Rotational Mechanics PYQs Set 10 (PDF)

Set 10 of 14 · Questions 271-300 of 410

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Mere Teacher · Previous Year Questions JEE Main Physics · Rotational Mechanics · Set 10 (Q271-300)
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271. $Four point masses, each of mass M are placed at the corners of a square of side L . The moment of inertia of the system about one of its diagonals is$ (1) $2 M L^{2}$ (2) $M L^{2}$ (3) $4 M L^{2}$ (4) $6 M L^{2}$ 272. $ABC is an equilateral triangle with O as its centre. F_{1}, F_{2} and F_{3} represent three forces acting along the sides AB, BC and AC respectively. If the total torque about O is zero then the magnitude of F_{3} is$ (1) $F_{1} + F_{2}$ (2) $F_{1} - F_{2}$ (3) $F_{1} + F_{2} 2$ (4) $2 (F_{1} + F_{2})$ 273. $Three solid spheres each of mass m and diameter d are stuck together such that the lines connecting the centres form an equilateral triangle of side of length d. The ratio Ι0/ΙA of moment of inertial Ι0 of the system about an axis passing the centroid and about center of any of the spheres ΙA and perpendicular to the plane of the triangle is :$ (1) $\frac{23}{13}$ (2) $\frac{13}{15}$ (3) $\frac{13}{23}$ (4) $\frac{15}{13}$ 274. $Two identical spheres A and B are free to move and to rotate about their centres. They are given the same impulse J. The lines of action of the impulses pass through the centre of A, and away from the centre of B.$ (1) $A and B will have the same speed.$ (2) $B will have greater kinetic energy than A.$ (3) $They will have the same kinetic energy, but the linear kinetic energy of B will be less than that of A.$ (4) $The kinetic energy of B will depend on the point of impact of the impulse on B.$ 275. $A car is moving with a constant speed of 20 m/s in a circular horizontal track of radius 40 m. A bob is suspended from the roof of the car by a massless string. The angle made by the string with the vertical will be : (Take g = 10 m/ s^{2})$ (1) $\frac{π}{6}$ (2) $\frac{π}{2}$ (3) $\frac{π}{4}$ (4) $\frac{π}{3}$ 276. $A square frame ABCD is formed by four identical rods each of mass ' m' and length ' l'. This frame is in X - Y plane such that side AB coincides with X -axis and side AD along Y -axis. The moment of inertia of the frame about X -axis is$ (1) $\frac{5 m l ^{2}}{3}$ (2) $\frac{2 m l ^{2}}{3}$ (3) $\frac{4 m l ^{2}}{3}$ (4) $\frac{m l ^{2}}{12}$ 277. $A uniform thin rod of 120 cm length and 1600 g mass is bent as shown in the figure. The moment of inertia of the bent rod about an axis passing through the point ' O' and perpendicular to the plane of the paper is\dots\dots\dots kg - m^{2} .$ (1) $0.084$ (2) $0.360$ (3) $0.018$ (4) $0.120$ 278. $A disk and a sphere of same radius but different masses roll off on two inclined planes of the same altitude and length. Which one of the two objects gets to the bottom of the plane first ?$ (1) $Depends on their masses$ (2) $Disk$ (3) $Sphere$ (4) $both reach at the same time$ 279. $Moment of inertia of a rod of mass ' M ' and length ' L ' about an axis passing through its center and normal to its length is ' α'. Now the rod is cut into two equal parts and these parts are joined symmetrically to form a cross shape. Moment of inertia of cross about an axis passing through its center and normal to plane containing cross is:$ (1) $α /4$ (2) $α /8$ (3) $α$ (4) $α /2$ 280. $Point masses m_{1} and m_{2} are placed at the opposite ends of a rigid rod of length L and negligible mass. The rod is to be set rotating about an axis perpendicular to it. The position of point P on this rod through which the axis should pass, so that the work required to set the rod rotating with angular velocity ω_{o} is minimum, is given by$ (1) $x = \frac{m _{1} L}{m _{1} + m _{2}}$ (2) $x = \frac{m _{1}}{m _{2}} L$ (3) $x = \frac{m _{2}}{m _{1}} L$ (4) $x = \frac{m _{2} L}{m _{1} + m _{2}}$ 281. $For a uniform rectangular sheet shown in the figure, the ratio of moments of inertia about the axes perpendicular to the sheet and passing through O (the centre of mass) and O ’ (corner point) is :$ (1) $1/2$ (2) $1/4$ (3) $1/8$ (4) $2/3$ 282. $A particle of mass m = 5 is moving with a uniform speed v = 32 in the XOY plane along the line Y = X + 4 . The magnitude of the angular momentum of the particle about the origin is$ (1) $60 units$ (2) $402 units$ (3) $zero$ (4) $7.5 units$ 283. $A circular disc of radius b has a hole of radius a at its centre (see figure). If the mass per unit area of the disc varies as (\frac{σ _{0}}{r}), then the radius of gyration of the disc about its axis passing through the centre is :$ (1) $\frac{a ^{2} + b ^{2} + ab}{2}$ (2) $\frac{a + b}{2}$ (3) $\frac{a ^{2} + b ^{2} + ab}{3}$ (4) $\frac{a + b}{3}$ 284. $From a disc of radius R and mass M, a circular hole of diameter R, whose rim passes through the centre is cut. What is the moment of inertia of the remaining part of the disc about at perpendicular axis, passing through the centre ?$ (1) $9 M R^{2} /32$ (2) $15 M R^{2} /32$ (3) $13 M R^{2} /32$ (4) $11 M R^{2} /32$ 285. $Three solid spheres each of mass m and diameter d are stuck together such that the lines connecting the centres form an equilateral triangle of side of length d . The ratio \frac{I _{0}}{I _{A}} of moment of inertial h of the system about an axis passing the centroid and about center of any of the spheres I_{A} and perpendicular to the plane of the triangle is :$ (1) $13/23$ (2) $11/19$ (3) $7/9$ (4) $13/11$ 286. $When a mass is rotating in a plane about a fixed point, its angular momentum is directed along$ (1) $a line perpendicular to the plane of rotation$ (2) $the lin2 making an angle of45o to the plane of rotation$ (3) $the radius$ (4) $the tangent to the orbit$ 287. $A particle of mass 20 g is released with an initial velocity 5 m/s along the curve from the point A, as shown in figure. The point A is at height h from point B. The particle sides along the frictionless surface. When the particle reaches point B, its angular momentum about O will be :$ (1) $3 kg- m^{2} /s$ (2) $6 kg- m^{2} /s$ (3) $2 kg- m^{2} /s$ (4) $8 kg- m^{2} /s$ 288. $Shown in the figure is a hollow ice-cream cone (it open at the top). If its mass is M, radius of its top, R and height, H, then its moment of inertia about its axis is:$ (1) $\frac{M ( R ^{2} + H ^{2} )}{4}$ (2) $\frac{M R ^{2}}{2}$ (3) $\frac{M R ^{2}}{3}$ (4) $\frac{M H ^{2}}{3}$ 289. $A metal coin of mass 5 g and radius 1 cm is fixed to a thin stick AB of negligible mass as shown in the figure The system is initially at rest. The constant torque, that will make the system rotate about AB at 25 rotations per second in 5 s, is close to:$ (1) $4.0 \times 10^{- 6} Nm$ (2) $1.6 \times 10^{- 5} Nm$ (3) $7.9 \times 10^{- 6} Nm$ (4) $2.0 \times 10^{- 5} Nm$ 290. $Ratio of radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for moment of Inertia about their diameter axis AB as shown in figure is 8/ x . The value of x is :$ (1) $51$ (2) $34$ (3) $17$ (4) $67$ 291. $Mass per unit area of a circular disc of radius a depends on the distance r from its centre as σ (r) = A + Br . The moment of inertia of the disc about the axis, perpendicular to the plane and passing through its centre is :$ (1) $2 π a^{4} (\frac{A}{4} + \frac{Ba}{5})$ (2) $2 π a^{4} (\frac{Aa}{4} + \frac{B}{5})$ (3) $π a^{4} (\frac{A}{4} + \frac{Ba}{5})$ (4) $2 π a^{4} (\frac{A}{5} + \frac{Ba}{4})$ 292. $A rod of length L revolves in a horizontal plane about the axis passing through its centre and perpendicular to its length. The angular velocity of the rod is ω . If A is the area of cross-section of the rod and ρ is its density, then the rotational kinetic energy of the rod is$ (1) $\frac{1}{3} A L^{3} ρ ω^{2}$ (2) $\frac{1}{2} A L^{3} ρ ω^{2}$ (3) $\frac{1}{24} A L^{3} ρ ω^{2}$ (4) $\frac{1}{18} A L^{3} ρ ω^{2}$ 293. $A rod of length 50 cm is pivoted at one end. It is raised such that if makes an angle of 30^{\circ} from the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in rads^{- 1}) will be (g = 10 ms^{- 2})$ (1) $\frac{30}{7}$ (2) $30$ (3) $\frac{20}{3}$ (4) $\frac{30}{2}$ 294. $Three identical spherical shells, each of mass m and radius r are placed as shown in figure. Consider an axis XX which is touching to two shells and passing through diameter of third shell. Moment of inertia of the system consisting of these three spherical shells about$ (1) $\frac{16}{5} m r^{2}$ (2) $4 m r^{2}$ (3) $\frac{11}{5} m r^{2}$ (4) $3 m r^{2}$ 295. $A mass of 10 kg is suspended vertically by a rope from the roof. When a horizontal force is applied on the rope at some point, the rope deviated at an angle of 45^{\circ} at the roof point. If the suspended mass is at equilibrium, the magnitude of the force applied is (g = 10 ms^{- 2})$ (1) $200 N$ (2) $140 N$ (3) $70 N$ (4) $100 N$ 296. $A solid spherical ball is rolled up an inclined plane of angle of inclination 30^{\circ} with an initial speed of 4 m / s at the bottom of the inclination. How far will the ball go up the plane. (Use g = 10 m / s^{2})$ (1) $56 cm$ (2) $112 cm$ (3) $224 cm$ (4) $120 cm$ 297. $A body of mass ' m' is projected with a speed ' u' making an angle of 45^{\circ} with the ground. The angular momentum of the body about the point of projection, at the highest point is expressed as \frac{2 mu ^{3}}{Xg} . The value of ' X^{’} is______$ 298. $A particle of mass 'm' is moving in time 't' on a trajectory given by r = 10 α t^{2} i^+ 5 β_{(} t - 5_{)} j^Where α and β are dimensional constants. The angular momentum of the particle becomes the same as it was for t=0 at time t= seconds.$ 299. $A cylinder is rolling down on an inclined plane of inclination 60^{\circ} . It's acceleration during rolling down will be \frac{x}{3} m / s^{2}, where x =_________(use (g = 10 m / s^{2}) .$ 300. $A solid sphere is rolling on a horizontal plane without slipping. If the ratio of angular momentum about axis of rotation of the sphere to the total energy of moving sphere is π : 22 then, the value of its angular speed will be_________rad s^{- 1} .$