STD 11 - System of particles and rotational motion PYQs: JEE Mains Physics – Page 2

Showing 51–100 of 369 questions

51. 51.
    Consider two blocks A and B of masses \(m_1=10 \mathrm{~kg}\) and \(m_2=5 \mathrm{~kg}\) that are placed on a frictionless table. The block A moves with a constant speed \(v=3 \mathrm{~m} / \mathrm{s}\) towards the block B kept at rest. A spring with spring constant \(\mathrm{k}=3000 \mathrm{~N} / \mathrm{m}\) is attached with the block B as shown in the figure. After the collision, suppose that the blocks A and B, along with the spring in constant compression state, move together, then the compression in the spring is _______. (Neglect the mass of the spring) 2025 Medium
52. 52. If \(\overrightarrow{\mathrm{L}}\) and \(\overrightarrow{\mathrm{P}}\) represent the angular momentum and linear momentum respectively of a particle of mass ' m ' having position vector \(\overrightarrow{\mathrm{r}}=\mathrm{a}(\hat{\mathrm{i}} \cos \omega \mathrm{t}+\hat{\mathrm{j}} \sin \omega \mathrm{t})\). The direction of force is 2025 Easy
53. 53. M and R be the mass and radius of a disc. A small disc of radius \(R / 3\) is removed from the bigger disc as shown in figure. The moment of inertia of remaining part of bigger disc about an axis AB passing through the centre O and perpendicular to the plane of disc is \(\frac{4}{x} M R^2\). The value of \(x\) is ________.
    2025 Hard
54. 54. Moment of inertia of a rod of mass ' M ' and length 'L' about an axis passing through its center and normal to its length is ' \(\alpha\) '. Now the rod is cut into two equal parts and these parts are joined symmetrically to form a cross shape. Moment of inertia of cross about an axis passing through its center and normal to plane containing cross is : 2025 Medium
55. 55. The center of mass of a thin rectangular plate (fig - x ) with sides of length \(a\) and \(b\), whose mass per unit area \((\sigma)\) varies as \(\sigma=\frac{\sigma_0 x}{a b}\) (where \(\sigma_0\) is a constant), would be \(\qquad\)
    
    2025 Medium
56. 56. The coordinates of a particle with respect to origin in a given reference frame is \((1,1,1)\) meters. If a force of \(\overrightarrow{\mathrm{F}}=\hat{i}-\hat{j}+\hat{k}\) acts on the particle, then the magnitude of torque (with respect to origin) in z-direction is ______. 2025 Easy
57. 57. The moment of inertia of a circular ring of mass \(M\) and diameter r about a tangential axis lying in the plane of the ring is : 2025 Easy
58. 58. The moment of inertia of a solid disc rotating along its diameter is 2.5 times higher than the moment of inertia of a ring rotating in similar way. The moment of inertia of a solid sphere which has same radius as the disc and rotating in similar way, is \(n\) times higher than the moment of inertia of the given ring. Here, \(\mathrm{n}=\)_________
    Consider all the bodies have equal masses. 2025 Hard
59. 59. The position vectors of two 1 kg particles, (A) and (B), are given by
    \(\overrightarrow{\mathrm{r}}_{\mathrm{A}}=\left(\alpha_1 \mathrm{t}^2 \hat{i}+\alpha_2 \mathrm{t} \hat{j}+\alpha_3 \mathrm{t} \hat{k}\right) \mathrm{m}\) and \(\overrightarrow{\mathrm{r}}_{\mathrm{B}}=(\beta_1 \mathrm{t} \hat{i}+\beta_2 \mathrm{t}^2 \hat{j}\) \(+\beta_3 \mathrm{t} \hat{k}) \mathrm{m}\), respectively; \((\alpha_1=1 \mathrm{~m} / \mathrm{s}^2, \alpha_2=3 \mathrm{n} \mathrm{m} / \mathrm{s},\) \(\alpha_3=2 \mathrm{~m} / \mathrm{s},\) \(\beta_1=2 \mathrm{~m} / \mathrm{s}, \beta_2=-1 \mathrm{~m} / \mathrm{s}^2, \beta_3=4 \mathrm{pm} / \mathrm{s})\), where t is time, n and p are constants. At \(t=1 \mathrm{~s},\left|\overrightarrow{V_A}\right|=\left|\vec{V}_B\right|\) and velocities \(\vec{V}_A\) and \(\vec{V}_B\) of the particles are orthogonal to each other. At \(t=1 \mathrm{~s}\), the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is \(\sqrt{\mathrm{L}} \mathrm{kgm}^2 \mathrm{~s}^{-1}\). The value of L is _______ . 2025 Hard
60. 60. The torque due to the force \((2 \hat{i}+\hat{j}+2 \hat{k})\) about the origin, acting on a particle whose position vector is \((\hat{i}+\hat{j}+\hat{k})\), would be 2025 Easy
61. 61.
    Three equal masses \(m\) are kept at vertices (A, B, C) of an equilateral triangle of side a in free space. At \(t=0\), they are given an initial velocity \(\vec{V}_A=V_0 \overrightarrow{A C}, \vec{V}_B=V_0 \overrightarrow{B A}\) and \(\vec{V}_C=V_0 \overrightarrow{C B}\). Here, \(\overrightarrow{A C}, \overrightarrow{C B}\) and \(\overrightarrow{B A}\) are unit vectors along the edges of the triangle. If the three masses interact gravitationally, then the magnitude of the net angular momentum of the system at the point of collision is : 2025 Hard
62. 62. Two iron solid discs of negligible thickness have radius \(R_1\) and \(R_2\) and moment of intertia \(I_1\) and \(I_2\), respectively. For \(R_2=2 R_1\), the ratio of \(I_1\) and \(I_2\) would be \(1 / x\), where \(\mathrm{x}=\) ________ - 2025 Easy
63. 63. Which of the following are correct expression for torque acting on a body?
    A. \(\vec{\tau}=\vec{r}\times\vec{L}\)
    B. \(\vec{\tau}=\frac{\mathrm{d}}{\mathrm{dt}}(\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{p}})\)
    C. \(\vec{\tau}=\overrightarrow{\mathrm{r}} \times \frac{\mathrm{d} \mathrm{p}}{\mathrm{dt}}\)
    D. \(\vec{\tau}=\mathrm{I} \vec{\alpha}\)
    E. \(\vec{\tau}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}\)
    ( \(\overrightarrow{\mathrm{r}}=\) position vector; \(\overrightarrow{\mathrm{p}}=\) linear momentum;
    \(\overrightarrow{\mathrm{L}}=\) angular momentum; \(\vec{\alpha}=\) angular acceleration;
    \(I=\) moment of inertia; \(\vec{F}=\) force; \(t=\) time)
    Choose the correct answer from the options given below : 2025 Easy
64. 64. A body of mass ' \(m\) ' is projected with a speed ' \(u\) ' making an angle of \(45^{\circ}\) with the ground. The angular momentum of the body about the point of projection, at the highest point is expressed as \(\frac{\sqrt{2} \mathrm{mu}^3}{\mathrm{Xg}}\). The value of ' \(\mathrm{X}\) ' is _______. 2024 Hard
65. 65. A body of mass \(5 \mathrm{~kg}\) moving with a uniform speed \(3 \sqrt{2} \mathrm{~ms}^{-1}\) in \(\mathrm{X}-\mathrm{Y}\) plane along the line \(\mathrm{y}=\mathrm{x}+4\).The angular momentum of the particle about the origin will be _______ \(\mathrm{kg}\  \mathrm{m} \mathrm{s}^{-1}\). 2024 Hard
66. 66. A circular disc reaches from top to bottom of an inclined plane of length \(l\). When it slips down the plane, if takes \(\mathrm{t} \mathrm{s}\). When it rolls down the plane then it takes \(\left(\frac{\alpha}{2}\right)^{1 / 2} \mathrm{t} s\), where \(\alpha\) is. _______. 2024 Hard
67. 67. A cylinder is rolling down on an inclined plane of inclination \(60^{\circ}\). It's acceleration during rolling down will be \(\frac{\mathrm{x}}{\sqrt{3}} \mathrm{~m} / \mathrm{s}^2\), where \(\mathrm{x}=\)___________. (use \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\) ). 2024 Hard
68. 68. 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 _______.
    2024 Hard
69. 69. A heavy iron bar of weight \(12 \mathrm{~kg}\) is having its one end on the ground and the other on the shoulder of a man. The rod makes an angle \(60^{\circ}\) with the horizontal, the weight experienced by the man is : 2024 Hard
70. 70. A heavy iron bar, of weight \(\mathrm{W}\) is having its one end on the ground and the other on the shoulder of a person. The bar makes an angle \(\theta\) with the horizontal. The weight experienced by the person is _______. 2024 Hard
71. 71. 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 \(\frac{x}{5}\). The value of \(x\) is _______. 2024 Hard
72. 72. A particle of mass \(m\) projected with a velocity ' \(u\) ' making an angle of \(30^{\circ}\) with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height \(\mathrm{h}\) is _______. 2024 Hard
73. 73. 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 \(\frac{7}{\mathrm{x}}\) where \(\mathrm{x}\) is___________. 2024 Hard
74. 74. A solid circular disc of mass \(50 \mathrm{~kg}\) rolls along a horizontal floor so that its center of mass has a speed of \(0.4 \mathrm{~m} / \mathrm{s}\). The absolute value of work done on the disc to stop it is _______ \(\mathrm{J}\). 2024 Hard
75. 75. 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 \(\frac{n}{10}\). The value of \(\mathrm{n}\) is __________. 2024 Hard
76. 76. A string is wrapped around the rim of a wheel of moment of inertia \(0.40 \mathrm{kgm}^2\) and radius \(10 \mathrm{~cm}\). The wheel is free to rotate about its axis. Initially the wheel is at rest. The string is now pulled by a force of \(40 \mathrm{~N}\). The angular velocity of the wheel after \(10\) \(\mathrm{s}\) is \(\mathrm{x} \mathrm{rad} / \mathrm{s}\), where \(\mathrm{x}\) is _______. 2024 Hard
77. 77. 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 \(\omega\). If another disc of same dimensions but of mass \(\mathrm{M} / 2\) is placed gently on the first disc co-axially, then the new angular velocity of the system is _______. 2024 Hard
78. 78. A uniform rod \(A B\) of mass \(2 \mathrm{~kg}\) and Length \(30 \mathrm{~cm}\) at rest on a smooth horizontal surface. An impulse of force \(0.2\  \mathrm{Ns}\) is applied to end \(B.\) The time taken by the rod to turn through at right angles will be \(\frac{\pi}{\mathrm{x}}\ \mathrm{s}\), where  X = _______. 2024 Hard
79. 79. A uniform thin metal plate of mass \(10 \mathrm{~kg}\) with dimensions is shown. The ratio of \(x\) and \(y\) coordinates of center of mass of plate in \(\frac{n}{9}\). The value of \(n\) is _______.
    2024 Hard
80. 80. Consider a Disc of mass \(5 \mathrm{~kg}\), radius \(2 \mathrm{~m}\), rotating with angular velocity of \(10 \mathrm{rad} / \mathrm{s}\) 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 _______ \(J\).
    2024 Hard
81. 81. Four particles each of mass \(1 \mathrm{~kg}\) are placed at four corners of a square of side \(2 \mathrm{~m}\). Moment of inertia of system about an axis perpendicular to its plane and passing through one of its vertex is_________ \(\mathrm{kgm}^2\).
    2024 Hard
82. 82. 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. 2024 Hard
83. 83. In a system two particles of masses \(m_1=3 \mathrm{~kg}\) and \(\mathrm{m}_2=2 \mathrm{~kg}\) are placed at certain distance from each other. The particle of mass \(m_1\) is moved towards the center of mass of the system through a distance \(2 \mathrm{~cm}\). In order to keep the center of mass of the system at the original position, the particle of mass \(\mathrm{m}_2\) should move towards the center of mass by the distance _______\(\mathrm{cm}\). 2024 Hard
84. 84. 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 \(A B\) as shown in figure is \(\sqrt{\frac{8}{x}}\). The value of \(x\) is _______.
    2024 Hard
85. 85. The identical spheres each of mass \(2 \mathrm{M}\) are placed at the corners of a right angled triangle with mutually perpendicular sides equal to \(4 \mathrm{~m}\) each. Taking point of intersection of these two sides as origin, the magnitude of position vector of the centre of mass of the system is \(\frac{4 \sqrt{2}}{x}\), where the value of \(x\) is _______. 2024 Hard
86. 86. Three balls of masses \(2 \mathrm{~kg}, 4 \mathrm{~kg}\) and \(6 \mathrm{~kg}\) respectively are arranged at centre of the edges of an equilateral triangle of side \(2 \mathrm{~m}\). The moment of inertia of the system about an axis through the centroid and perpendicular to the plane of triangle, will be _______\(\mathrm{kg} \mathrm{m}{ }^2\). 2024 Hard
87. 87. Two discs of moment of inertia \(I_1=4 \mathrm{~kg} \mathrm{~m}^2\) and \(\mathrm{I}_2=2 \mathrm{~kg} \mathrm{~m}^2\) about their central axes & normal to their planes, rotating with angular speeds \(10 \mathrm\ {rad} / \mathrm{s}\) & \(4  \mathrm\ {rad} / \mathrm{s}\) 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 _______ \(\mathrm{J}\). 2024 Hard
88. 88. Two identical spheres each of mass \(2 \mathrm{~kg}\) and radius \(50 \mathrm{~cm}\) are fixed at the ends of a light rod so that the separation between the centers is \(150 \mathrm{~cm}\). Then, moment of inertia of the system about an axis perpendicular to the rod and passing through its middle point is \(\frac{x}{20} \mathrm{~kg} \mathrm{~m}^2\), where the value of \(\mathrm{x}\) is _______. 2024 Hard
89. 89. A child of mass \(5\,kg\) is going round a merry-goround that makes \(1\) rotation in \(3.14\,s\). The radius of the merry-go-round is \(2\,m\). The centrifugal force on the child will be \(.......\,N\) 2023 Medium
90. 90. A circular plate is rotating in horizontal plane, about an axis passing through its center and perpendicular to the plate, with an angular velocity \(\omega\). A person sits at the center having two dumbbells in his hands. When he stretches out his hands, the moment of inertia of the system becomes triple. If \(E\) be the initial Kinetic energy of the system, then final Kinetic energy will be \(\frac{E}{x}\).The value of \(x\) is \(....\) 2023 Medium
91. 91. A force of \(- P \hat{k}\) acts on the origin of the coordinate system. The torque about the point \((2,-3)\) is \(P(a \hat{i}+b \hat{j})\), the ratio of \(\frac{a}{b}\) is \(\frac{x}{2}\). The value of \(x\) is 2023 Medium
92. 92. A hollow spherical ball of uniform density rolls up a curved surface with an initial velocity \(3\, m / s\) (as shown in figure). Maximum height with respect to the initial position covered by it will be \(...........cm\).
    2023 Hard
93. 93. 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.....\( {rad \,s}{ }^{-2}\). 2023 Medium
94. 94. 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 \(\sqrt{\frac{2}{x}}\). The value of \(x\) is \(.......\) 2023 Medium
95. 95. A solid cylinder is released from rest from the top of an inclined plane of inclination \(30^{\circ}\) and length \(60\,cm\). If the cylinder rolls without slipping, its speed upon reaching the bottom of the inclined plane is \(...........ms ^{-1}\). (Given \(g =10\,ms ^{-2}\))
    2023 Medium
96. 96. 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 \(\left( k _{\text {sph }}: k _{ cyl }\right)\) is \(2: \sqrt{ x }\), then value of \(x\) is ............. 2023 Medium
97. 97. A solid sphere is rolling on a horizontal plane without slipping. If the ratio of angular momentum about axis of rotation of the sphere to the total energy of moving sphere is \(\pi: 22\) then, the value of its angular speed will be \(...........\,rad / s\). 2023 Medium
98. 98. A solid sphere of mass \(1\,kg\) rolls without slipping on a plane surface. Its kinetic energy is \(7 \times 10^{-3}\,J\). The speed of the centre of mass of the sphere is \(.........cm s ^{-1}\). 2023 Medium
99. 99. A solid sphere of mass \(2\,kg\) is making pure rolling on a horizontal surface with kinetic energy \(2240\,J\). The velocity of centre of mass of the sphere will be \(..........ms ^{-1}\). 2023 Medium
100. 100. 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 \(x \times 10^{-2}\) times its angular momentum about the diameter. Then the value of \(x\) will be .............. 2023 Medium

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