon is _________ × 10–1^ kg m^2. [24 Feb, 2021 (Shift-II)] 37. Consider a badminton racket with length scales as shown in the figure. If the mass of the linear and circular portions of the badmin- ton racket are same (M) and the mass of the threads are negligible, the moment of inertia of the racket about an axis perpendicular to
the handle and in the plane of the ring at, 2
r (^) distance from the end
A of the handle will ____________ Mr^2. [26 Aug, 2021 (Shift-I)]
P
6 r 2 r
A
r
38. Consider two uniform discs of the same thickness and different radii R 1 = R and R 2 = α R made of the same material. If the ratio of their moments of inertia I 1 and I 2 , respectively, about their axes is I 1 : I 2 = 1 : 16 then the value of α is: [4 Sep, 2020 (Shift-II)]
( a ) 4 ( b ) 2 ( c ) 2 ( d ) (^2 )
39. The radius of gyration of a uniform rod of length , about an axis
passing through a point 4
(^) away from the centre of the rod, an
perpendicular to it is: [7 Jan, 2020 (Shift-I)]
( a ) 1 8
( b )^7 48
( c )^
( d )^3 8
40. Mass per unit area of a circular disc of radius a depends on the distance r from its centre as σ( r ) = A + Br. The moment of inertia of the disc about the axis, perpendicular to the plane passing through its centre is: [7 Jan, 2020 (Shift-II)]
( a ) 2 4 4 5
aA B π a ^ +
( b ) 2 4 4 5
A aB π a ^ +
( c ) 4 4 5
π a ^^ A^ + aB ^ ( d )^
π a ^^ A^ + B
80 cm
60 cm
y
x
O
O ′ [4 Sep, 2020 (Shift-II)]
For a uniform rectangular sheet shown in the figure, the ratio of moments of inertia about the axes perpendicular to the sheet and passing through O (the centre of mass) and O ′ (corner point) is: ( a ) 1 / 2 ( b ) 2 / 3 ( c ) 1 / 4 ( d ) 1 / 8
42. Shown in the figure is a hollow icecream cone (it is open at the top). If its mass is M , radius of its top, R and height, H , then its moment of inertia about its axis is: [6 Sep, 2020 (Shift-I)]
R
H
( a )
2
2
MR
( b )
( )
2 2
4
M R + H
( c )
2
3
MH (^) ( d ) 2 3
MR
43. Three solid spheres each of mass m and diameter d are stuck together such that the lines connecting the centres form an equilateral triangle of side of length d. The ratio I O / I A of moment of inertia I O of the system about an axis passing the centroid and about center of any of the spheres I A and perpendicular to the plane of the triangle is: [9 Jan, 2020 (Shift-I)]
B (^) A
C
O
d
( a )
( b )
( c )
( d )
44. Moment of inertia of a cylinder of mass M , length L and radius R about an axis passing through its centre and perpendicular to the
axis of the cylinder is I =
2 2 . 4 12
M ^^ R^ + L
If such a cylinder is to
be made for a given mass of a material, the ratio L / R for it to have minimum possible I is [3 Sep, 2020 (Shift-I)]
( a ) 2 3
( b ) 3 2
( c ) 3 2
( d ) 2 3
45. The linear mass density of a thin rod AB of length L varies from A
to B as (^) ( ) x 0 1 x L
λ = λ ^ +
, where x is the distance from A. If M is
the mass of the rod then its moment of inertia about an axis passing through A and perpendicular to the rod is: [6 Sep, 2020 (Shift-II)]
( a ) 5 2 12
ML ( b )^2
ML ( c ) 2 2 5
ML ( d )^2
ML
46. ABC is a plane lamina of the shape of an equilateral triangle. D , E are mid points of AB , AC and G is the centroid of the lamina. Moment of inertia of the lamina about an axis passing through G and perpendicular to the plane ABC is I 0. If part ADE is removed, the moment of inertia of the remaining part about the same axis is 0 16
NI (^) where N is an integer. Value of N is ______.
[4 Sep, 2020 (Shift-I)]
Rotational Motion^161
A
B C
D E
G
47. A massless equilateral triangle EFG of side ‘ a ’ (As shown in figure) has three particles of mass m situated at its vertices. The moment of inertia of the system about the line EX perpendicular to EG in the plane of EFG is (^) ma^2 20
N (^) where N is an integer. The value
of N is _______. [3 Sep, 2020 (Shift-II)] X F
E a G
48. The moment of inertia of a solid sphere, about an axis parallel to its diameter and at a distance of x from it, is ‘ I ( x )’. Which one of the graphs represents the variation of I ( x ) with x correctly? [12 Jan, 2019 (Shift-II)]
( a )
O x
I ( x )
( b )
O x
I ( x )
( c )
O (^) x
I ( x )
( d )
O x
I ( x )
49. A solid sphere of mass M and radius R is divided into two unequal
parts. The first part has a mass of 7 8
M (^) and is converted into a
uniform disc of radius 2R. The second part is converted into a uniform solid sphere. Let I 1 be the moment of inertia of the disc about its axis and I 2 be the moment of inertia of the new sphere about its axis. The ratio I 1 / I 2 is given by: [10 April, 2019 (Shift-II)] ( a) 185 ( b ) 65 ( c ) 285 ( d ) 140
50. A circular disc of radius b has a hole of radius a at its centre (see figure). If the mass per unit area of the disc varies as ρ = ρ 0 / r then the radius of gyration of the disc about its axis passing through the centre is: [12 April, 2019 (Shift-I)]
a
b
( a ) 2
a + b ( b ) 3
a + b
( c )
2 2
2
a + b + ab ( d )
2 2
3
a + b + ab
51. Two identical spherical balls of mass M and radius R each are stuck on two ends of a rod of length 2 R and mass M (see figure). The moment of inertia of the system about the axis passing perpendicularly through the centre of the rod is: [10 Jan, 2019 (Shift-II)] 2R R R
( a ) 137 15
MR^2 ( b ) 17 15
MR^2 ( c ) 209 15
MR^2 ( d ) 152 15
MR^2
52. A circular disc D 1 of mass M and radius R has two identical discs D 2 and D 3 of the same mass M and radius R attached rigidly as its opposite ends (see figure). The moment of inertia of the system about the axis ‘ OO ’, passing through the centre of D 1 as shown in the figure, will: [11 Jan, 2019 (Shift-II)]
O
O '
D 2 D 3
D 1
( a ) MR^2 ( b ) 3 MR^2 ( c ) 4 2 5
MR ( d )^2
MR
53. A thin disc of mass M and radius R has mass per unit area σ( r ) = kr^2 where r is the distance from its centre. Its moment of inertia about an axis going through its centre of mass and perpendicular to its plane is: [10 April, 2019 (Shift-I)]
( a )
2
2
MR
( b )
2
3
MR
( c )
2
6
MR
( d )
MR
54. The equilateral triangle ABC is cut from a thin solid sheet of wood. (See figure) D , E and F are the mid points of its sides as shown and G is the centre of the triangle. The moment of inertia of the triangle about an axis passing through G and perpendicular to the plane of the triangle is I 0. If the smaller triangle DEF is removed from ABC , the moment of inertia of the remaining figure about the same axis is I. Then: [11 Jan, 2019 (Shift-I)] A
E
B F C
D
.G
( a ) (^0)
I = I ( b )^0
I = I
( c ) 90
I = I ( d )^0
I
I =
55. Let the moment of inertia of a hollow cylinder of length 30 cm (inner radius 10 cm and outer radius 20 cm), about its axis be I. The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also I , is: [12 Jan, 2019 (Shift-I)]
( a ) 12 cm ( b ) 16 cm ( c ) 14 cm ( d ) 18 cm
Rotational Motion^163
70. A force (^) F = ( i ˆ^ + 2 ˆ j +3 ) k ˆ
N acts at a point (4 i ˆ^ + 3 ˆ j − k ˆ) m. Then the magnitude of torque about the point (ˆ^ i^^ +^2 ˆ j^ +^ k ˆ) m will be (^) xN - m. The value of x is [5 Sep, 2020 (Shift-I)] ( a ) 195 ( b ) 165 ( c ) 105 ( d ) 135
71. A rigid massless road of length 3 l has tow masses attached at each end as shown in the figure. The rod is pivoted at point P on the horizontal axis (see figure). When released from initial horizontal position, its instantaneous angular acceleration will be : [10 Jan, 2019 (Shift-II)]
5 M o P 2 M o
l 2 l
( a ) 13
g l
( b ) 3
g l
( c ) 2
g l
( d ) 7 3
g l
72. A particle of mass m is moving along a trajectory given by x = x 0 + a cos ω 1 t y = y 0 + b sin ω 2 t The torque, acing on the particle about the origin, at t = 0 is: [10 April, 2019 (Shift-I)] ( a ) my a 0^ ω 12 k ˆ ( b ) m (^^ −^ x b 0^ +^ y a 0^ )ω 12 k ˆ ( c ) (^) − m ( − x b 0 ω −^22 y a 0 ω^21 ) k ˆ ( d ) Zero
Rotational/Equilibrium &Toppling
73. A heavy iron bar of weight 12 kg is having its one end on the ground and the other on the shoulder of a man. The rod makes an angle 60° with the horizontal, the weight experienced by the man is: [27 JAN, 2024 (Shift-II)]
( a ) 6 kg ( b ) 12 kg ( c ) 3 kg ( d ) 6 3 kg
74. A heavy iron bar, of weight W is having its one end on the ground and the other on the shoulder of a person. The bar makes an angle θ with the horizontal. The weight experienced by the person is: [ April, 2024 (Shift-I)] ( a ) W /2 ( b ) W ( c ) W cos θ ( d ) W sin θ 75. An object of mass 8 kg is hanging from one end of a uniform rod CD of mass 2 kg and length 1m pivoted at its end C on a vertical wall as shown in figure below. It is supported by a cable AB such that the system is in equilibrium. The tension in the cable is : (Take g = 10m/s^2 ) [25 Jan, 2023 (Shift-I)]
8 kg
60 cm 40 cm
D
Wall
C
B
A
axis
( a ) 240 N ( b ) 90 N ( c ) 300 N ( d ) 30 N
76. A metre scale is balanced on a knife edge at its centre. When two coins, each of mass 10 g are put one on the top of the other at the 10.0 cm mark the scale is found to be balanced at 40.0 cm mark. The mass of the metre scale is found to be x × 10–2^ kg. The value of x is _______. [24 June, 2022 (Shift-I)] 77. A uniform cylinder of mass M and radius R is to be pulled over a step of height a ( a < R ) by applying a force F at its centre ‘ O ’ perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of F required is: [2 Sep, 2020 (Shift-I)] F
a
O
R
( a )
2 (^1 )
a Mg R
− ( b )
2 1
R a Mg R
( c )
2 (^) − 1 −
Mg R R a
( d ) (^) Mg a R
78. Consider uniform cubical box of side a on a rough floor that is to be moved by applying minimum possible force F at a point b above its centre of mass (see figure). If the coefficient of friction is μ = 0.4,
the maximum possible value of 100 b a
× for box not to topple before
moving is____. [7 Jan, 2020 (Shift-II)]
F
79. Shown in the figure is rigid and uniform one meter long rod AB held in horizontal position by two strings tied to its ends and attached to the ceiling. The rod is of mass ‘m’ and has another weight of mass 2m hung at a distance of 75 cm from A. The tension in the string at A is: [2 Sep, 2020 (Shift-I)]
A B
2 m
( a ) 0.75 mg ( b ) 1 mg ( c ) 2 mg ( d ) 0.5 mg
80. Put a uniform meter scale horizontally on your extended index fingers with the left one at 0.00 cm and the right one at 90.00 cm. When you attempt to move both the fingers slowly towards the center, initially only the left finger slips with respect to the scale and the right finger does not. After some distance the left finger stops and the right one starts slipping. Then the right finger stops at a distance xR from the center (50.00 cm) of the scale and the let one starts slipping again. This happens because of the difference in the frictional forces on he two fingers. If the coefficients of static and dynamic friction between the fingers and the scale are 0.40 and 0.32, respectively, the value of x (^) R (in cm) is _______ [JEE Adv, 2020]
164 JEE Main and Advanced Physics PYQs
81. A uniform rod of length ‘ l ’ is pivoted at one of its ends on a vertical shaft of negligible radius. When the shaft rotates at angular speed ω the rod makes an angle with it (see figure). To find θ equate the rate of change of angular momentum (direction going into the paper) 2 2 12
ml ω sinθcosθ^ about the centre of mass ( CM ) to the torque
provided by the horizontal and vertical forces FH and FV about the CM. The value of 8 is then such that: [3 Sep, 2020 (Shift-II)]
FV
FH l θ
ω
( a ) (^) cos g 2 l
θ = ω
( b ) (^) cos (^22) 3
g l
θ = ω ( c ) (^) cos (^32) 2
g l
θ = ω
( d ) (^) cos (^2) 2
g l
θ = ω
82. An L-shaped object, made of thin rods of uniform mass density, is suspended with a string as shown in figure. If AB = BC, and the angle made by AB with downward vertical is θ, then: [9 Jan, 2019 (Shift-I)]
z
x
q B
A
C
( a ) tanθ = 1 2 3
( b ) tanθ = 1 2
( c ) tanθ = 2 3
( d ) tanθ = 1 3
Conservation of Angular Momentum
83. A body of mass 5 kg moving with a uniform speed 3 2 ms–1^ in
X – Y plane along the line y = x + 4. The angular momentum of the particle about the origin will be ______ kg m^2 s –1. [29 Jan, 2024 (Shift-II)]
84. A particle of mass m projected with a velocity ‘ u ’ making an angle of 30º with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height h is: [30 Jan, 2024 (Shift-I)]
( a )
mu
g
( b )
mu
g
( c )
3
mu
g
( d ) zero
85. Consider a Disc of mass 5 kg, radius 2m, rotating with angular velocity of 10 rad/s about an axis perpendicular to the plane of rotation. An identical disc is kept gently over the rotating disc along the same axis. The energy dissipated so that both the discs continue to rotate together without slipping is _____ J. [30 Jan, 2024 (Shift-I)]
ω = 10 rad/sec Mass = 5 kg
2m
86. Two discs of moment of inertia I 1 = 4 kg m 2 and I 2 = 2kg m 2 about their central axes & normal to their planes, rotating with angular speeds 10 rad/s & 4 rad/s respectively are brought into contact face to face with their axe of rotation coincident. The loss in kinetic energy of the system in the process is___________J. [30 Jan, 2024 (Shift-II)] 87. A body of mass 'm' is projected with a speed ‘u’ making an angle of 45° with the ground. The angular momentum of the body about
the point of projection, at the highest point is expressed as
2 mu^3 Xg The value of 'X' is_______. [31 Jan, 2024 (Shift-II)]
88. If the radius of earth is reduced to three-fourth of its present value without change in its mass then value of duration of the day of earth will be ______ hours 30 minutes. [06 April, 2024 (Shift-I)] 89. A thin circular disc of mass M and radius R is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with angular velocity ω. If another disc of same dimensions but of mass M /2 is placed gently on the first disc co-axially, then the new angular velocity of the system is: [08 April, 2024 (Shift-II)]
( a )
ω ( b )
ω ( c )
ω ( d )
ω
90. A particle of mass 100 g is projected at time t = 0 with a speed 20 ms –1^ at an angle 45o^ with the horizontal as given in the figure. The magnitude of the angular momentum of the particle about the starting point at time t = 2 s is found to be (^) K kgm^2^ / s. The value of K is _______ (Take g = 10 ms –2) [29 Jan, 2023 (Shift-II)]
v = 20 m/s
45° X
Y
91. If the earth suddenly shrinks to 1 64
th of its original volume with its
mass remaining the same, the period of rotation of earth becomes 24 h x
. The value of x is [10 April, 2023 (Shift-I)] 92. A solid sphere of mass 500 g and radius 5 cm is rotated about one of its diameter with angular speed of 10 rad s–1. If the moment of inertia of the sphere about its tangent is x × 10 –2^ times its angular momentum about the diameter. Then the value of x will be _________. [11 April, 2023 (Shift-I)]
166 JEE Main and Advanced Physics PYQs
the rod rotating with an angular velocity Ω and the disc rotating about its vertical axis with angular velocity 4 Ω. The total angular
momentum of the system about the point O is
2
48
Ma n
. The
value of n is. [JEE Adv, 2021]
a
O M
W
a /
M 4W
105. A thin circular ring of mass M and radius r is rotating about its axis with an angular speed ω. Two particles having mass m each are now attached at diametrically opposite points. The angular speed of the ring will become: [18 March, 2021 (Shift-I)]
( a ) M^ 2m M 2m
ω
( b ) M^ 2m M
ω
( c ) M M m
ω
( d ) M M 2m
ω
106. Two discs have moments of intertia I 1 and I 2 about their respective axes perpendicular to the plane and passing through the centre. They are rotating with angular speeds, ω 1 and ω 2 respectively and are brought into contact face to face with their axes of rotation coaxial. The loss in kinetic energy of the system in the process is given by: [27 Aug, 2021 ( Shift-II ) ]
( a )
2 1 2 1 2 1 2
I I
I I
− ω ω
( b ) 1 2 1 22 1 2
I I
I I
ω − ω
( c ) 1 2 1 22 1 2
I I
I I
ω − ω
( d )
2 1 2 1 2
2( I I )
ω − ω
107. Four point masses, each of mass m , are fixed at the corners of a square of side . The square is rotating with angular frequency ω, about an axis passing through one of the corners of the square and parallel to its diagonal, as shown in the figure. The angular momentum of the square about this axis is [06 Sep, 2020 (Shift-I)] ( a) 3 m ^2 ω ( b ) 5 m ^2 ω ( c ) m ^2 ω ( d ) 2 m ^2 ω 108. Consider a uniform rod of mass M = 4m and length pivoted about its centre. A mass m moving with velocity v making angle 4
π θ = to the rod’s long axis collides with one end of the rod and sticks to it. The angular speed of the rod-mass system just after the collision is: [8 Jan, 2020 (Shift-I)]
( a ) 4 7
v
( b ) 3 2 7
v
( c ) 3 7
v
( d ) 3 7 2
v
109. A thin rod of mass 0.9 kg and length 1 m is suspended, at rest, from one end so that it can freely oscillate in the vertical plane. A particle of mass 0.1 kg moving in a straight line with velocity 80 m/s hits the rod at its bottom most point and sticks to it (see figure). The angular speed (in rad/s) of the rod immediately after the collision will be [05 Sep, 2020 (Shift-II)]
1 m
o v
110. A person of 80 kg mass is standing on the rim of a circular platform of mass 200 kg rotating about its axis at 5 revolutions per minute (rpm). The person now starts moving towards the centre of the platform. What will be the rotational speed (in rpm) of the platform when the person reaches its centre ______. [03 Sep, 2020 (Shift-I)] 111. A circular disc of mass M and radius R is rotating about its axis with angular speed ω 1. If another stationary disc having radius 2
R
and same mass M is dropped co-axially on to the rotating disc. Gradually both discs attain constant angular speed ω 2. The energy lost in the process is p % of the initial energy. Value of p is _______. [04 Sep, 2020 (Shift-I)]
112. A rod of mass m and length L , pivoted at one of its ends, is hanging vertically. A bullet of the same mass moving at speed v strikes the rod horizontally at a distance x from its pivoted end and gets embedded in it. The combined system now rotates with angular speed ω about the pivot. The maximum angular speed ω m is achieved for x = xM. Then [JEE Adv, 2020]
x L
v
r
( a ) (^2 )
vx L x
ω =
( b ) (^2 )
vx L x
ω =
( c ) 3
M
L
x = (^) ( d ) (^3) M 2
v L
ω =
113. A wheel is rotating freely with an angular speed ω on a shaft. The moment of inertia of the wheel is I and the moment of inertia of the shaft is negligible. Another wheel of moment of inertia 3 I initially at rest is suddenly coupled to the same shaft. The resultant fractional loss in the kinetic energy of the system is [5 Sep, 2020 (Shift-I)] ( a ) 5 6
( b ) 1 4
( c ) 0 ( d ) 3 4
114. Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres. The moment of inertia and angular velocity of the first disc are 0. kg-m^2 and 10 rad s–1^ respectively while those for the second one are 0.2 kg-m^2 and 5 rad s–1^ respectively. At some instant they get stuck together and start rotating as a single system about their common axis with some angular speed. The kinetic energy of the combined system is [2 Sep, 2020 (Shift-II)]
( a ) (^20) J 3
( b ) (^5) J 3
( c ) (^10) J 3
( d ) (^2) J 3
115. A block of mass m = 1 kg slides with velocity v = 6 m/s on a frictionless horizontal surface and collides with a uniform vertical rod and sticks to it as shown. The rod is pivoted about O and swings as a result of the collision making angle θ before momentarily coming to rest. If the rod has mass M = 2 kg and length l = 1 m, the value of θ is approximately (take g = 10 m/s^2 ) [3 Sep, 2020 (Shift-I)]
Rotational Motion^167
m m
m
M, l
O
v
( a ) 49° ( b ) 55° ( c ) 63° ( d ) 69°
116. A particle of mass 20 g is released with an initial velocity 5 m/s along the curve from the point A , as shown in the figure. The point A is at height h from point B. The particle slides along the frictionless surface. When the particle reaches point B , its angular momentum about O will be: [Take g = 10 m/s^2 ] [12 Jan, 2019 (Shift-II)]
h =10m
a =10m
O
A
B
( a ) 2 kg-m^2 /s ( b ) 8 kg-m^2 /s ( c ) 6 kg-m^2 /s ( d ) 3 kg-m^2 /s
117. A thin smooth rod of length L and mass M is rotating freely with angular speed ω 0 about an axis perpendicular to the rod and passing through its centre. Two beads of mass m and negligible size are at the centre of the rod initially. The beads are free to slide along the rod. The angular speed of the system, when the beads reach the opposite ends of the rod, will be: [9 April, 2019 (Shift-II)] ( a)^0 3
M
M m
ω
( b ) M^0 M m
ω
( c ) 0 2
M
M m
ω
( d ) 0 6
M
M m
ω
118. A person of mass M is, sitting on a swing of length L and swinging with an angular amplitude θ 0. If the person stands up when the swing passes through its lowest point, the work done by him, assuming that his centre of mass moves by a distance ( << L ), is close to [12 April, 2019 (Shift-I)] ( a ) Mg ^ ( b ) Mg (1 + θ^20 )
( c ) Mg ( 1 − θ^20 ) ( d )
2 1 0 2
Mg
θ +
119. Two coaxial discs, having moments of inertia I 1 and 1 2
I ,
are rotating with respectively angular velocities ω 1 and 1 2
ω , about
their common axes. They are brought in contact with each other and thereafter they rotate with a common angular velocity. If Ef and E i are the final and initial total energies, then ( Ef – Ei ) is: [10 April, 2019 (Shift-I)]
( a) –
2 1 1 12
I ω (^) ( b ) 2 1 1
I ω ( c)^
2 1 1 6
I ω (^) ( d ) – 1 12 24
I ω
Combined Rotation and
Translation, Rolling Motion
120. A ring and a solid sphere roll down the same inclined plane without slipping. They start from rest. The radii of both bodies are identical
and the ratio of their kinetic energies is 7
x
where x is ______.
[27 Jan, 2024 (Shift-II)]
121. A cylinder is rolling down on an inclined plane of inclination
60°. It's acceleration during rolling down will be
x
m/s^2 , where
x = _____. (use g = 10 m/s^2 ). [29 Jan, 2024 (Shift-I)]
122. A solid circular disc of mass 50 kg rolls along a horizontal floor so that its center of mass has a speed of 0.4 m/s. The absolute value of work done on the disc to stop it is ______ J. [31 Jan, 2024 (Shift-I)] 123. A disc of radius R and mass M is rolling horizontally without slipping with speed v. It then moves up an inclined smooth surface as shown in figure. The maximum height that the disc can go up the incline is: [1 Feb, 2024 (Shift-II)]
( a )
v^2
g
( b )
v
g
( c )
v
g
( d )
v
g
124. A solid sphere and a hollow cylinder roll up without slipping on same inclined plane with same initial speed v. The sphere and the cylinder reaches upto maximum heights h 1 and h 2 , respectively,
above the initial level. The ratio h 1 : h 2 is
n
. The value of n
is_____. [04 April, 2024 (Shift-I)]
125. A hollow sphere is rolling on a plane surface about its axis of symmetry. The ratio of rotational kinetic energy to its total kinetic energy is x /5. The value of x is ________. [05 April, 2024 (Shift-II)] 126. A circular disc reaches from top to bottom of an inclined plane of length l. When it slips down the plane, if takes t s. When it rolls
down the plane then it takes α 2
1 2
/ t s , where α is ………..
[09 April, 2024 (Shift-II)]
127. A disc is rolling without slipping on a surface. The radius of the disc is R. At t = 0, the top most point on the disc is A as shown in figure. When the disc completes half of its rotation, the displacement of point A from its initial position is [13 April, 2023 (Shift-I)]
( a ) R ( π^2 + 4) ( b ) R ( π^2 +1)
( c ) 2 R ( d ) 2 R ( 1 + 4 π^2 )
128. A solid cylinder is released from rest from the top of an inclined plane of inclination 30° and length 60 cm. If the cylinder rolls without slipping, its speed upon reaching the bottom of the inclined plane is ________ ms–1. (Given g = 10 ms–1) [1 Feb, 2023 (Shift-I)]
Rotational Motion^169
142. A solid cylinder length is suspended symmetrically through two massless strings, as shown in the figure. The distance from the initial rest position, the cylinder should by unbinding the strings to achieve a speed of 4 ms–1, is ____________cm. (take g = 10 ms–2^ ) [27 July, 2022 (Shift-II)]
T T
143. A solid disc of radius ‘ a ’ and mass ‘ m ’ rolls down without slipping on an inclined plane making an angle θ with the horizontal. The acceleration of the disc will be (^2) g sin b
θ where^ b^ is (Round off to
the Nearest Integer) ( g = acceleration due to gravity, θ = angle as shown in figure) [16 March, 2021 (Shift-II)]
a P
q
144. A body rolls down an inclined plane without slipping. The kinetic energy of rotation is 50% of its translational kinetic energy. The body is: [20 July, 2021 (Shift-II)] ( a ) Hollow cylinder ( b ) Ring ( c ) Solid sphere ( d ) Solid cylinder 145. A 2 kg steel rod of length 0.6 m is clamped on a table vertically at its lower end and is free to rotate in vertical plane. The upper end is pushed so that the rod falls under gravity. Ignoring the friction due to clamping at its lower end, the speed of the free end of rod when it passes through its lowest position is _________ ms–1. [1 Sept, 2021 (Shift-II)] 146. In the given figure, two wheels P and Q are connected by a belt B. The radius of P is three times as that of Q. In case of same
rotational kinetic energy, the ratio of rotational inertias 1 2
I
I
æ ö çè ÷ø will be x : 1. The value of x will be _________. [27 July, 2021 (Shift-II)]
P
B
Q
3 R
R
147. A boy is rolling a 0.5 kg ball on the frictionless floor with the speed of 20 ms –1^. The ball gets deflected by an obstacle on the way. After deflection it moves with 5% of its initial kinetic energy. What is the speed of the ball now? [17 March, 2021 (Shift-I)] ( a ) 14.41ms–1^ ( b ) 19.0 ms– ( c ) 1.00ms–1^ ( d ) 4.47 ms– 148. A solid cylinder of mass m is wrapped with an inextensible light string and, is placed on a rough inclined plane as shown in the figure. The frictional force acting between the cylinder and the inclined plane is: [The coefficient of static friction, μ s is 0.4] [18 March, 2021 (Shift-II)]
( a ) 0 ( b ) 7 2
mg (^) ( c ) 5 mg ( d ) 5
mg
149. A rod of mass M and length L is lying on a horizontal frictionless surface. A particle of mass ‘ m ’ travelling along the surface hits at one end of the rod with a velocity ‘ u ’ in a direction perpendicular to the rod. The collision is completely elastic. After collision, particle
comes to rest. The ratio of masses m M
^ is^
x
. The value of ‘ x ’ will
be _______. [20 July, 2021 (Shift-I)]
150. A football of radius R is kept on a hole of radius r ( r < R ) made on a plank kept horizontally. One end of the plank is now lifted so that it gets tilted making an angle θ from the horizontal as shown in the figure below. The maximum value of θ so that the football does not start rolling down the plank satisfies (figure is schematic and not drawn to scale) [JEE Adv, 2020]
Plank R
q
2 r
( a ) (^) sin r R
θ = ( b )^ tan
r R
θ = ( c )^ sin 2
r R
θ = ( d )^ cos 2
r R
θ =
151. A horizontal force F is applied at the centre of mass of a cylindrical object of mass m and radius, perpendicular to its axis as shown in the figure. The coefficient of friction between the object and the ground is μ. The centre of mass of the object has an acceleration a. The acceleration due to gravity is g. Given that the object rolls without slipping, which of the following statement(s) is(are) correct? [JEE Adv, 2020]
R
F
( a ) For the same F , the value of a does not depend on whether the cylinder is solid or hollow ( b ) For a solid cylinder, the maximum possible value of a is 2μ g ( c ) The magnitude of the frictional force on the object due to the ground is always μ mg ( d ) For a thin-walled hollow cylinder, 2
F
a m
170 JEE Main and Advanced Physics PYQs
152. A small roller of diameter 20 cm has an axle of diameter 10 cm (see figure below on the left). It is on a horizontal floor and a meter scale is positioned horizontally on its axle with one edge of the scale on top of the axle (see figure on the right). The scale is now pushed slowly on the axle so that it moves without slipping on the axle, and the roller starts rolling without slipping. After the roller has moved 50 cm, the position of the scale will look like (figures are schematic and not drawn to scale) [JEE Adv, 2020]
( a )
x = 0 x = 50 cm
( b )
x = 0 x = 75 cm
( c )
x = 0
x = 25 cm
( d )
x = 0 x = 100 cm
153. A uniform sphere of mass 500 g rolls without slipping on a plane horizontal surface with its centre moving at a speed of 5.00 cm/s. Its kinetic energy is: [8 Jan, 2020 ( Shift-II ) ] ( a ) 8.75 × 10–4^ J ( b ) 8.75 × 10–3^ J ( c ) 6.25 × 10–4^ J ( d ) 1.13 × 10–3^ J 154. One end of a straight uniform 1 m long bar is pivoted on horizontal table. It is released from rest when it makes an angle 30° from the horizontal (see figure). Its angular speed when it hits the table is given as (^) ns −^1 , where n is an integer. The value of n is __________. [9 Jan, 2020 (Shift-I)]
30 o
155. A uniformly thick wheel with moment of inertia I and radius R is free to rotate about its centre of mass (see figure). A massless string is wrapped over its rim and two blocks of masses m 1 and m 2 ( m 1 > m 2 ) are attached to the ends of the string. The system is released from rest. The angular speed of the wheel when m 1 descents by a distance h is: [9 Jan, 2020 (Shift-II)]
( a )
1 1 2 2 2 ( 1 2 )
m m gh m m R I
+^ +
( b ) (^ )^
1 1 2 2 2 1 2
m m gh m m R I
+^ +
( c ) (^ )^
1 1 2 2 2 1 2
m m gh m m R I
+^ +
( d ) (^ )^
1 1 2 2 2 ( 1 2 )
m m gh m m R I
+^ +
m 1
m 2
156. The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal plane: ( i ) a ring of radius R, ( ii ) a solid cylinder of radius 2
R (^) and ( iii ) a solid sphere of radius
R (^). If in each case, the speed of the centre of mass at bottom of the
incline is same, the ratio of the maximum heights they climb is: [9 April, 2019 (Shift-I)]
( a ) 4 : 3 : 2 ( b ) 14 : 15 : 20 ( c ) 10 : 15 : 7 ( d ) 2 : 3 : 4
157. A string is wound around a hollow cylinder of mass 5 kg and radius 0.5m. If the string is now pulled with a horizontal force of 40 N. and the cylinder is rolling without slipping on a horizontal surface (see figure), then the angular acceleration of the cylinder will be (Neglect the mass and thickness of the string) [11 Jan, 2019 (Shift-II)]
40N
( a ) 20 rad/s^2 ( b ) 16 rad/s 2 ( c ) 12 rad/s^2 ( d ) 10 rad/s 2
158. A homogeneous solid cylindrical roller of radius R and mass M is pulled on a cricket pitch by a horizontal force. Assuming rolling without slipping, angular acceleration of the cylinder is : [10 Jan, 2019 (Shift-I)]
( a ) 3 2
F
mR
( b ) 3
F
mR
( c ) 2
F
mR
( d ) 2 3
F
mR
159. A solid sphere and solid cylinder of identical radii approach an incline with the same linear velocity (see figure). Both roll without slipping all throughout. The two climb maximum heights hsph and
h (^) cyl on the incline. The radio sph cyl
h h
is given by: [8 April, 2019 (Shift-II)]
( a ) 1 ( b ) 4 5
( c )
( d )
160. Moment of inertia of a body about a given axis is 1.5 kg m 2 Initially the body is at rest. In order to produce a rotational kinetic energy of 1200 J, the angular acceleration of 20 rad/s 2 must be applied about the axis of rotation for a duration of: [9 April, 2019 (Shift-II)] ( a ) 2 s ( b ) 5 s ( c ) 2.5 s ( d ) 3 s
172 JEE Main and Advanced Physics PYQs
A
B
C
( a ) (^) VC − V (^) A = 2( V (^) C − VB )
( b ) (^) VC − V (^) B = ( VB − VA )
( c ) | VC − VA |= 2 | VC − VB |
( d ) (^) VC − V (^) A =4 | VB |
7. The figure shows a system consisting of ( i ) a ring of outer radius 3 R rolling clockwise without slipping on a horizontal surface with angular speed ω and ( ii ) an inner disc of radius 2 R rotating anti- clockwise with angular speed ω/2 The ring and disc are separated by frictionless ball bearings. The system is in the x - z plane. The point P on the inner disc is at a distance R from the origin, where OP makes an angle of 30° with the horizontal. Then with respect to the horizontal surface, (IIT-JEE 2012) Z
X
2 R
3 R R
ω/
ω
( a ) the point O has a linear velocity 3 R ω i
( b ) the point P has a linear velocity
R ω + i R ω k
( c ) the point P has a linear velocity
R ω i − R ω k
( d ) the point P has a linear velocity
R i R k
ω + (^) − (^) ω
Numerical Types/Integer Types
8. A uniform circular disc of mass 1.5kg and radius 0.5m is initially at rest on a horizontal frictionless surface. Three forces of equal magnitude F = 0.5N are applied simultaneously along the three sides of an equilateral triangle F XYZ with its vertices on the perimeter of the disc (see figure). One second after applying the forces, the angular speed of the disc in rad s –1^ is (IIT-JEE 2012) F
F
F
Y
X
Z
O
Subjective
9. A man pushes a cylinder of mass m 1 with the help of a plank of mass m 2 as shown. There is no slipping at any contact. The horizontal component of the force applied by the man is F. Find
F m 2
m (^1)
( a ) the accelerations of the plank and the centre of mass of the cylinder and ( b ) the magnitudes and directions of frictional forces at contact points. (IIT-JEE 1999)
Fill in the Blanks
10. A uniform disc of mass m and radius R is rolling up a rough inclined plane which makes an angle of 30° with the horizontal. If the coefficients of static and kinetic friction are each equal to μ and the only forces acting are gravitational and frictional, then the magnitude of the frictional force acting on the disc is ......... and its direction is. (write up or down) the inclined plane. (IIT-JEE 1997) 11. A smooth uniform rod of length L and mass M has two identical beads of negligible size, each of mass m, which can slide freely along the rod. Initially, the two beads are at the centre of the rod and the system is rotating with an angular velocity ω 0 about an axis perpendicular to the rod and passing through the midpoint of the rod (see figure).
L /
ω 0
L /
There are no external forces. When the beads reach the ends of the rod, the angular velocity of the system is....... (IIT-JEE 1988)
True/False
12. A thin uniform circular disc of mass M and radius R is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with an angular velocity ω. Another disc of the same dimensions but of mass M /4 is placed gently on the first disc coaxially. The angular velocity of the system now is 2 ω 5 (JEE Adv. 1986)
Moment of Inertia
Single Correct
13. A solid sphere of radius R has moment of inertia I about its geometrical axis. It is melted into a disc of radius r and thickness t. If it's moment of inertia about the tangential axis (which is perpendicular to plane of the disc) is also equal to I , then the value of r is equal to (IIT-JEE 2006) l r
( a ) 2 15
R ( b )^
R ( c ) 3 15
R ( d )^3 15
R
Rotational Motion^173
14. A circular disc of radius R and mass 9 M , a small disc of radius R / is removed from the disc. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through O is (IIT-JEE 2005) R /
R O
( a ) 4 MR^2 ( b ) 2
MR ( c ) 10 MR^2 ( d ) 37 2 9
MR
15. One quarter section is cut from a uniform circular disc of radius R. This section has a mass M. It is made to rotate about a line perpendicular to its plane and passing through the center of the original disc. Its moment of inertia about the axis of rotation is
(IIT-JEE 2001)
( a ) 1 2 2
MR ( b )^12 4
MR ( c )^2
MR ( d ) 2 MR^2
16. A thin wire of length l , and uniform linear mass density ρ bent into a circular loop with centre at as shown. The moment of inertia of the loop about the axis is (IIT-JEE 2000)
90°
O
X X ′
( a )
3
82
ρ l π
( b )
3
162
ρ l π
( c )
3 2
ρ l π
( d )
3 2
ρ l π
17. Let I be the moment of inertia of a uniform square plate about an axis AB that passes through its centre and is parallel to two of its sides. CD is a line in the plane of the plate that passes through the centre of the plate and makes an angle θ with AB. The moment of inertia of the plate about the axis CD is then equal to (IIT-JEE 1998) E
D
A θ B
C
F ( a ) I ( b ) I sin^2 θ ( c ) I cos^2 θ ( d ) I cos^2 (θ/2)
18. Two point masses of 0.3kg and 0.7kg are fixed at the ends of a rod of length 1.4m and of negligible mass. The rod is set rotating about an axis perpendicular to its length with a uniform angular speed. The point on the rod through which the axis should pass in order that the work required for rotation of the rod is minimum, is located at a distance of (IIT-JEE 1992)
( a ) 0.42m from mass of 0.3 kg ( b ) 0.70m from mass of 0.7 kg ( c ) 0.98m from mass of 0.3 kg ( d ) 0.98m from mass of 0.7 kg
Multiple Correct
19. The moment of inertia of a thin square plate ABCD , of uniform thickness about an axis passing through the centre O and perpendicular to the plane of the plate is
A
D
B
C
where, I 1 , I 2 , I 3 and I 4 are respectively moments of inertia about axes 1, 2, 3 and 4 which are in the plane of the plate (IIT-JEE 1992) ( a ) I 1 + I 2 ( b ) I 3 + I 4 ( c ) I 1 + I 3 ( d ) I 1 + I 2 + I 3 + I 4
Numerical Types/Integer Types
20. A lamina is made by removing a small disc of diameter 2 R from a bigger disc of uniform mass density and radius 2 R , as shown in the figure. The moment of inertia of this lamina about axes passing through O and P is I (^) O and I (^) P , respectively. Both these axes are perpendicular to the plane of the lamina. The ratio IP / IO to the nearest integer is (IIT-JEE 2012)
2 R
O^2 R P
21. Four solid spheres each of diameter 5 cm and mass 0.5 kg are placed with their centres at the comers of a square of side 4 cm. The moment of inertia of the system about the diagonal of the square is N×10–4^ kg-m^2 , then N is (IIT-JEE 2011)
Fill in the Blanks
22. Put a uniform meter scale horizontally on your extended index fingers with the left one at 0.00 cm and the right one at 90.00 cm. When you attempt to move both the fingers slowly towards the center, initially only the left finger slips with respect to the scale and the right finger does not. After some distance the left finger stops and the right one starts slipping. Then the right finger stops at a distance xR from the center (50.00 cm) of the scale and the let one starts slipping again. This happens because of the difference in the frictional forces on the two fingers. If the coefficients of static and dynamic friction between the fingers and the scale are 0.40 and 0.32, respectively, the value of xR (in cm) is _______ (JEE Adv. 2020)
Rotational Motion^175
Multiple Correct
30. A particle of mass M = 0.2 kg is initially at rest in the xy -plane at a point ( x = – l, y = – h ) where l = 10 m and h = 1 m. The particle is accelerated at time t = 0 with a constant acceleration a = 10 m/s^2 along the positive x -direction. Its angular momentum and torque with respect to the origin, in SI units, are represented by L
and τ^ ^ , respectively. ˆ ˆ i , j and k ˆ are unit vectors along the positive x, y and z- directions, respectively. If k ˆ^ = ˆ i^ × ˆ j then which of the following statement(s) is(are) correct? C-18.94 W-11.64 UA-42.06 PC-27.37 (JEE Adv. 2021) ( a ) The particle arrives at the point ( x = l, y = – h ) at time t = 2s ( b ) (^) τ =^ ^2 k ˆwhen the particle passes through the point ( x = l, y = – h ) ( c ) (^) L = 4 k ˆ
when the particle passes through the point ( x = l, y = – h ) ( d ) (^) τ =^ ^ k ˆwhen the particle passes through the point ( x = 0 , y = – h )
31. Consider a body of mass 1.0 kg at rest at the origin at time
t = 0. A force F = ( α ti ^ + β i )^ is applied on the body, where α = 1.0Ns–1^ and b = 1.0N. The torque acting on the body about the origin at time τ = 1.0s is τ. Which of the following statements is (are) true? C-16.3 W-41.89 UA-27.15 PC-14.67 (JEE Adv. 2018) ( a ) |τ| = 1/3 N-m ( b ) The torque τ is in the direction of the unit vector + k ( c ) The velocity of the body at t =1s is v = 1/2 ( i^ +2 ) j ms– ( d ) The magnitude of displacement of the body at t = 1s is 1/6 m
32. A wheel of radius R and mass M is placed at the bottom of a fixed step of height R as shown in the figure. A constant force is continuously applied on the surface of the wheel so that it just climbs the step without slipping. Consider the torque τ about an axis normal to the plane of the paper passing through the point Q. Which of the following options is/are correct? C-20.63 W-34.31 UA-39.42 PC-5.63 (JEE Adv. 2017)
P
R
S
Q
( a ) If the force is applied to the circumference at point P , then τ net is zero ( b ) If the force is applied tangentially at point S , then τ net ≠ 0 but the wheel never climbs the step ( c ) If the force is applied at point P tangentially, then τ net decreases continuously as the wheel climbs ( d ) If the force is applied normal to the circumference at point X , then τ net is constant
33. A rigid uniform bar AB of length L is slipping from its vertical position on a frictionless floor (as shown in the figure). At some instant of time, the angle made by the bar with the vertical is θ. Which of the following statements about its motion is/are correct?
L
B
O
θ
C-9.75 W-29.75 UA-28.15 PC-32.35 (JEE Adv. 2017)
( a ) Instantaneous torque about the point in contact with the floor is proportional to sinθ ( b ) The trajectory of the point A is parabola ( c ) The mid-point of the bar will fall vertically downward ( d ) When the bar makes an angle θ with the vertical, the displacement of its mid-point from the initial position is proportional to (1 – cosθ)
34. The torque τ on a body about a given point is found to be equal to A × L , where A is a constant vector and L is the angular momentum of the body about that point. From this it follows that (IIT-JEE 1998) ( a ) dL / dt is perpendicular to L at all instants of time ( b ) The component of L in the direction of A does not change with time ( c ) The magnitude of L does not change with time ( d ) L does not change with time
Match the Column
35. In the List-I below, four different paths of a particle are given as functions of time. In these functions, α and b are positive constants of appropriate dimensions and α ≠ b. In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned: p is the linear momentum, L is the angular momentum about the origin, K is the kinetic energy, U is the potential energy and E is the total energy. Match each path in List-I with those quantities in List-II, which are conserved for that path. C-27.33 W-34.48 UA-38.19 (JEE Adv. 2018) List-I List-II P r ( t ) = α ti + b tj 1. P Q r ( t ) = αcosω ti + bsinω tj 2. L R r ( t ) = α(cosω ti + sinω tj ) 3. K S r ( t ) = α ti + b/2 t 2 j 4. U
- E
( a ) P → 1, 2, 3, 4, 5; Q → 2, 5; R → 2, 3, 4, 5; S → 5 ( b ) P → 1, 2, 3, 4, 5; Q → 3, 5; R → 2, 3, 4, 5; S → 2, 5 ( c ) P → 2, 3, 4; Q → 5; R → 1, 2, 4; S → 2, 5 ( d ) P → 1, 2, 3, 5; Q → 2, 5; R → 2, 3, 4, 5; S → 2, 5
Numerical Types/Integer Types
36. A boy is pushing a ring of mass 2 kg and radius 0.5 m with a stick as shown in the figure. The stick applies a force of 2 N on the ring and rolls it without slipping with an acceleration of 0.3 m/s^2. The coefficient of friction between the ground and the ring is large enough that rolling always occurs and the coefficient of friction between the stick and the ring is ( P /10). The value of P is Stick
Ground (IIT-JEE 2011)
176 JEE Main and Advanced Physics PYQs
Subjective
37. Three particles A , B and C , each of mass m, are connected to each other by three massless rigid rods to form a rigid, equilateral triangular body of side l. This body is placed on a horizontal frictionless table ( x - y plane ) and is hinged to it at the point A , so that it can move without friction about the vertical axis through A (see figure). The body is set into rotational motion on the table about A with a constant angular velocity ω.
F B C
A
l
y x ω
( a ) Find the magnitude of the horizontal force exerted by the hinge on the body. ( b ) At time T , when the side BC is parallel to the X -axis, a force F is applied on B along BC (as shown). Obtain the x -component and the y -component of the force exerted by the hinge on the body, immediately after time T. (IIT-JEE 2005)
38. A rod AB of mass M and length L is lying on a horizontal frictionless surface. A particle of mass m travelling along the surface hits the end A of the rod with a velocity v 0 in a direction perpendicular to AB. The collision is elastic. After the collision, the particle comes to rest. (2000) ( a ) Find the ratio m / M. ( b ) A point P on the rod is at rest immediately after collision. Find the distance AP. ( c ) Find the linear speed of the point P a time π L /3 v 0 after the collision. (IIT-JEE 2000) 39. Two thin circular discs of mass 2 kg and radius 10 cm each are joined by a rigid massless rod of length 20 cm. The axis of the rod is along the perpendicular to the planes of the disc through their centres. This object is kept on a truck in such a way that the axis of the object is horizontal and perpendicular to the direction of motion of the truck. Its friction with the floor of the truck is large enough, so that the object can roll on the truck without slipping. Take X -axis as the direction of motion of the truck and Z -axis as the vertically upwards direction. If the truck has an acceleration 9 m / s^2 , calculate
O
20 cm ( a ) the force of friction on each disc and ( b ) the magnitude and direction of the frictional torque acting on each disc about the centre of mass O of the object. Express the torque in the vector form in terms of unit vectors (^) i j ,^ and k in x , y and z -directions. (IIT-JEE 1997)
40. A homogeneous rod AB of length L = 1.8m and mass M is pivoted at the centre O in such a way that it can rotate freely in the vertical plane (figure). The rod is initially in the horizontal position. An insect S of the same mass M falls vertically with speed v on the point C , midway between the points O and B. Immediately after falling, the insect moves towards the end B such that the rod rotates with a constant angular velocity ω. S
A O C B
L/ 2 L/ 4 L/ 4
v
( a ) Determine the angular velocity ω in terms of v and L ( b ) If the insect reaches the end B when the rod has turned through an angle of 90°, determine v. (IIT-JEE 1992)
41. A cylinder of mass M and radius R is resting on a horizontal platform (which is parallel to the x - y plane) with its axis fixed along the Y -axis and free to rotate about its axis. The platform is given a motion in the x -direction given by x = A cos(ω t ). There is no slipping between the cylinder and platform. The maximum torque acting on the cylinder during its motion is (IIT-JEE 1988) 42. A uniform cube of side a and mass m rests on a rough horizontal table. A horizontal force F is applied normally to one of the faces at a point that is directly above the center of the face, at a height 3 a /4 above the base. The minimum value of F for which the cube begins to tip about the edge is. (Assume that the cube does not slide). (IIT-JEE 1984)
Fill in the Blanks
43. A rod of weight W is supported by two parallel knife edges A and B and is in equilibrium in horizontal position. The knives are at a distance d from each other. The centre of mass of the rod is at distance x from A. The normal reaction on A is ……… and on B is ........... (IIT-JEE 1997)
Angular Momentum and
Conservation of Angular Momentum
Single Correct
44. A bar of mas M = 1.00 kg and length L = 0.20 m is lying on a horizontal frictionless surface. One end of the bar is pivoted at a point about which it is free to rotate. A small mass m = 0.10 kg is moving on the same horizontal surface with 5.00 ms – speed on a path perpendicular to the bar. It hits the bar at a distance L /2 from the pivoted end and returns back on the same path with speed v. After this elastic collision, the bar rotates with an angular velocity ω. Which of the following statement is correct? C-10.89 W-29.13 UA-59.98 (JEE Adv. 2023) ( a ) ω = 6.98 rad s–1^ and v = 4.30 m s– ( b ) ω = 3.75 rad s –1^ and v = 4.30 m s– ( c ) ω = 3.75 rad s –1^ and v = 10.0 m s– ( d ) ω = 6.80 rad s –1^ and v = 4.10 m s–
