Rotational Mechanics PYQs Set 7 (PDF)

Set 7 of 14 · Questions 181-210 of 410

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Mere Teacher · Previous Year Questions JEE Main Physics · Rotational Mechanics · Set 7 (Q181-210)
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181. $A ceiling fan rotates about its own axis with some angular velocity. When the fan is switched off, the angular velocity becomes (1/4)^{th} of the original in time t and n revolutions are made in that time. The number of revolutions made by the fan during the time interval between switch off and rest are (Angular retardation is uniform)$ (1) $\frac{4 n}{15}$ (2) $\frac{8 n}{15}$ (3) $\frac{16 n}{15}$ (4) $\frac{32 n}{15}$ 182. $The time dependence of the position of a particle of mass m = 2 is given by r \to (t) = 2 t i^- 3 t^{2} j^. Its angular momentum, with respect to the origin, at time t = 2 is :$ (1) $48 (i^+ j^)$ (2) $36 k^$ (3) $- 34 (k^- i^)$ (4) $- 48 k^$ 183. $For a hollow cylinder and a solid cylinder rolling without slipping on an inclined plane, then which of these reaches earlier?$ (1) $solid cylinder$ (2) $hollow cylinder$ (3) $both simultaneously$ (4) $can’t say anything.$ 184. $Moment of inertia of a body about two perpendicular axes X and Y in the plane of lamina are 20 kg m^{2} and 25 kg m^{2} respectively. Its moment of inertia about an axis perpendicular to the plane of the lamina and passing through the point of intersection of X and Y axes is$ (1) $5 kg m^{2}$ (2) $45 kg m^{2}$ (3) $12.5 kg m^{2}$ (4) $500 kg m^{2}$ 185. $A thin metal wire of length Land uniform linear mass density ρ is bent into a circular coil with O as centre. The moment of inertia of a coil about the axis X X^{’} is$ (1) $3 ρ L^{3} /8 π^{2}$ (2) $ρ L^{3} /4 π^{2}$ (3) $3 ρ L^{2} /4 π^{2}$ (4) $ρ L^{3} /8 π^{2}$ 186. $A mass m moving horizontally (along the x-axis) with velocity v collides and sticks to mass of 3m moving vertically upward (along the y-axis) with velocity 2v. The final velocity of the combination is :$ (1) $\frac{1}{4} v i^+ \frac{3}{2} v j^$ (2) $\frac{1}{3} v i^+ \frac{2}{3} v j^$ (3) $\frac{2}{3} v i^+ \frac{1}{3} v j^$ (4) $\frac{3}{2} v i^+ \frac{1}{4} v j^$ 187. $A block of mass m = 1 kg slides with velocity v = 6m/s on a frictionless horizontal surface and collides with a uniform vertical rod and sticks to it as shown. The rod is pivoted about O and swings as a result of the collision making angle θ before momentarily coming to rest. If the rod has mass M = 2kg and length I = lm, the value of θ is approximately (take g = 10m/s²)$ (1) $49°$ (2) $55°$ (3) $63°$ (4) $69°$ 188. $Two discs have moments of intertia I_{1} and I_{2} about their respective axes perpendicular to the plane and passing through the centre. They are rotating with angular speeds, ω_{1} and ω_{2} respectively and are brought into contact face to face with their axes of rotation coaxial. The loss in kinetic energy of the system in the process is given by :$ (1) $\frac{I _{1} I _{2}}{( I _{1} + I _{2} )} (ω_{1} - ω_{2})^{2}$ (2) $\frac{( I _{1} - I _{2} ) ^{2} ω _{1} ω _{2}}{2 ( I _{1} + I _{2} )}$ (3) $\frac{I _{1} I _{2}}{2 ( I _{1} + I _{2} )} (ω_{1} - ω_{2})^{2}$ (4) $\frac{( ω _{1} - ω _{2} ) ^{2}}{2 ( I _{1} + I _{2} )}$ 189. $A small mass attached to a string rotates on a frictionless table top as shown. If the tension in the string is increased by pulling the string causing the radius of the circular motion to decrease by a factor of 2, the kinetic energy of the mass will$ (1) $decrease by a factor of 2$ (2) $remain constant$ (3) $increase by a factor of 2$ (4) $increase by a factor of 4$ 190. $A solid cylinder of mass 50 kg and radius 0.5 m is free to rotate about the horizontal axis. A massless string is wound round the cylinder with one end attached to it and other hanging freely. Tension in the string required to produce an angular acceleration of 2 revolutions s^{- 2} is$ (1) $25 N$ (2) $50 N$ (3) $78.5 N$ (4) $157 N$ 191. $Three identical spherical shells, cach of mass m and radius r are placed as shown in figure. Consider an axis XX' which is touching to two shells and passing through diameter of third shell. Moment of inertia of the system consisting of these three sphereical shell about XX' axis is :$ (1) $3 mr2$ (2) $\frac{16}{5} m r^{2}$ (3) $4 mr2$ (4) $\frac{11}{5} m r^{2}$ 192. $Four spheres each of diameter 2 a and mass m are placed in a way that their centres lie on the four corners of a square of side b . Moment of inertia of the system about an axis along one of the sides of the square is$ (1) $\frac{8}{5} m a^{2}$ (2) $\frac{4}{5} m a^{2} + 5 m b^{2}$ (3) $\frac{4}{5} m a^{2} + 2 m b^{2}$ (4) $\frac{8}{5} m a^{2} + 2 m b^{2}$ 193. $The moment of inertia of a circular disc of radius 2 m and mass 1 kg about an axis passing through the centre of mass but perpendicular to the plane of the disc is 2 kg m^{2} . Its moment of inertia about an axis parallel to this axis but passing through the edge of the disc is (See the given figure).$ (1) $8 kg m^{2}$ (2) $4 kg m^{2}$ (3) $10 kg m^{2}$ (4) $6 kg m^{2}$ 194. $A circular platform is mounted on a frictionless vertical axle. Its radius R = 2m and its moment of inertia about the axle is 200 kgm2. it is initially at rest. A 50 kg man stands on the edge of the platform and begins to walk along the edge at the speed of 1ms-1 relative to the ground. Time taken by the man to complete one revolution is :$ (1) $π s$ (2) $\frac{3 π}{2} s$ (3) $2π s$ (4) $\frac{π}{2} s$ 195. $The torque of a force 5 i^+ 3 j^- 7 k^about the origin is τ . If the force acts on a particle whose position vector is 2 i^+ 2 j^+ k^, then the value of τ will be :$ (1) $11 i^+ 19 j^- 4 k^$ (2) $- 11 i^+ 9 j^- 16 k^$ (3) $- 17 i^+ 19 j^- 4 k^$ (4) $17 i^+ 9 j^+ 16 k^$ 196. $A solid sphere of mass m = 500 gm is rolling without slipping on a horizontal surface. Find kinetic energy of a sphere if velocity of centre of mass is 5 cm/ sec.$ (1) $\frac{35}{2} \times 1 0^{- 4} J$ (2) $\frac{35}{4} \times 1 0^{- 4} J$ (3) $35 \times 10^{- 4} J$ (4) $35 \times 10^{- 3} J$ 197. $A sphere of mass 2 kg and radius is rolling 0.5 m with an initial speed of 1 ms^{- 1} goes up an inclined plane which makes an angle of 30^{\circ} with the horizontal plane, without slipping. How low will the sphere take to return to the starting point A ?$ (1) $0.60 s$ (2) $0.52 s$ (3) $0.57 s$ (4) $0.80 s$ 198. $As shown in the figure, a bob a mass m is tied by a massless string whose other end portion is wound on a fly wheel (disc) of radius r and mass m. When released from rest the bob starts falling vertically. When it has covered a distance of h, the angular speed of the wheel will be :$ (1) $r \frac{3}{4 g h}$ (2) $\frac{1}{r} \frac{2 g h}{3}$ (3) $r \frac{3}{2 g h}$ (4) $\frac{1}{r} \frac{4 g h}{3}$ 199. $Four identical thin rods each of mass M and length l, form a square frame. Moment of inertia of this frame about an axis through the centre of the square and perpendicular to its plane is$ (1) $\frac{2}{3} M l^{2}$ (2) $\frac{13}{3} M l^{2}$ (3) $\frac{1}{3} M l^{2}$ (4) $\frac{4}{3} M l^{2}$ 200. $Moment of inertia of a square plate of side about the axis passing through one of the corner and perpendicular to the plane of square plate is given by:$ (1) $\frac{M l ^{2}}{6}$ (2) $M l^{2}$ (3) $\frac{M l ^{2}}{12}$ (4) $\frac{2}{3} M l^{2}$ 201. $Three point masses each of mass m are kept at the corners of an equilateral triangle of side L . The system rotates about the center of the triangle without any change in the separation of masses during rotation. The period of rotation is directly proportional to (cos 30^{\circ} = sin 60^{\circ} = 3 /2)$ (1) $L$ (2) $L^{3/2}$ (3) $L$ (4) $L^{- 2}$ 202. $A thin horizontal circular disc is rotating about a vertical axis passing through its centre. An insect is at rest at a point near the rim of the disc. The insect now moves along a diameter of the disc to reach its other end. During the journey of the insect, the angular speed of the disc$ (1) $remains unchanged$ (2) $continuously decreases$ (3) $continuously increases$ (4) $first increases and then decreases$ 203. $The moment of the force, F = 4 i + 5 j - 6 k at (2, 0, - 3), about the point (2, - 2, - 2), is given by$ (1) $- 8 i - 4 j - 7 k$ (2) $- 4 i - j - 8 k$ (3) $- 7 i - 8 j - 4 k$ (4) $- 7 i - 4 j - 8 k$ 204. $A circular disc has radius R_{1} and thickness T_{1} . Another circular disc made of the same material has radius R_{2} and thickness T_{2} . If the moment of inertia of both discs are same and \frac{R _{1}}{R _{2}} = 2 then \frac{T _{1}}{T _{2}} = \frac{1}{α} . The value of α is____.$ 205. $For rolling spherical shell, the ratio of rotational kinetic energy and total kinetic energy is \frac{x}{5} . The value of x is_____.$ 206. $Consider a Disc of mass 5 kg, radius 2 m, rotating with angular velocity of 10 rad / s about an axis perpendicular to the plane of rotation. An identical disc is kept gently over the rotating disc along the same axis. The energy dissipated so that both the discs continue to rotate together without slipping is ______J.$ 207. $A metre scale is balanced on a knife edge at its centre. When two coins, each of mass 10 g are put one on the top of the other at the 10.0 cm mark the scale is found to be balanced at 40.0 cm mark. The mass of the metre scale is found to be x x 1 0^{- 2} kg. The value of x is$ 208. $M and R be the mass and radius of a disc. A small disc of radius R /3 is removed from the bigger disc as shown in figure. The moment of inertia of remaining part of bigger disc about an axis AB passing through the center O and perpendicular to the plane of disc is \frac{4}{x} MR^{2} . The value of x is __________.$ 209. $A, B and C are disc, solid sphere and spherical shell respectively with same radii and masses. These masses are placed as shown in figure. The moment of inertia of the given system about PQ axis is \frac{x}{15} I, where I is the moment of inertia of the disc about its diameter. The value of x is______.$ 210. $Two identical spheres each of mass 2 kg and radius 50 cm are fixed at the ends of a light rod so that the separation between the centers is 150 cm . Then, moment of inertia of the system about an axis perpendicular to the rod and passing through its middle point is \frac{x}{20} kg m^{2}, where the value of x is______$