& o} 1S) — oO a ise} Y Vv SUOISAHdwSaral Contents 1 Module Description 2. Homework Index 3. Exercise | 4. Exercise LA 5. Exercise 2 6. Exercise 2A 7. Answer Key Note Page - ii Page - ii Page - 1 Page - 6 Page - 8 Page - 14. Page - 23 Detailed solutions are available on the eSaral App. Rotational Motion wSaral @ at aa area SIYS. aral Rotational Motion Module Description sm gts wet Forn stering the concepts only learning is not sufficient. We have to practice and apply those concepts in problem solving. This sheet does just that. It contains a collection of problems segregated in the following exercises to help you master the concepts in a systematic and organized way. “Practice makes a man perfect” 1. Concept builder— 1 & 1A As soon as you have finished learning the concept do the problems from these exercises first. These exercises contains easy level questions to help you build your concepts. 1 —» Contains Single Correct Type questions 1A — Contains pattern based questions incorporating the latest JEE Advanced based patterns like more than one correct, matching lis match the column, ete. 2. Brain Booster — 2 & 2A Now that you have built your concepts it’s time to master them by solving Brain Boosting problems. Don’t hurry through these problems. Take time to solve & lea from them, These exer contains Medium & Tough level problems. Do questions from 2 & 2A after attempting 1 & 1A 2 — Single Correct Type questions, 2A ——® Pattern Based questions. Simulator — JM & JA Contains questions from previous year JEE Mains & JEE Advanced questions in exercise JM & exercise JA respectively. Get the real taste & feel of the type of questions being asked in JEE. It’s a great tool for simulating your mind with JEE problems. These exercis are not included in the module but are provided separately, IM —» JEE Mains previous years topic questions. JA —+ JEE Advance previous years topie wise questions. Get Top Ranks aT SewSaral Home Work Index Problem solving is an integral part of learning. Find questions to solve after each video in the homework Index. Make sure that you attempt all the problems (in Ex 1 to 2A) after learning a topic from the videos. For example if' you have finished topic 8, first attempt all the problems listed in the index corresponding to topic & before proceeding to the video Rotational Motion of topic 9. Sr.No.|Topie Name Ext |Ex1A| Ex2 Ex2A 1 [Introduction 8.9 2 [Kinematics of Pure Rotational Motion] —1,2,3 10,11,12, 3 [Moment of Inertia 4 ic 1 4 [Perpendicular Axis Theorem 56,789] 1 5 {Parallel Axis Theorem a; ee 2 1,2,3,24-29,37 6 [Torque 15 7 [Equilibrium 34,5 4 8 — Torque about an Axis 18 6 5 9 |Inertial Pulley 7 10 [Work Energy Theorem 15,16 ll |Angular Momentum 19,20,21 2,17 8,9,10,11 22,23,24, 12 |Angular Momemtum Conservation 25,2627 18,19 12,13,14 67,8 13 |Collsion 28 15,16 38 14 |Toppling 17,18 9 15 |CRTM 10 i 19.20.21, 16 [Rolling 5 22,23 11,39 - 24,25,26,.27 17 |Dynamics of CRTM (28,29,30 12,13,14,15 18 |Kinetic Energy in CRTM_ 29 6,7 31,32,33,34] 16,30,31,36 17,18,19, 19 |Angular Momentum in CRTM 30 34 35 20,21,40 20 |ICORIAOR 31 41 21 |Cross Product 32,33 22 | Afier Full Chapter 36,37 | 22,23,32-35 wSaral 2 at aa ae SIwSaral Exercise - 1 Rotational Motion Concept Builder SINGLE CORRECT TYPE Each question has FOUR options for correct | QA answer. ONLY ONE of these four options is correct option. For each question, choose the correct option 10 answer the question. Q2 Qs Two gear wheels which are meshed together have radii of 0.50 em and 0.15 em. The number of revolutions does the smaller turns when the larger turns through 3 revolution is (A) 5 revolution (B) 20 revolution (©) 1 revolution (D) 10 revolution Qs The rotating rod starts from rest and acquires a rotational speed n= 600 revolution/minute in2 seconds with constant angular accelera- tion. The angular acceleration of the rod is (A) 10 x rad/s? (B) 5 x rads (©) 15 x radis* (D) None of these A particle starts from the point (0m, 8m) and moves with uniform velocity of 37 m/s. | Q.6 After 5 seconds, the angular velocity of the particle about the origin will be : 3 m/s 8n Q7 a x 8g (A) Fg5 tadis 3 289 (B) g rad/s 24 (©) sp rads 8 389 OF rad/s Moment of Inertia Four similar point masses (cach of mass m) are placed on the circumference of a disc of mass M and radius R. The M.1. of the system about the normal axis through the centre O will be:- (A) MR? + 4mR? 1 (B) 5 MR? + 4mr* 8 (©) MR? + = mR? (D) none of these Perpendicular Axis Theorem Three thin uniform rods each of mass M and length L are placed along the three axis of a cartesian coordinate system with one end of each rod at the origin, The M. I. of the system about z-axis ML 2ML? A> Bz ML? ; oO (D) ML? By the theorem of perpendicular axes, if a body be in X-Z-plane then :~ (AL -L= 1 (B) 1 +1 =1, CL+h=1 (Dy, +1, The axis X and Z in the plane of a dise are mutually perpendicular and Y-axis is perpendicular to the plane of the disc. Ifthe moment of inertia of the body about X and Y axes is respectively 30 kg m? and 40 kg, n? then M.I. about Z-axis in kg nm will be:~ (A) 70 (B) 50 (C) 10 (D) Zero wSaral 2 at aa aea SIwSaral Rotational Motion Q.8 Let I be the moment of inertia of'a uniform | Q.11 Three solid spheres of mass M and radius square plate about an axis AB that passes R are shown in the figure. The moment of through its centre and is parallel to two of inertia of the system about XX’ axis will be:~ its sides. CD is a line in the plane of the plate that passes through the centre of the plate and makes an angle 0 with AB. The moment of inertia ofthe plate about the axis CD is then equal to:- * Ze 4 * (a) EM? ® Sue ' 2 AH +e 16 a (Cc) “5 MR’ (D) “5 MR’ Cc B D (ar (By! Q.12 The moment of inertia of a ring of mass M and radius R about PQ axis will be :- (©)1eos' (vy teos{ 3} Q.9 The moment of inertia ofa thin square plate ABCD of uniform thickness about an axis passing through its centre and perpendicular to its plane will be :- MR? (A) MR? B) 2 3 (© MR? (D) 2MR (Ay +1 (B)1,-1 Q.13. The moment of inertia of a thin uniform rod ine poe of mass M and length ¢ about an axis per- OL+h+h (D)T, +1, +1, pendicular to the rod through its centre is I. ‘The moment of inertia of the rod about an axis perpendicular to the rod through its end Parallel Axis Theorem Q.10 The moment of inertia of a uniform circular “ .¢ of radius 'R’ and mass 'M' about an axis, (21 (Al passing from the edge of the disc and normal to the disc is- Q14 One quarter section is cut from a uniform circular disc of radius R. This section has a (a) dvr? (B) Jur? mass M. Jt is made to ceauriie line 2 2 perpendicular to its plane and passing through the centre of the original disc. Its (© SMR? (D) MR? moment of inertia about the axis of rotation 2 is Get Top Ranks in IIT-JEE/NEET/10" * with eSaral APP pp>Getiton,wSaral a ay LMR? ) LMR? > ® 5 © gM? (D) V2MR* Torque Q.15 A wheel having moment of inertia 2 kg-m? about its vertical axis, rotates at the rate of 60 rpm about the axis. The torque which can stop the wheel's rotation in one minute would be :- A) EN B) =N. Nm ®) {3m ©) =n b) 22N (© jgh™ () [3 ™ Q6 A door 1.6 m wide req to be applied at the free end to open or close it, The force that is required at a point 0.4 m distant from the hinges for opening or closing the door is (A) 12N (C()24N (B) 3.6N (D)4N Q.17 Four equal and parallel force are acting on a rod (as shown in figure at distances of 20 cm, 40 cm, 60 cm and 80 cm respectively from one end of the rod. Under the influence of these forces the rod F E lao. 60, Rotational Motion (A) is at rest (B) experiences a torque (C) experiences a linear motion (D) Experiences a torque and also a linear motion RTIAL PULLEY Qu18 A constant torque of 1000 N-mtumsa wheel of moment of inertia 200 kg-m? about an axis thorugh its centre. Its angular velocity afler 3 seconds is- (in rad/sec.) (AT (B)5 (15 (D) 10 ANGULAR MOMENTUM Q.19 Find the angular momentum of a disc about the z-axis. Q.20 A particle of mass m is rotating in a plane in a circular path of radius r, Its angular momentum is L. The centripetal force acting on the particle is ‘m, (B) - oe mr mr wSaral 2 at aa aea SIwSaral Q.21 When a mass is rotating in a plane about a fixed point, its angular momentum is directed along (A) the radius (B) the tangent to orbit (©) line at an angle of 45° to the plane of rotation (D) the axis of rotation Angular Momemtum Conservation Q.22. Adancer on ice spins faster, when she folds her arms. This is due to (A) increase in energy and increase in angular momentum (B) decrease in friction at the skates (©) constant angular momentum and incre- ase in kinetic energy (D) increase in kinetic energy and decrease angular momentum. Q23 If the earth were to suddenly contract to Lhor p jth ofits present radius without any change in its mass then the duration of the new day will be nearly- 24 (A) =) hour (B) 24n hour 24 (©) & hour (D) 24n? hour Q.24 A particle undergoes uniform circular motion. About which point on the plane of the circle, will the angular momentum of the particle remain conserved ? (A) Centre of circle (B) On the circumference of the circle (C) Inside the circle (D) Outside the circle Get Top Ranks in IIT-JEE/NEET/10" Rotational Motion Q.25 A child is standing with folded hands at the centre ofa platform rotating about its ¢: axis. The kinetic energy of the sys and moment of inertia is I. The child now stretches his arms so that the moment of inertia of the system doubles. The kinetic energy of the system now is ~ K B> (A) 2K > K OZ OAK Q.26 A circular turn table has a block of ice placed at its centre, The system rotates with an angular speed @ about an axis passing through the centre of the table. If the ice melts on its own without any evaporation, the speed of rotation of the system- (A) becomes zero (B) remains constant at the same value « (C) increases to value greater than © (D) decreases to a value less than © Q27 A thin circular ring of mass M and radius ‘r’ is rotating about its axis with a constant angular velocity «, Four objects each of mass m, are kept gently to the opposite ends of two perpendicular diameters of the ring The angular velocity of the ring will be- Mo Mo am ®) Mam o Mame (M+ 4m)o © M M+ 4m Q.28 A mass m is moving at speed v perpendicular to arod of length d and mass M = 6m which pivots around a ffictionless axle running through its centre. It strikes and sticks to the end of the rod. The moment of inertia of the rod about its centre is Mc?/12. Then the angular speed of the system right after the collision is (A) 2vid (B) 2v/3d (wd (D) 3v/2d rT PTUs aad ied LoreewSaral etic Energy in CRTM Q.29 A disc of mass M and radius R rolls on a horizontal surface and then rolls up an inclined plane as shown in the figure. If the velocity of the disc is v, the height to which the dise will rise will be h yo i O56 ® a yX ye © 4 Ox Angular Momentum in CRTM Q30 A disc of mass M and radius R is rolling with angular speed @ on a horizontal plane as shown. The magnitude of angular momentum of the disc about the origin 0 (B) MR (D) 2MR%o Rotational Motion ICOR, IAOR A dise of radius R = 2m moves as shown in the figure, with a velocity of translation of 6v, of its centre of mass and an angular velocity Q31 2 of $3. The distance (in m) of instantaneous axis of rotation from its centre of mass is - ae ous (a3 (B)4 os (D6 Cross Product Q32 A =301,B = 40k then A x B=? (A) 12001 (B) 1200} (©) -12005 (D) 1200k Q33 A= 21+ 4j-k, B= 4j+3k then Ax B (A) 161 - 6j- 8k (B) 161 + 6}-8k (©) 161-6] +8k (D) None of These wSaral 2 at aa aea SIwSaral Exercise - 1A Rotational Motion Concept Builder ONE OR MORE TH. ONE CORRECT TY Each question has FOUR options for correct answer(s). ONE OR MORE THAN ONE of these four option(s) is (are) correct option(s). For each question, choose the correct option(s) to answer the question. Q.1 The moment of inertia of a thin square plate ABCD, of uniform thickness about an axis passing through the centre O and perpendi- cular to the plane of the plate is (where I,, 1, I, and I, are respectively moments of inertia about axis 1, 2, 3 and 4 which are in the plane of the plate) (A141, ©lL+h (B)1,+1, (D)L, +4414], Q2 A particle falls freely near the surface of the earth. Consider a fixed point O (not vertically below the particle) on the ground (A) Angular momentum of the particle about O is increasing. (B) Torque of the gravitational force on the particle about O is decreasing. (©) The moment of inertia of the part about O is decreasing (D) The angular velocity of the particle about O is increasing. A smooth tube of certain mass is rotated in gravity free space and released. The two balls shown in the figure move towards ends of the tube. For the whole system which of the following quantity is not conserved :- Get Top Ranks in IIT-JEE/NEET/10" (A) Angular momentum (B) Linear momentum (©) Kinetic energy (D) Angular speed Q4 A uniform disc is rolling on a horizontal surface. At a certain instant B is the point of contact and A is at height 2R from ground, where R is radius of disc A (A) The magnitude of the angular momen- tum of the disc about B is thrice that about A. (B) The angular momentum of the dise about A is anticlockwis (C) The angular momentum of the dise about B is clockwise (D) The angular momentum of the disc about A is equal to that about B. MATCH THE COLUMN TYPE Following questions contain statements given in two columns, which have to be matched. The statements in Column-I are labelled as A, B, C and D while the statements in Column-IT are labelled as (P), (Q), (R) and (S). Any given statement in Column-T can have correct matching with ONE OR MORE statement(s) in Column-IT, rT PTUs aad ied LtwSaral Rotational Motion QS A rigid cylinder is kept on a smooth horizontal surface as shown, If Column-I indicates velocities of various points (3-centre of cylinder, 2- top point, 4-bottom point, 1- on the level of 3 at the rim) onit shown, choose correct state of motion from Column-II (R) Rolling without slippi Column-I (P) Pure rotation about centre (Q) Rolling without slipping to left to right (S) Not possible NUMERICAL TYPE SUBJECTIVE TYPE The answer to each question is a NUMERICAL | QS VALUE. For each question, find the correct numerical value (in decimal notation, truncated/ rounded-off to the second decimal place; e.g. 6.25, 7.00, -0.33, -.30, 30.27, -127.30) Q.6 A hollow cylinder is rolling down on an inclined plane which is inelined at an angle misof 30° to the horizontal. Find the speed after travelling distance of 10 metre? | gg (take g = 9.8 mvs) Q.7 Two uniform similar discs roll down on two inclined planes of length $ and 2S respectively as shown in the figure. Find the | ¢ 19 ratio of velocities of two discs at the point A and B of the inclined planes ? A disks rotates about a fixed axis. Its angular velocity « varies with time according to equation @=at +b. At the instant ¢ = 0 its angular velocity is 1.0 rad/s and angular position is 2 rad and at the instant ¢ = 2 s, angular velocity is 5.0 rad/s. Determine angular position 0 and angular acceleration aatr=4s. A rigid body rotate about a fixed axis with variable angular speed = A-Bt where A and B are constant, Find the angle through which it rotate before it comes to rest. The moment of inertia of a ring and a dise ofsame mass about their diameters are same. What will be the ratio of their radii ? Two bodies of mass iKg and 2Kg are attached to the ends ofa 2 metre long weight less rod. This system is rotating about an axis, passing through middle point of rod. Calculate M.L. of system. wSaral 2 at aa aea SIwSaral Qa2z Q13 Qu4 Q.16 About which axis would a uniform cube have its minimum rotational inertia Three mass m,, m, and m, are located at the vertices of an equilateral triangle of length a. What is moment of inertia of the system about an axis along the altitude of the triangle passing through m, ? The density ofrod AB continiously inereases from A to B. Is it easier to set it in rotation by clamping it at A and applying a perpendicular force at B or by clamping it at B and applying the force at A. Explain your answer. ‘A wheel of moment of inertia 10 kg-m? is rotating at 10 rotations per minute. Find the work done in increasing its angular speed to 5 times of its initial value. (take x? = 10) A wheel is rotating about a fixed axis. Find the moment of inertia of the wheel about the axis of rotation, when its angular speed is 30 radian/sec and its kinetic energy is 360 joule ? Qi7 Quis Quis Rotational Motion A uniform circular dise of mass 200g and radius 4-Ocm is rotated about one of its diameter at an angular speed of 10 rad/sec. Find the kinetic energy of the dise & its angular momentum about the axis of rotation ? A rotating platform of moment of inertia 100 kg-m? completes one rotations in 10 seconds. A person of mass 50 kg. is standing at the centre of platform. Ifthe person moves at a distance of 2m, from the centre along the radius then find the angular velocity of the platform. Two disc have moments of inertia I, and 1, about their respective axis (normal to the dise and passing the centre) and rotating with angular speeds ©, and @, are brought into contact face to face with their axis of rotation coincident and directon of rotation is same then find the angular speed of the two dise system, Exercise - 2 Brain Booster SINGLE CORRECT TYPE Each question has FOUR options for correct answer ONLY ONE of these four options is correct option. For each question, choose the correct option to answer the question. Qu Two loops P and Q are made from a uniform wire, The radii of P and Q are r, and r, respectively, and their moments of inertia L are 1, and I, respectively. If +*=4 then a equals- Ae Bs! Ose (Ee Get Top Ranks in IIT-JEE/NEET/10" Q2 SU Cas The ratio of the radii of gyration ofa circular dise about a tangential axis in the plane of the disc and of a circular ring of the same radius about a tangential axis in the plane of the ring is =~ (A2:1 (B) V5: V6 (py: V2 (C233 A right triangular plate ABC of mass m is free to rotate in the vertical plane about a fixed horizontal axis through A. It is supported by a string such that the side AB is horizontal. The reaction at the support A. in equilibrium is Get iton z Cee teewSaral Q4 Qs Q6 es apt Ja ' t mg 2mg ws oF oy (0) mg Q7 A uniform rod of length / is placed symmetrically on two walls as shown in figure. The rod is in equilibrium. If N, and N, are the normal forces exerted by the walls on the rod then = yw 1 (NPN, (B) N, Vp A uniform circular disc placed on a rough horizontal surface has initially a velocity v, and an angular velocity «, as shown in the figure. The dise comes to rest after moving some distance in the direction of motion. wSaral 2 at aa aea SIwSaral Rotational Motion Q.27 Calculate the ratio of the times taken by a Then jg - uniform solid sphere and a disc of the same Fy mass and the same diameter to roll down through the same distance from rest on a oF smooth inclined plane. a (A) 15: 14 (B) Vis: Vi4 (©) 18? 2 18 (@) 1:1 1 “Ws (B)1 Q.28 A body of mass m slides down an incline and reaches the bottom with a velocity v. 3 Ifthe same mass were in the form of a ring Os, (D)2 which rolls down this incline, the velocity of the ring at the bottom would have been Q.24 A uniform thin stick of length ¢ and mass Ay (B) v2 mis held horizontally with its end B hinged at a point B on the edge of a table. Point A © © pew is suddenly released. The acceleration of the 5 centre of mass of the stick at the time of release, is :- Q.29 A solid cylinder of mass M and radius R rolls without slipping down an inclined pha plane of length L and height h. What is the speed of its centre of mass when the cylinder reaches its bottom- (A) J2ah (B) ) Ze a ‘ © Jaen D) J4gh 5 © 458 (D) 4g Cy Sr Ops Q.30 Inthe following figure, a sphere of radius 3 mrolls on a plank. The accelerations of the sphere and the plank are indicated. The value of ais - Q.25 Aspherical shell and a solid cylinder ofsame radius rolls down an inclined plane. The ratio of their accelerations will be:- (A) 15:14 (B) 9:10 (23 (D) 3:5 Q.26 Arring takes time t, and t, for sliding down ay 3 ms? and rolling down an inclined plane of length 3 rad/s? L respectively for reaching the bottom. The (A) 3 rads? ratio of t, and t, is : (B) 6 radis? : (C) 3 rad/s* (opposite to the direction shown (Ayy2 1 (By 1: V2 in figure) r:2 (2:1 (D) I rads? eta CULL | Pca scsttey SewSaral Q.31 The moment of inertia of a solid cylinder | Q.36 about its axis is given by (1/2)MR?. If this cylinder rolls without slipping, the ratio of its rotational kinetic energy to its transla~ tional kinetic energy is - (A)L:1 (B)2:2 (1:2 D)1:3 Q.32__A dise rolls down a plane of length L and inclined at angle 0, without slipping. Its velocity on reaching the bottom will be w fetsind ® petsind © ostsin’ (D) /4eLsind Q.33 If rotational kinetic energy is 50% of total kinetic energy then the body will be ~ (A) ring (B) cylinder Q37 (C) hollow sphere (D) solid sphere Q.34 Ahollow smooth uniform sphere A of mass ‘m’ rolls without sliding on a smooth horizontal surface. It collides head on elastically with another stationary smooth solid sphere B of the same mass m and same radius. The ratio of kinetic energy of 'B’ to that of ‘A’ just after the collision is: (i A (B)2:3 (D) None Q35_ A solid sphere is placed on a horizontal surface, A horizontal impulse I is applied at a distance h above the central line as shown in the figure. Soon after giving the impulse the sphere starts rolling. The ratio h/R would be- 1 »t @o2 ot ot M7 OF OF MO; Rotational Motion A solid sphere of radius R is placed on smooth horizontal surface. A horizontal force ‘F” is applied at height *h’ from the lowest point. For the maximum acceleration of centre of mass, which is correct Lowest Point (Ah=R (B)h=2R (Qh=0 (D) No relation between h and R An equilateral triangle ABC formed from a uniform wire has two small identical beads initially located at A. The triangle is set rotating about the vertical axis AO. Then the beads are released from rest simultancously and allowed to slide down, one along AB and other along AC as shown, Neglecting frictional effects, the quantities that are conserved as the beads slide down are : (A) angular velocity and total energy (kinetic and potential) (B) total angular momentum and total energy (C) angular velocity and moment of inertia about the axis of rotation (D) total angular momentum and moment of inertia about the axis of rotation wSaral 2 at aa ae SIwSaral Exercise - 2A Rotational Motion Brain Booster ONE OR MORE THAN ONE CORRECT TYPE Each question has FOUR options for correct answer(s). ONE OR MORE THAN ONE of these four option(s) is (are) correct option(s). For each question, choose the correct option(s) 10 answer the question. Qu Q2 From a circular disc of radius R and mass 9M, a small dise of radius = is removed from the disc. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the dise and passing through Ois:- R 2 z tp a 40 (A) 4MR° (8) MR* (©) 1OMR? A solid sphere of radius R has moment of inertia I about its geometrical axis. If it is melted into a dise of radius r and thickness t. If it’s moment of inertia about the tangential axis (which is perpendicular to plane of the disc) is also equal to I, then the value of r is equal to := 2 2 (A) sk RR 3 v3 (© 75k ) FSR Qs Q4 Be ai The figure shows a uniform rod lying along the x-axis. The locus of all the points lying on the xy-plane, about which the moment of inertia of the rod is same as that about O is y. (A) an ellipse (B) acircle (©) a parabola (D) a straight line A heavy seesaw (ie., not massless) is out of balance. A light git! sits on the end that is tilted downward, and a heavy body sits on the other end so that the seesaw now balances. If they both move forward so that they are one-half their original distance from the pivot point (the fulcrum) what will happen to the seesaw ? (A) The side the body is sitting on will tit downward (B) The side the girl is sitting on will once again tilt downward (C) Nothing ; the seesaw will still be balanced (D) It is impossible to say without knowing the masses and the distances A sphere is placed rotating with its centre initially at rest in a comer (a (B) as shown in figure (A) & (B). Coefficient of friction between all surfaces and the 1 sphere is =. Find the ratio of the frictional force $f by ground in situations (A) & (B) “al (C) 10/9 (B) 9/10 (D) None TT SewSaral Q6 Qs A smooth uniform rod of length L and mass M has two identical beads (1 and 2) of negligible size, each of mass m, which can slide freely along the rod. Initially the two beads are at the centre of the rod and the system is rotating with angular velocity «, about an axis perpendicular to the rod and is passing through its mid-point. There are no external forces when the beads reach the ends of the rod, the angular velocity of the | 29 system is eds ay Mos 5) Mee ( ) Mero (B) m Mo, © vem (D) 0, The torque ona body about a given point is found to be equal to AxL where A is a constant vector and {, is the angular momentum of the body about that point. From this it follows that : (A) dL/dt is perpendicular to L at all instants of time (B) the components of L in the direction of A does not change with time (C) the magnitude of L does not change with time (D) L does not change with time A block of mass m moves on a horizontal rough surface with initial velocity v, The height of the centre of mass of the block is h from the surface. Consider a point A on the surface. Q.10 Rotational Motion (A) angular momentum about A is mvh initially (B) the velocity of the block decreases at time passes. (C) torque of the forces acting on block is zero about A (D) angular mometum is not conserved about A. A uniform? kg cylinder rests on a laboratory cart as shown. The coefficient of static friction between the cylinder and the cart is 0.5. If the cylinder is 4 cm in diameter and 10 cm in height, which of the following is closest to the maximum acceleration of the cart such that cylinder neither slips nor tips over? (2 ms (Sms (B)4 (D) 6 mis” A small sphere A of mass m and radius rolls without slipping inside a large fixed hemispherical bow! of radius R (>> r) as shown in figure. If the sphere starts from rest at the top point of the hemisphere find the normal force exerted by the small sphere ‘on the hemisphere when it is at the bottom B of the hemisphere. 10 () > me 7 (D) me 5 (© 5 mg wSaral 2 at aa aea SIwSaral Q.11 Ifa cylinder is rolling down the long incline with sliding (A) after some time it may start pure rolling (B) after some time it will start pure rolling (©) it may be possible that it will never start pure rolling (D) None of these Q.12 A hollow sphere of radius R and mass m is fully filled with non viscous liquid of mass m, It is rolled down a horizontal plane such that its centre of mass moves with a veloc- ity v. Ifit purely rolls. 5 (A) Kinetic energy of the sphere is =m? 4 (B) Kinetic energy of the sphere is = mv* (C) Angular momentum of the sphere about . _ 8 a fixed point on ground is mvR (D) Angular momentum ofthe sphere about 14 a fixed point on ground is “> mvR Qu3 A small object of uniform density rolls up a curved surface with an initial velocity v. av It reaches up to a maximam height of “g- with respect to the initial position. The object is = (A) ring (©) hollow sphere (B) solid sphere (D) dise Q.14 A disc of circumference s is at rest at a point Aon a horizontal surface when a constant horizontal force begins to act on its centre. Between A and B there is sufficient friction Get Top Ranks in IIT-JEE/NEET/10" Rotational Motion to prevent slipping and the surface is smooth to the right of B. The disc moves from A to B in time T. To the right of B, x SB (A) the angular acceleration of the dise will disappear, linear acceleration will remain unchanged (B) linear acceleration of the dise will increase (C) the dise will make one rotation in time 12 (D) the disc will cover a distance greater than s in further time T. Q.15 A plank with a uniform sphere placed on it, rests on a smooth horizontal plane. Plank is pulled to right by a constant force F. If the sphere does not slip over the plank. 0 (A) Acceleration of centre of sphere is less than that of the plank. (B) Acceleration of centre of sphere is greater than the plank because friction acts rightward on the sphere. (C) Acceleration of the centre of sphere may be towards left. (D) Acceleration of the centre of sphere relative to plank may be greater than that of the plank relative to floor br Q.16 A ring rolls without slipping on the ground. Its centre C moves with a constant speed P is any point on the ring. The speed of P with respect to the ground is v. (Ayosve2u (B) v=u, if CP is horizontal. (C) v=u, if CP makes an angle of 30° with the horizontal and P is below the horizontal level of C. (D) v= ¥2u, if CP is horizontal. rT PTUs aad iedwSaral Q.17 A sphere of mass M and radius R is attached Qus by a light rod of length / to a point P. The sphere rolls without slipping on a circular track as shown, It is released from the hori- zontal position, The angular momentum of the system about P when the rod becomes vertical is : (Ay M a U+R] 10 2 B) M,|—g/ | /+=R “) ver [3R] 10 7 ©) Mf 2g | 142 (© M, ult zr] 10 2 O M/Fe! [:-28| A thin table cloth covers a horizontal table and a uniform body of round shape lies on top oft. The table cloth is pulled from under the body, and friction causes the body to slide and rotate. What is the body’s final motion on the table? (Assume that the table is so large that the body does not fall off it.) -—_ © —_O (A) Body will finally roll towards left (B) Body will finally roll towards right (© Body will finally come to rest (D) Any of the above is possible depending on shape of body Q.20 Rotational Motion Q.19 A circular platform is free to rotate in a horizontal plane about a verti. through its centre. A tortoise is s edge of the platform. Now the platform is given an angular velocity «,. When the tortoise move along a chord of the platform with a constant velocity (with respect to the platform). The angular velocity of the platform oo(t) will vary with time t as :- oftif (a) A uniform rod AB of mass m and length /is at rest on a smooth horizontal surface. An impulse J is applied to the end B, perpendicular to the rod in the horizontal direction. Speed of particle P at a distance 1 & Homthe centre towards ofthe rod after time t= 2 ime t= Ty (A) ot B On ® Dm J J a ad Ox ) 2- wSaral 2 at aa ae SIwSaral Q.21 A uniform rod of length J and mass M rotating about a fixed vertical axis on a smooth horizontal table. It elastically strikes a particle placed at a distance 1/3 fiom its axis and stops. Mass of the particle is - wt 3M oOo; © 4M 3 3M (A)3M_(B) Q.22__ Inthe given figure a uniform wheel ofradius 30cm rests against a rigid rectangular block 15cm high. The wheel weighs 1000 N. The minimum pull P through the center which will tum the wheel over the block is (A) 5003 N (C) 1000 N (B) 1000V3 N (D) 4003 N Inthe figure shown, the plank is being pulled to the right with a constant speed v. If the cylinder does not slip then - (A) the speed of the centre of mass of the cylinder is 2v (B) the speed of the centre of mas cylinder is zero of the (© the angular velocity of the cylinder is wR (D) the angular velocity of the eylinder is zero Get Top Ranks in IIT-JEE/NEET/10" Rotational Motion Paragraph Type This section contains PARAGRAPHS. Based on each paragraph, there are questions. Each question has FOUR options. ONLY ONE of these four options corresponds to the correct answer. For each question, choose the option corresponding 10 the correct answer Paragraph for Question No, 24 to 37 The figure shows an isosceles triangular plate of mass M and base L. The angle at the apex is 90°, ‘The apex lies at the origin and the base is parallel to X-axis Q.24 The moment of inertia of the plate about the zeaxis is :- MI ML (A) Te (B) 24 2 © “ (D) none of these Q2s ‘The moment of inertia of the plate about the ML ore (D) none of these Q.26 The moment of inertia of the plate about the x-axis is MU? ML? A B A (B) 32 ML* ML* OW Ore Pca scsttey SewSaral Q.27 The moment of inertia of the plate about its base parallel to the x-axis is =~ ML? p) ME 8) 3 (D) none of these Paragraph for Question on 28 and 29 In the treatment of moments of inertia, introductory textbooks often present two theorems, generally called the parallel axis theorem and the perpendicular axis theorem. There is another theorem of this same genre, which is not usually included, but which is interesting and useful. It is 141,41, =2)my Here, 1,1, and I, are the moments of inertia about three mutually perpendicular intersecting axes, m, is the mass of the i particle and r, is the distance from the intersection. The proof is simple: Taking. the three axes as coordinate axes, we have A +h+1= 2m (vi+zi)+Lim(zt +x}) +L (sity) = 2D m (xi yi +23) =2Eme One important application is the calculation of the moment of inertia I, of a uniform thin-walled spherical shell, of mass M and radius R, about a diameter. Taking the centre as the origin of coordinates, we have I, =I, = I, = Iy andr, = R. ‘The theorem gives 31, = 2(2,m)R? = 2MR3,where 1, = 2MR° Q.28 Consider a solid cube of mass M and side L. What will be the value of Em, for this body when the point of intersection of axes is the centre of the cube Mi ML? B “ 2 p) ME O's Rotational Motion Q.29 Find the moment of inertia of ring of mass m and radius R about an axis passing through its centre and making an angle 45° with its plane : ay ae pm AZ (B) ~~ 3 aR? a cc) pmR (D) mR? Paragraph for Question No.30 and 31 A small sphere of mass 1 kg is rolling without slipping ona rough stationary base with linear speed 200 |= mis It eaves the inclined plane at point C. c ia A B Q30 Find its linear speed at point C =~ (A) peuae (A) ms ., [i00 2 © yas ™s () 35S Q31 Find ratio of rotational and translational kinetic energy of the sphere whe the ground after leaving from point C 2 2 Ns B)3 1 1 OG OZ wSaral 2 at aa aea SIwSaral Paragraph for Question on 32 to 35 A spring having initial unstretched length /, is lying on a smooth table. I's one end is fixed and the other one is fastened to a small particle of mass m. The particle is imparted an initial speed v, horizontally in a direction perpendicular to the spring. In the course of the motion in horizontal plane, the maximum elongation of the spring is A= 6/10. (Given = 0.1 kg, (,= 1m, v, = 11 mis). Q32 In the course of motion, which of the quantities relating to spring block system are conserved ? (A) kinetic energy (B) momentum (C) angular momentum (D) potential energy Q.33 Which of the following is correct about ini- tial situation and situation at maximum elogation ? Q3. 4 Rotational Motion (A) The orientiation of spring in both positions should be perpendicular to each other. (B) The velocity at maximum extension should be zero (C) The velocity at maximum extension as well as at initial position should be perpendicular to spring. (D) Acceeration should be zero at the ‘maximum extension as well as at initial position. Student-A: at maximum extension mv (,* AO) = mv, Student-B : at maximum extension where v, is velocity at instant of maximum extension (A) Only Student-A and B are correct (B) Only Student-A and C are correct (©) Only Student-B and C are correct (D) Allare correct The value of spring constant (in N/m) is : (A) 100 (B)210- (C420. (D) 105 MATCH THE COLUMN TYPE Following questions contain statements given in two columns, which have to be matched. The statements in Column-1 are labelled as A, B, C and D while the statements in Column-Il are labelled as (P), (Q). (R) and (S). Any given statement in Column-I can have correct matching with ONE OR MORE statement(s) in Column-IL Q.36 Column-I depicts various situatio: where some sudden events are taking place. Column-II describes changes in various parameters of systems immediately after the events taking place in column-I. (Platform is hinged at A & free to rotate) Get Top Ranks in IIT-JEE/NEET/10" SU Cas TT SewSaral Column-t (A) Joker is standing on revolving platform and batman throws the ball and joker catches the ball while it was moving horizontally, Se Ae ° Joker, ball and platform is system. (B) Joker throws the ball horizontally and perpendicular to his motion while standing on the revolving platform, S Joker, ball and platform is system, (© Joker jumps horizontally towards right from the cart which is moving at speed v on smooth horizontal floor. v th gmooth Joker and cart is the system (D) Joker drops himself vertically from the moving cart with no horizontal velocity relative to cart. v tr smooth Joker and cart is the system wSaral 2 at aa ae SI Rotational Motion Column- (P) Linear momentum remains conserved. (Q) Mechanical energy is conserved (R) Mechanical energy increases. (S) Mechanical energy decreases. (1) vor @ changeswSaral SUBJECTIVE TYPE, Q37 If the moment of inertia of a dise about an axis tangentially and parallel to its surface be I, what will be the moment of inertia about the axis tangential but perpendicular to the surface. A uniform rod of mass 8m and length 6a is lying ona horizontal table. Two point masses m and 2m moving with speed 2v and v respectively strike the rod and stick to it as shown in figure then- ra =. (a) Calculate the speed of centre of mass of rod after the collision. (b) Calculate angular velocity of the rod about an axis passing through its centre of mass. (c) Kinetic energy of system after collision. A tangential force F acts at the top ofa thin spherical shell of mass m and radius R. Find the acceleration of the shell if it rolls without slipping. Get Top Ranks in IIT-JEE/NEET/10" Rotational Motion Q.40 A cylinder of radius R and weight IV is to Qa Be ai be raised against a step of height / by applying a horizontal force at its center as shown in the figure. Find the required minimum magnitude of this foree. Assume sufficient friction between the cylinder and the corner of the step to prevent slipping. (given that R = 2h) A 100 em rod is moving on a horizontal surface. At an instant, when it is parallel to the x-axis its ends A and B have velocities 30 em’s and 20 ems as shown in the figure 20 cm/s (a) Find its angular velocity and velocity of its center. (b) Locate its instantaneous axis of rotation, TT SewSaral Rotational Motion Answer Key Ex-1A LABOQ ACD) 3.0) 4. (ABC) “oy 5. (A)-R, (B)-S, (C)-P, (D)-Q. 6. Tm/s 7 wy : Ro 8. a =2 rad/s*, 0 = 22 rad. 2B 10. R, V2 11. 3 kg-n? 12. Body diagonal 13, om, +m) 14, See Explanation 15, 133.3) 16.1= 0.8 ke-m? 17.4 10°F, 8% 104Es 18, Zerad see. 19, @=Netho, Lek, wSaral 2 at aa ae SIwSa ral Rotational Motion Ex-2A LA 2A 3.B 4.B 5B 6A 7. ABC 8. A,B,D 9B 10. B 1. AC 12.C 13. D 14, B,C.D 15.A 16. A,C,D 17.D 18.C 19. 20.D 21.B 2A 23. B,C 25.C 26.4 21.€ 28. B 29.€ 30.A 31.C 32.C 33.¢ 34.B 35.B 36, (A) ST (B) R (©) PRT (D) PQ 16 y § 37, '=21 38. (a) v= 0, (b) @=— anticlockwise. (c) KI 5 Sa 6F 9. 40.F= 3W 41. (a) @ = 0.5 rad/s, V.=-20j+ 0.5k x 50i = 5.0jem/s (b) 40cm from A. Get Top Ranks in IIT-JEI ort od