Title 1B. Rotational Dynamics (Rigid body dynamics) ( 25 - 56 ).pdf ✅

NISHITH Multimedia India (Pvt.) Ltd., 2 5 JEE ADVANCED - VOL-III ROTATIONAL DYNAMICS NISHITH Multimedia India (Pvt.) Ltd., LEVEL - VI SINGLE ANSWER QUESTIONS 1. The point P of a string is pulled up with an acceleration g. then the acceleration of the hanging disc over which the string is warpped, is m P (a) 2 3 g  (b) 3 g  (c) 4 3 g  (d) 3 g  2. A sphere of mass m1 is placed on a plank of mass m2 . The coeffcient of friction between the plank and sphere is  . If the inclined plane is smooth, the frictional force between the plank and sphere : m1 m2   =0 (a) depends on m1 (b) depends on m2 (c) 0 (d) 1    m g cos 3. A hanging uniform rod is in equilibrium. The tension in the strings in T1 , say. If the strings is cut, just after cutting, the tension in the other string becomes T2 . Then 2 1 T T: is: m (a) 1: 2 (b) 1:1 (c) 3: 4 (d) 1: 4 4. Four beadseach of mass m are glued at the top, bottom and the ends of the horizontal diameter of a ring of mass m . If the ring rolls without sliding with the velocity v of its , the kinetoc energy of the system (beads +ring) is: m m m m (a) 2 5mv (b) 2 4mv (c) 2 2mv (d) none of these 5. A rolling body is connected with a trolley car by a spring of stiffness k . It does not side and remains in equilibrium relative to the accelerating trolley car. If the trolly car is stopped after a time 0 t t  : k c . m a (a) 0 0 C    for ct t (b) 0 f for t t   0 (c) ma x k  , where x = deformation of the spring (d)   2 2 max 0 1 2 KE ma t  , where  max KE is the maxi- mum KE of the rolling body 6. A linear impulse Fdt  acts at a point C of the smoothe rod AB . The value of x is so that the end A remains stationary just after the impact is : x Fdt c B A (a) 4 l (b) 3 l (c) 6 l (d) 5 l 7. A uniform cube of side ‘b’ and mass M rests on a rough horizontal table. A horizontal force F is applied normal to one of the face at a point at a height 3 / 4 b above the base. What should be the coefficient of friction   between cube and table so that it will tip about an edge before it starts s l i p - ping? (a) 2 3   (b) 1 3   (c) 3 2   (d)  1
Jr Chemistry E/M ROTATIONAL DYNAMICS 2 6 NISHITH Multimedia India (Pvt.) Ltd., JEE ADVANCED - VOL-III NISHITH Multimedia India (Pvt.) Ltd., 8. Two light vertical springs with equal natural lengths and spring constants 1 k and 2 k are sparated by a distance l . Their upper ends are fixed to the ceiling and their lower ends to the ends A and B of a light horizontal rod AB. A vertical downwards force F is applied at point C on the rod. AB will remain horizontal in equilibrium if the distance AC is : 1 k 2 k A C B (a) 2 l (b) 1 1 2 lk k k  (c) 2 1 lk k (d) 2 1 2 lk k k  9. A rod of length / is travelling with velocity cm u and rotating with angular velocity  such that / 2 CM u l  . The distance covered by the point B when the rod completes one full ro- tation is : (a) l (b) 8l (c) 4l (d) 2l 10. A string is wrapped around a cylinder of mass m and radius R. The string is pulled vertically upward to prevent the centre of mass from falling as the cylinder unwinds the string. the length of the string unwound when the cylinder has reached a speed  will be : (a) 4 R g  (b) 2 2 4 R g  (c) 8 R g  (d) 2 2 8 R g  11. 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  with AB. Then the moment of inertia of the plate abotu the axis CD is equal to : (a) I (b) 2 Isin  (c) 2 I cos  (d)   2 I cos / 2  12. Two point masses A of mass M and B of mass 4M are fixed at the ends of a rod of length l and of negligible mass. The rod is set rotation about an axis perpendicular to its length with a uniform anguular speed. The work required for rotating the rod will be minimum when the distance of axis of rotation from the mass A is at (a) 2 5 l (b) 8 5 l (c) 4 5 l (d) 5 l 13. A horizontal uniform beam AB of length 4m and a mass of 20 kg is supported at the end B by means of a string which passes over a fixed, smooth pulley supporting a counterbalancing weight of 8kg on the other side. What force, F, when applied at the point A in a suitable direction, will hold the beam in static equilibrium? 4m A B 30 20kg 0 (a) F = 160 N (b) F = 40 19 N (c) F= 200-80 3 N   (d) no force F as specified can hold the beam in static equilibrium 14. x,y, z are the Cartesian axes of an inertial frame of reference. A particle of mass 1 kg moves with a uniform velocity of 1m/s from A(2, 3m, 0) to B(4m, 0, 0). The motion of the particle is observed from a frame K which rotates with constant angular v3elocity  about the z-axis by an observer O located at (0,0,7m) in xyz stystem. What the magnitude of average pseudo force that observer O should consider as actiong on the particle during its motion from A toB, if  = 3 10  k rad/ s;k being a unit vector in the direction of positive z- axis? (a) Zero (b) 3 10 N  ‘ (c) 3 50 N  (d) 5 2 2 3 10        N
NISHITH Multimedia India (Pvt.) Ltd., 2 7 JEE ADVANCED - VOL-III ROTATIONAL DYNAMICS NISHITH Multimedia India (Pvt.) Ltd., 15. A ring of mass M and radius R lies in x-y plane with its centre at origin as shown. The mass distribution of rings is non-uniform such that at any point P on the ring, the mass per unit length is given by    0 2 cos  ( where 0 is a appositive constant). Then the moment of inertia of the ring about z- axis is M R P x y  (a) 2 MR (b) 1 2 2 MR (c) 2 MR  (d) 5 MR  16. As shown in figure, the hinges A and B hold a uniform 400 n door in place. the upper hinge supports the entire weight of the door. find the resultant force exerted on the door at the hinges . the width of the door is 2 h , where h is the distance between the hinges. A B M 400N y (a) 312 N (b) 280 N (c) 412 N (d) 480 N 17. A light t bar, 10cm on each arm, rests between two vertical walls, as shown in figure, the feft wall is smooth, the coefficients of static friction between the bar and floor, and between the bar and right wall, are 0.32 and 0.50 respectively, the bar is subjected to a vertical load of 1 N, as shown. what is the smallest value of the vertical force F for which the bar will be in static equilibrium in the position shown? 8cm 5cm 5cm 5cm 8cm 5cm 6cm 6cm IN 4cm 5cm 2   0 .3 5  1  0 3   0 .5 0 (a) 1 19 N (b) 2 19 N (c) 1 15 N (d) 20 19 N 18. An automobile of mass m is going around a curve in an arc of a circle of radius R at a speed v. the curve is banked at an angle  to the horizontal and the coefficient of static friction between the tites and the road is s if  is not vbery big, the maximum speed the car can be moving without skidding is (a) sin cos  cos sin s s gR         (b) cos sin  cos sin s s gR         (c) sin cos  sin cos s s gR         (d) cos sin  sin cos s s gR         19. A box of mass 1 kg is mounted with two cylinders each of mass 1kg, moment of inertia0.5kg 2 m and radius 1m as shown in figure, Cylinders are mounted on their control axis of rotation and this system is placed on a rough horizontal surface, the rear cylinder is connected to battery operated motor which provides a torque of 100n-m to this vcylinder via a belt as shown. if sufficient friction is present between cylinder and horizontal surface for pure rolling, find acceleration of the vehicle in 2 m s . ( Neglect mass of motor, belt and other accessories of vehicle). m Electric motor (a) 20 2 m s (b) 10 2 m s (c) 25 2 m s (d) 30 2 m s
Jr Chemistry E/M ROTATIONAL DYNAMICS 2 8 NISHITH Multimedia India (Pvt.) Ltd., JEE ADVANCED - VOL-III NISHITH Multimedia India (Pvt.) Ltd., 20. Two identical rings Aand Bare acted upon by torques TA and TB respectively.A is rotating about an axis passing through the centre of mass and perpendicular to the plane of the ring. B is rotating about a chord at a distance 1 2 times the radius from the centre of the ring. if the angular acceleration of the rings is the same, then (a) A B    (b) A B    (c) A B    (d) Nothing can be said about A  and B  as data are insufficient 21. A uniform plank of weight W and total length 2L is placed as shown in figure with its ends in contact with the inclined planes. the angle.of friction is 0 15 . determine the maximum value of the angle a at which slipping impends. 600 450 W L L  (a) 0 18.1 (b) 0 48.4 (c) 0 36.2 (d) 0 88.8 22. A uniform rod AB of length three times the radius of a hemisphered bowl remains in equilibrium in the bowl as shown. Neglecting friction find the inclination of the rod with the horizontal. r 3r  C B A (a) 1 sin (0.92) (b) 1 cos  (0.92) (c) 1 cos  (0.49) (d) 1 tan (0.92) 23. Aparticle of mass m is released from rest at point A in the figure falling freely under gravity parallel to the vertical Y-axis. the magnitude of angular momentum of particle about point Owhen it reaches B is ( whereOA=b and AB=h) B h A b Y O  (a) mh bg (b) mb gh 2 (c) mb gh 3 (d) 2mb gh 24. Three rings each of mass m and radius r are so placed that they touch each other. The radius of gyration of the system about the axis as shown in the figure is (XX’ is the plane of the rings) X X ' (a) 6 5 r (b) 5 6 r (c) 6 7 r (d) 7 6 r 25. The end B of the rod AB which makes an angle  with the floor is being pulled with a constant velocity V0 as shown in the figure. The length of the rod is l . At the instant when 0  37 A Y O B X l  V0 (a) Velocity of end A is 0 2 3 V downwards (b) angular velocity of rod is 0 5 3 V l (c) angular velocity of rod is constant (d) velocity of end A is constant