Rotational Mechanics PYQs Set 13 (PDF)

Set 13 of 14 · Questions 361-390 of 410

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Mere Teacher · Previous Year Questions JEE Main Physics · Rotational Mechanics · Set 13 (Q361-390)
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361. $The deceleration of a car traveling on a straight highway is a function of its instantaneous velocity v given by ω = a v, where a is a constant. If the initial velocity of the car is 60 km / h, the distance of the car will travel and the time it takes before it stops are$ (1) $\frac{2}{3} m, \frac{1}{2} s$ (2) $\frac{3}{2 a} m, \frac{1}{2 a} s$ (3) $\frac{3 a}{2} m, \frac{a}{2} s$ (4) $\frac{. 2}{3 a} m, \frac{2}{a} s$ 362. $Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres. The moment of inertia and angular velocity of the first disc are0.1kg - m^{2} and 10 rad s^{- 1} respectively while those for the second one are 0.2kg - m^{2} and 5rad s^{- 1} respectively. Atsome instant they get stuck together and start rotating as a single system about their common axis with some angular speed. The kinetic energy of the combined system is:$ (1) $\frac{20}{3} J$ (2) $\frac{5}{3} J$ (3) $\frac{10}{3} J$ (4) $\frac{2}{3} J$ 363. $A circular disk of moment of inertia I_{t} is rotating in a horizontal plane, about its symmetry axis, with a constant angular speed ω_{i} . Another disk of moment of inertia I_{b} is dropped coaxially onto the rotating disk. Initially the second disk has zero angular speed. Eventually both the disks rotate with a constant angular speed ω_{f} . The energy lost by the initially rotating disc to friction is$ (1) $\frac{1}{2} \frac{I _{b}^{2}}{( I _{t} + I _{b} )} ω_{i}^{2}$ (2) $\frac{1}{2} \frac{I _{t}^{2}}{( I _{t} + I _{b} )} ω_{i}^{2}$ (3) $\frac{I _{b} - I _{t}}{I _{t} + I _{b}} ω_{i}^{2}$ (4) $\frac{1}{2} \frac{I _{b} I _{t}}{( I _{t} + I _{b} )} ω_{i}^{2}$ 364. $A couple produces$ (1) $linear and rotational motion$ (2) $no motion$ (3) $purely linear motion$ (4) $purely rotational motion.$ 365. $Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R. Assertion A: Moment of inertia of a circular disc of mass 'M' and radius 'R' about X, Y axes (passing through its plane) and Z-axis which is perpendicular to its plane were found to be I_{x}, I_{y} and I_{z} respectively. The respective radii of gyration about all the three axes will be the same. Reason R: A rigid body making rotational motion has fixed mass and shape. In the light of the above statements, choose the most appropriate answer from the options given below:$ (1) $Both A and R are correct but R is NOT the correct explanation of A.$ (2) $A is not correct but R is correct.$ (3) $A is correct but R is not correct.$ (4) $Both A and R are correct and R is the correct explanation of A.$ 366. $Two discs are rotating about their axes, normal to the discs and passing through the centres of the discs. Disc D_{1} has 2 kg mass and 0.2 m radius and initial angular velocity of 50 rad s^{- 1} . Disc D_{2} has 4 kg mass, 0.1 m radius and initial angular velocity of 200 rad s^{- 1} . The two discs are brought in contact face to face, with their axes of rotation coincident. The final angular velocity (in rad s^{- 1}) of the system is$ (1) $60$ (2) $100$ (3) $120$ (4) $40$ 367. $A uniform rod of length 6 L and mass 8 m is pivoted at its centre C . Two masses m and 2 m with speed 2 v, v as shown strikes the rod and stick to the rod. Initially the rod is at rest. Due to impact, If it rotates with angular velocity ω_{1} then ω will be$ (1) $\frac{v}{5 L}$ (2) $Zero$ (3) $\frac{8 v}{6 L}$ (4) $\frac{11 v}{3 L}$ 368. $A uniformly thick wheel with moment of inertial I and radius R is free to rotate about its centre of mass (see figure). A massless string is wrapped over its rim and two blocks of masses m_{1} and m_{2} (m_{1} > m_{2}) are attached to the ends of the string. The system is released from rest. The angular speed of the wheel when m_{1} descents by a distance h is :$ (1) $[\frac{m _{1} + m _{2}}{( m _{1} + m _{2} ) R ^{2} + I}]^{1/2} g h$ (2) $[\frac{2 ( m _{1} - m _{2} ) g h}{( m _{1} + m _{2} ) R ^{2} + I}]^{1/2}$ (3) $[\frac{2 ( m _{1} + m _{2} ) g h}{( m _{1} + m _{2} ) R ^{2} + I}]^{1/2}$ (4) $[\frac{( m _{1} + m _{2} )}{( m _{1} + m _{2} ) R ^{2} + I}]^{1/2} g h$ 369. $An object of radius R and mass M is rolling horizontally without slipping with speed v . It then rolls up the hill to a maximum height h = \frac{3 v ^{2}}{4 g} . The moment of inertia of the object is (g = acceleration due to gravity)$ (1) $\frac{2}{5} M R^{2}$ (2) $\frac{1}{2} M R^{2}$ (3) $M R^{2}$ (4) $\frac{3}{2} M R^{2}$ 370. $A spherical shell of 1 kg mass and radius R is rolling with angular speed ω on horizontal plane (as shown in figure). The magnitude of angular momentum of the shell about the origin O is \frac{a}{3} R^{2} ω . The value of a will be :$ (1) $2$ (2) $3$ (3) $5$ (4) $4$ 371. $Consider a uniform rod of mass M = 4 m and length L pivoted about its centre. A mass m moving with velocity v making angle θ = \frac{π}{4} to the rods long axis collides with one end of the rod and sticks to it. The angular speed of the rod mass system just after the collision is:$ (1) $\frac{3 2 V}{7 L}$ (2) $\frac{2 V}{7 L}$ (3) $\frac{2 V}{3 L}$ (4) $\frac{3 V}{2 L}$ 372. $A solid sphere of mass M and radius R is divided into two unequal parts. The first part has a mass of \frac{7 M}{8} and is converted into a uniform disc of radius 2 R . The second part is converted into a uniform solid sphere. Let I_{1} be the moment of inertia of the disc about its axis and I_{2} be the moment of inertia of the new sphere about its axis. The ratio I_{1} / I_{2} is given by :$ (1) $185$ (2) $140$ (3) $285$ (4) $65$ 373. $Three equal masses m are kept at vertices (A, B, C) of an equilateral triangle of side a in free space. At t = 0, they are given an initial velocity V_{A} = V_{0} AC, V_{B} = V_{0} BA and V_{C} = V_{0} CB . Here, AC, CB and BA are unit vectors along the edges of the triangle. If the three masses interact gravitationally, then the magnitude of the net angular momentum of the system at the point of collision is :$ (1) $3 amV_{0}$ (2) $\frac{3}{2} a mV_{0}$ (3) $\frac{3}{2} a mV_{0}$ (4) $\frac{1}{2} a mV_{0}$ 374. $The magnitude of torque on a particle of mass 1 kg is 2.5 Nm about the origin. If the force acting on it is 1 N, and the distance of the particle from the origin is 5 m, the angle between the force and the position vector is (in radians) :$ (1) $\frac{π}{6}$ (2) $\frac{π}{4}$ (3) $\frac{π}{8}$ (4) $\frac{π}{3}$ 375. $The pulley shown in figure is made using a thin rim and two rods of length equal to diameter of the rim. The rim and each rod have a mass of M . Two blocks of mass of M and m are attached to two ends of a light string passing over the pulley, which is hinged to rotate freely in vertical plane about its center. The magnitudes of the acceleration experienced by the blocks is____(assume no slipping of string on pulley).$ (1) $\frac{( M - m ) g}{( ( \frac{13}{6} ) M + m )}$ (2) $\frac{( M - m ) g}{( ( \frac{8}{3} ) M + m )}$ (3) $\frac{( M - m ) g}{2 M + m}$ (4) $\frac{( M - m ) g}{M + m}$ 376. $A mass M hangs on a massless rod of length l which rotates at a constant angular frequency. The mass M moves with steady speed in a circular path of constant radius. Assume that the system is in steady circular motion with constant angular velocity ω . The angular momentum of M about point A is L_{A} which lies in thepositive z direction and the angular momentum of M about B is L_{B} . The correct statement for this system is:$ (1) $L_{A} and L_{B} are both constant in magnitude and direction$ (2) $L_{B} is constant in direction with varying magnitude$ (3) $L_{B} is constant, both in magnitude and direction$ (4) $L_{A} is constant, both in magnitude and direction$ 377. $A solid sphere of mass ' m' and radius ' r' is allowed to roll without slipping from the highest point of an inclined plane of length ' L' and makes an angle 30^{\circ} with the horizontal. The speed of the particle at the bottom of the plane is v_{1} . If the angle of inclination is increased to 45^{\circ} while keeping L constant. Then the new speed of the sphere at the bottom of the plane is v_{2} . The ratio v_{1}^{2} : v_{2}^{2} is$ (1) $1 : 2$ (2) $1 : 3$ (3) $1 : 3$ (4) $1 : 2$ 378. $Match List-I with List-II List - I A Moment of inertia of solid sphere of radius R about any tangent B Moment of inertia of hollow sphere of radius (R) about any tangent C Moment of inertia of circular ring of radius (R) about its diameter. D Moment of inertia of circular disc of radius (R) about any diameter List - II I \frac{5}{3} M R^{2} II \frac{7}{5} M R^{2} III \frac{1}{4} M R^{2} IV \frac{1}{2} M R^{2} Question: Choose the correct answer from the options given below$ (1) $A-II, B-I, C-IV, D-III$ (2) $A-I,B-II,C-IV,D-III$ (3) $A-II, B-I, C-III, D-IV$ (4) $A-I, B-II, C-III, D-IV$ 379. $A ball of mass 0.5 kg is attached to a string of length 50 cm . The ball is rotated on a horizontal circular path about its vertical axis. The maximum tension that the string can bear is 400 N . The maximum possible value of angular velocity of the ball in rad / s is,:$ (1) $1600$ (2) $40$ (3) $1000$ (4) $20$ 380. $A uniform cylinder of radius 1 m, mass 1 kg spins about its axis with an angular velocity 20 rad / s . At certainmoment, the cylinder is placed into a corner as shown in the figure. The coefficient of friction between the horizontal wall and the cylinder is μ, whereas the vertical wall is frictionless. If the number of rounds made by the cylinder is 5 before it stops, then the value of μ is (acceleration due to gravity, g = 10 m / s^{2})$ (1) $\frac{3}{π}$ (2) $\frac{2}{π}$ (3) $\frac{1}{π}$ (4) $\frac{0.4}{π}$ 381. $A thin circular ring of mass M and radius R is rotating with a constant angular velocity 2 r a d s^{- 1} in a horizontal plane about an axis vertical to its plane and passing through the center of the ring. If two objects each of mass m be attached gently to the opposite ends of a diameter of ring, the ring will then rotate with an angular velocity (in r a d s^{- 1}).$ (1) $\frac{M}{( M + m )}$ (2) $\frac{( M + 2 m )}{2 M}$ (3) $\frac{2 M}{( M + 2 m )}$ (4) $\frac{2 _{(} M + 2 m _{)}}{M}$ 382. $As shown in the figure, a metallic rod of linear density 0.45 kg m^{- 1} is lying horizontally on a smooth incline plane which makes an angle of 45° with the horizontal. The minimum current flowing in the rod required to keep it stationary, when 0.15 T magnetic field is acting on it in the vertical upward direction, will be : (Use g=10m/ s^{2})$ (1) $30 A$ (2) $15A$ (3) $10A$ (4) $3A$ 383. $A system consists of two identical spheres each of mass 1.5kg and radius 50cm at the end of light rod. The distance between the centres of the two spheres is 5m. What will be the moment of inertia of the system about an axis perpendicular to the rod passing through its midpoint?$ (1) $18.75 k g m^{2}$ (2) $1.905 \times 1 0^{5} k g m^{2}$ (3) $19.05 k g m^{2}$ (4) $1.875 \times 1 0^{5} k g m^{2}$ 384. $The position vectors of two 1 kg particles, (A) and (B), are given by r_{A} = (α_{1} t^{2} i + α_{2} t j α_{3} t k) m and r_{B} = (β_{1} t i + β_{2} t^{2} j + β_{3} t k) m, respectively; (α_{1} = 1 m / s^{2}, α_{2} = 3 nm / s, α_{3} = 2 m / s, β_{1} = 2 m / s, β_{2} = - 1 m / s^{2}, β_{3} = 4 pm / s), where t is time, n and p are constants. At t = 1 s, (V_{A}) = (V_{B}) and velocities V_{A} and V_{B} of the particles are orthogonal to each other. At t = 1 s, the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is L kgm^{2} s^{- 1} . The value of L is .$ 385. $A thin uniform rod of length 2m. cross sectional area 'A' and density 'd' is rotated about an axis passing through the centre and perpendicular to its length with angular velocity ω . If value of ω in terms of its rotational kinetic energy E is \frac{α E}{A d} then the value of is$ 386. $Moment of Inertia (M.I.) of four bodies having same mass 'M' and radius '2R' are as follows: I_{1} = M.I. of solid sphere about its diameter; I_{2} = M.I. of solid cylinder about its axis; I_{3} = M.I. of solid circular disc about its diameter; I_{4} = M.I. of thin circular ring about its diameter. If 2 (I_{2} + I_{3}) + I_{4} = x \cdot I_{1} then the value of x will be ______$ 387. $A ring and a solid sphere rotating about an axis passing through their centres have same radii of gyration. The axis of rotation is perpendicular to plane of ring. The ratio of radius of ring to that of sphere is \frac{2}{x} . The value of x is______.$ 388. $A disc of mass 1 kg and radius R is free of rotate about a horizontal axis passing through its centre and perpendicular to the plane of disc. A body of same mass as that of disc is fixed at the highest point of the disc. Now the system is released, when the body comes to the lowest position, its angular speed will be 4 \frac{x}{3 R} rad s^{- 1} where x =______ (g = 10 m s^{- 2})$ 389. $Consider a 20 kg uniform circular disk of radius 0.2 m . It is pin supported at its center and is at rest initially.The disk is acted upon by a constant force F = 20 N through a massless string wrapped around its peripheryas shown in the figure. Suppose the disk makes n number of revolutions to attain an angular speed of 50 rads^{- 1} . The value of n, to thenearest integer, is____ [Given: In one complete revolution, the disk rotates by 6.28 rad]$ 390. $A solid cylinder is released from rest from the top of an inclined plane of inclination 30° and length 60 cm. If the cylinder rolls without slipping, its speed upon reaching the bottom of the inclined plane is__________m s^{- 1} . (Given g = 10 m s^{- 2})$