PREPARED BY: ER. VINEET LOOMBA (B.TECH. IIT ROORKEE)

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1 Cass XI TARGET : JEE Main/Adv PREPARED BY: ER. VINEET LOOMBA (B.TECH. IIT ROORKEE) ALP ADVANCED LEVEL LPROBLEMS ROTATION- Topics Covered: Rigid body, moment of inertia, parae and perpendicuar axes theorems, moment of inertia of uniform bodies with simpe geometrica shapes; Anguar momentum; Torque; Conservation of anguar momentum; Dynamics of rigid bodies with fixed axis of rotation; Roing without sipping of rings, cyinders and spheres; Equiibrium of rigid bodies; Coision of point masses with rigid bodies. Q. In absence of torque the rotationa frequency of a body changes from cy/sec to 6 cy/sec, then ratio of radius of gyration in two cases wi be : : : : 4 : Q. A partice of mass m is rotating in a pane is a circuar path of radius r, its anguar momentum is L. The centripita force acting on the partice is : L mr L m r L L mr mr Q. For the same tota mass which of the foowing wi have the argest moment of inertia about an axis passing through its centre of mass and perpendicuar to the pane of the body a disc of radius a a ring of radius a a square amina of side a four rods forming a square of side a Q.4 Find the moment of inertia of a pate cut in shape of a right anged triange of mass M, side AC = BC = a about an axis perpendicuar to the pane of the pate and passing through the mid point of side AB Ma Ma 6 Ma Ma Q.5 Three identica thin rods each of mass m & ength are paced aong x, y & z-axis respectivey they are paced such that, one end of each rod is at origin 'O'. Then moment of inertia of this system about z-axis is m m m m 4 Q.6 Two rods of equa mass m and ength ie aong the x axis and y axis with their centres origin. What is the moment of inertia of both about the ine x=y : m m 4 m m 6

2 Cass (XI) Q.7 Moment of inertia of a rectanguar pate about an axis passing through P and perpendicuar to the pate is I. Then moment of PQR about an axis perpendicuar to the pane of the pate: about P = I/ about R = I/ about P > I/ about R > I/ Q.8 Let I, I and I be the moment of inertia of a uniform square pate about axes AOC, xdx' and yby' respectivey as shown in the figure. The moments of inertia of the pate I : I : I are in the ratio. : 7 : 7 : 7 : 7 : : 7 7 : 7 : 7 Q.9 A thin uniform rod of mass M and ength L has its moment of inertia I about its perpendicuar bisector. The rod is bend in the form of a semicircuar arc. Now its moment of inertia through the centre of the semi circuar arc and perpendicuar to its pane is I. The ratio of I : I wi be < > = can t be said Q.0 The moment of inertia of semicircuar pate of radius R and mass M about axis AA in its pane passing through its centre is MR MR cos 4 MR Q. In the trianguar sheet given PQ = QR =. If M is the mass of the sheet. What is the moment of inertia about PR M 4 M Q. Moment of inertia of a thin semicircuar disc (mass = M & radius = R) about an axis through point O and perpendicuar to pane of disc, is given by : M 6 sin MR 4 M 4 MR MR 8 MR MR Q. Moment of inertia of a semicircuar ring of radius R and mass M ; about an axis passing through A and perpendicuar to the pane of the paper is MR MR 5 MR MR Q.4 A square sheet of edge ength L and uniform mass per unit area is used to form a hoow cyinder. The moment of inertia of this cyinder about the centra axis is 4 L 4 L 4 8 L L 4 Prepared By: Er. Vineet Loomba (IIT Roorkee)

3 Jupiter (XI) Q.5 A rigid body can be hinged about any point on the x-axis. When it is hinged such that the hinge is at x, the moment of inertia is given by I = x x + 7 The x-coordinate of centre of mass is x = x = 0 x = x = Q.6 A square pate of mass M and edge L is shown in figure. The moment of inertia of the pate about the axis in the pane of pate passing through one of its vertex making an ange 5 from horizonta is. ML ML 4 7ML none Q.7 A wire of mass M and ength L is bent in the form of a circuar ring. The moment of inertia of the ring about its axis is 8 ML (8 )ML 4 ML (4 )ML Q.8 The figure shows a uniform rod ying aong the x-axis. The ocus of a the points ying on the xy-pane, about which the moment of inertia of the rod is same as that about O is an eipse a circe a paraboa a straight ine Q.9 Consider the foowing statements Assertion : The moment of inertia of a rigid body reduces to its minimum vaue as compared to any other parae axis when the axis of rotation passes through its centre of mass. Reason (R): The weight of a rigid body aways acts through its centre of mass in uniform gravitationa fied. Of these statements: both A and R are true and R is the correct expanation of A both A and R are true but R is not a correct expanation of A A is true but R is fase A is fase but R is true For more such free Assignments visit Question No. 0 to (4 questions) The figure shows an isoscees trianguar pate of mass M and base L. The ange at the apex is 90. The apex ies at the origin and the base is parae to X axis Q.0 The moment of inertia of the pate about the z-axis is ML ML ML 4 6 Q. The moment of inertia of the pate about the x-axis is none of these ML ML ML ML Q. The moment of inertia of the pate about its base parae to the x-axis is Prepared By: Er. Vineet Loomba (IIT Roorkee)

5 Jupiter (XI) 5 side AB is horizonta. The reaction at the support A is: mg mg mg mg Q. In an experiment with a beam baance on unknown mass m is baanced by two known mass m is baanced by two known masses of 6 kg and 4 kg as shown in figure. The vaue of the unknown mass m is 0 kg 6 kg 8 kg kg Q. A uniform cube of side b and mass M rest on a rough horizonta tabe. A horizonta force F is appied norma to one of the face at a point, at a height b/4 above the base. What shoud be the coefficient of friction ( ) between cube and tabe so that is wi tip about an edge before it starts sipping? > > > none Q. A homogeneous cubica brick ies motioness on a rough incined surface. The haf of the brick which appies greater pressure on the pane is : eft haf right haf both appies equa pressure the answer depend upon coefficient of friction Q.4 Find minimum height of obstace so that the sphere can stay in equiibrium. R cos R ( sin ) R sin R ( cos ) Q.5 A haow cone of radius R and height R is paced on an incined pane of incination. If is increased graduay, at what vaue of the cone wi toppe. Assume sufficient friction is present to prevent sipping. tan (/) tan (/) sin (/) cos (/) For more such free Assignments visit Q.6 A smooth rod of ength is kept inside a troey at an ange as shown in the figure. What shoud be the acceeration a of the troey so that the rod remains in equiibrium with respect to it? g tan g cos g sin g cot Q.7 A uniform adder of ength 5m is paced against the wa as shown in the figure. If coefficient of friction is the same for both the was, what is the minimum vaue of for it not to sip? = = 4 = = 5 Q.8 A uniform cyinder rests on a cart as shown. The coefficient of static friction between the cyinder and Prepared By: Er. Vineet Loomba (IIT Roorkee)

6 Jupiter (XI) 6 the cart is 0.5. If the cyinder is 4 cm in diameter and 0 cm in height, which of the foowing is the minimum acceeration of the cart needed to cause the cyinder to tip over? m/s 4 m/s 5 m/s the cyinder woud side before it begins to tip over. Q.9 A uniform rod of ength L and weight W is suspended horizontay by two vertica ropes as shown. The first rope is attached to the eft end of the rod whie the second rope is attached a distance L 4 from the right end. The tension in the second rope is W 4 W W W Q.40 The spoo shown in figure is paced on rough horizonta surface and has inner radius r and outer radius R. The ange between the appied force and the horizonta can be varied. The critica ange ( ) for which the spoo does not ro and remains stationary is given by r r r r = cos = cos R = cos R = sin R R Q.4 Two ight vertica springs with equa natura engths and spring constants k and k are separated by a distance. Their upper ends are fixed to the ceiing and their ower ends to the ends A and B of a ight horizonta rod AB. A vertica downwards force F is appied at point C on the rod. AB wi remain horizonta in equiibrium if the distance AC is k k Q.4 Consider the foowing statements Assertion : A cycist aways bends inwards whie negotiating a curve Reason(R) : By bending he owers his centre of gravity Of these statements, both A and R are true and R is the correct expanation of A both A and R are true but R is not the correct expanation of A A is true but R is fase A is fase but R is true Q.4 A cone of radius r and height h rests on a rough horizonta surface, the coefficient of friction between the cone and the surface being. A graduay increasing horizonta force F is appied to the vertex of the cone. The argest vaue of for which the cone wi side before it toppes is = r h = r 5h k k k k k k = h r = r h Q.44 A uniform rod of mass m and ength hinged at its end is reeased from rest when it is in the horizonta position. The norma reaction at the hinge when the rod becomes vertica is : Prepared By: Er. Vineet Loomba (IIT Roorkee)

8 Jupiter (XI) 8 mass m o hangs from the end of a ight string wound round the cyinder which does not sip over it. When the system is aowed to move, the acceeration of the descending mass wi be mog m m o mog m m o mog m m o mog m m o Q.5 A uniform rod of ength, hinged at the ower end is free to rotate in the vertica pane. If the rod is hed verticay in the beginning and then reeased, the anguar acceeration of the rod when it makes an ange of 45 o with the horizonta (I = m /) g 6g Q.5 A bock of mass m is attached to a puey disc of equa mass m, radius r by means of a sack string as shown. The puey is hinged about its centre on a horizonta tabe and the bock is projected with an initia veocity of 5 m/s. Its veocity when the string becomes taut wi be m/s.5 m/s 5/ m/s 0/ m/s Q.54 A sma bead of mass m moving with veocity v gets threaded on a stationary semicircuar ring of mass m and radius R kept on a horizonta tabe. The ring can freey rotate about its centre. The bead comes to rest reative to the ring. What wi be the fina anguar veocity of the system? v/r v/r v/r v/r Q.55 A sma object is attached to a ight string which passes through a hoow tube. The tube is hed by one hand and the string by the other. The object is stet into rotation in a circe of radius r. The string is then pued down, shortening the radius of the circe to r. The ratio of the new kinetic energy to origina kinetic energy is r r g r r g r r Q.56 A man, sitting firmy over a rotating stoo has his arms streched. If he fods his arms, the work done by the man is zero positive negative may be positive or negative. Q.57 A partice of mass kg ocated at the position ( î ĵ) m has a veocity ( î ĵ kˆ ) m/s. Its anguar momentum about z-axis in kg-m /s is: zero +8 8 Q.58 A partice is moving in a circuar orbit of radius r with an anguar veocity. It jumps to another circuar orbit of radius r and attains an anguar veocity. If r = 0.5 r and assuming that no externa torque is appied to the system, then the anguar veocity, is given by : = 4 = = = Prepared By: Er. Vineet Loomba (IIT Roorkee)

9 Q.59 A partice of mass m is projected with a veocity u making an ange 45 with the horizonta. The magnitude of the torque due to weight of the projectie, when the partice is at its maximum height, about the point of projectie mu 4 mu 4 mu mu Q.60 Three bodies have equa masses m. Body A is soid cyinder of radius R, body B is a square amina of side R, and body C is a soid sphere of radius R. Which body has the smaest moment of inertia about an axis passing through their centre of mass and perpendicuar to the pane (in case of amina) A B C A and C both Q.6 A point mass m A is connected to a point mass m B by a massess rod of ength as shown in the figure. It is observed that the ratio of the moment of inertia of the system about the two axes BB and AA, which is parae to each other and perpendicuar to the rod is IBB I =. The distance of the centre of mass of the system from the mass A is AA (/4) (/) (/) (/4) Q.6 A chid with mass m is standing at the edge of a disc with moment of inertia I, radius R, and initia anguar veocity. See figure given beow. The chid jumps off the edge of the disc with tangentia veocity v with respect to the ground. The new anguar veocity of the disc is I mv I I mvr I (I mr ) mv I (I mr ) mvr I Q.6 A partice of mass 0.5 kg is rotating in a circuar path of radius m and centrepeta force on it is 9 Newtons. Its anguar momentum (in J sec) is: Question No. 64 & 65 ( questions) A uniform rod is fixed to a rotating turntabe so that its ower end is on the axis of the turntabe and it makes an ange of 0 to the vertica. (The rod is thus rotating with uniform anguar veocity about a vertica axis passing through one end.) If the turntabe is rotating cockwise as seen from above. Q.64 What is the direction of the rod's anguar momentum vector (cacuated about its ower end)? verticay downwards down at 0 to the horizonta up at 0 to the horizonta verticay upwards Q.65 Is there a torque acting on it, and if so in what direction? yes, verticay yes, horizontay yes at 0 to the horizonta no Q.66 A straight rod of ength L is reeased on a frictioness horizonta foor in a vertica position. As it fas + sips, the distance of a point on the rod from the ower end, which foows a quarter circuar ocus is L/ L/4 L/8 None Q.67 Two partices of mass m each are fixed at the opposite ends of a massess rod of ength 5m which is oriented verticay on a smooth horizonta surface and reeased. Find the dispacement of the ower

10 mass on the ground when the rod makes an ange of 7 with the vertica..5 m m.5 m.5 m Q.68 A partice starts from the point (0m, 8m) and moves with uniform veocity of i m/s. After 5 seconds, the anguar veocity of the partice about the origin wi be : 8 89 rad/s 8 rad/s 4 89 rad/s 8 7 rad/s Q.69 A hinged construction consists of three rhombs with the ratio of sides 5::. Vertex A moves in the horizonta direction at a veocity v. Veocity of A is.5 V V.5 V 0.8 V Q.70 A whee of radius r roing on a straight ine, the veocity of its centre being v. At a certain instant the point of contact of the whee with the grounds is M and N is the highest point on the whee (diametricay opposite to M). The incorrect statement is: The veocity of any point P of the whee is proportiona to MP. Points of the whee moving with veocity greater than v form a arger area of the whee than points moving with veocity ess than v. The point of contact M is instantaneousy at rest. The veocities of any two parts of the whee which are equidistant from centre are equa. Q.7 Two points of a rigid body are moving as shown. The anguar veocity of the body is: R R R Q.7 There is rod of ength. The veocities of its two ends are v and v in opposite directions norma to the rod. The distance of the instantaneous axis of rotation from v is: zero v v v v v v R / Q.7 A thin rod of ength L is paced at ange to vertica on a frictioness horizonta foor and reeased. If the center of mass has acceeration = A, and the rod an anguar acceeration = at initia moment, then A = (L ).sin A/ = (L ).sin A = (L ).sin A = L Q.74 A disc of radius R is roing purey on a fat horizonta surface, with a constant anguar veocity. The ange between the veocity and acceeration vectors of point P is zero 45 5 tan (/) Q.75 A adder of ength L is sipping with its ends against a vertica wa and a horizonta foor. At a certain moment, the speed of the end in contact with the horizonta foor is v and the adder makes

11 an = 0 with the horizonta. Then the speed of the adder s center must be v v/ v None ange Q.76 In the previous question, if dv/dt = 0, then the anguar acceeration of the adder when = 45 is v /L v /L [v L ] None Q.77 A ring of radius R ros without siding with a constant veocity. The radius of curvature of the path foowed by any partice of the ring at the highest point of its path wi be R R 4R None Q.78 Two spheres are roing with same veocity (for their C. M.) their ratio of kinetic energy is : & radius ratio is :, their mass ratio wi be : : 4 : 8 : : Q.79 Two identica circuar oops are moving with same kinetic energy one ros & other sides. The ratio of their speed is : : : : 5 : Q.80 Inner and outer radii of a spoo are r and R respectivey. A thread is wound over its inner surface and paced over a rough horizonta surface. Thread is pued by a force F as shown in fig. then in case of pure roing Thread unwinds, spoo rotates anticockwise and friction act eftwards Thread winds, spoo rotates cockwise and friction acts eftwards Thread winds, spoo moves to the right and friction act rightwards Thread winds, spoo moves to the right and friction does not come into existence. For more such free Assignments visit Q.8 Portion AB of the wedge shown in figure is rough and BC is smooth. A soid cyinder ros without sipping from A to B. The ratio of transationa kinetic energy to rotationa kinetic energy, when the cyinder reaches point C is : /4 5 7/5 8/ Q.8 A pank of mass M is paced over smooth incined pane and a sphere is aso paced over the pank. Friction is sufficient between sphere and pank. If pank and sphere are reeased from rest, the frictiona force on sphere is: up the pane down the pane horizonta zero Q.8 A pank with a uniform sphere paced on it rests on a smooth horizonta pane. Pank is pued to right by a constant force F. If sphere does not sip over the pank. Which of the foowing is incorrect. Acceeration of the centre of sphere is ess than that of the pank. Work done by friction acting on the sphere is equa to its tota kinetic energy.

13 Q.9 A sender uniform rod of ength is baanced verticay at a point P on a horizonta surface having some friction. If the top of the rod is dispaced sighty to the right, the position of its centre of mass at the time when the rod becomes horizonta : ies at some point to the right of P ies at some point to the eft of P must be / to the right of P ies at P Q.9 A soid sphere with a veocity (of centre of mass) v and anguar veocity is genty paced on a rough horizonta surface. The frictiona force on the sphere: must be forward (in direction of v) must be backward (opposite to v) cannot be zero none of the above Q.9 A cyinder is pure roing up an incine pane. It stops momentariy and then ros back. The force of friction on the cycinder is zero throughout the journey is directed opposite to the veocity of the centre of mass throughout the journey is directed up the pane throughout the journey is directed down the pane throughout the journey Q.94 A uniform circuar disc paced on a rough horizonta surface has initiay a veocity v 0 and an anguar veocity 0 as shown in the figure. The disc comes to rest after moving some distance in v the direction of motion. Then 0 r is 0 Q.95 On a soid sphere ying on a horizonta surface a force F is appied at a height of R/ from the centre of mass. The initia acceeration of a point at the top of the sphere is (there is no sipping at any point) 5F 7M 5F 4M 0 F 7M M F Q.96 A ba ros down an incined pane, figure. The ba is first reeased from rest from P and then ater from Q. Which of the foowing statement is/ are correct? (i) The ba takes twice as much time to ro from Q to O as it does to ro from P to O. (ii) The acceeration of the ba at Q is twice as arge as the acceeration at P. (iii) The ba has twice as much K.E. at O when roing from Q as it does when roing from P. i, ii ony ii, iii ony i ony iii ony Question No. 97 to 0 (6 questions) In the foowing probems, indicate the correct direction of friction force acting on the cyinder, which is pued on a rough surface by a constant force F. Q.97 A cyinder of mass M and radius R is pued horizontay by a force F. The

14 friction force can be given by which of the foowing diagrams (B ) cannot be interpreted Q.98 A cyinder is pued horizontay by a force F acting at a point beow the centre of mass of the cyinder, as shown in figure. The friction force can be given by which of the foowing diagrams (B ) cannot be interpreted Q.99 A cyinder is pued horizontay by a force F acting at a point above the centre of mass of the cyinder, as shown in figure. The friction force can be given by which of the foowing diagrams (B ) cannot be interpreted Q.00 A cyinder is paced on a rough pank which in turn is paced on a smooth surface. The pank is pued with a constant force F. The friction force can be given by which of the foowing diagrams canot be interpreted For more such free Assignments visit Maths Notes and Assignments for IIT-JEE: Physics Notes and Assignments for IIT-JEE: Chemistry Notes and Assignments for IIT-JEE: New Chapters and Assignments are being reguary updated. Share it with your friends because Sharing is Caring. Discuss among yoursef or with your teachers in case of doubts. You can post your doubts on website comment section too and I wi try to answer as eary as possibe. SHARING IS CARING!! I AM SHARING WITH YOU. YOU HAVE TO SHARE WITH YOUR FRIENDS AND HELP THEM.

15 ANSWER KEY Q. D Q. D Q. D Q.4 B Q.5 B Q.6 C Q.7 C Q.8 D Q.9 A Q.0 D Q. B Q. B Q. D Q.4 B Q.5 D Q.6 B Q.7 C Q.8 B Q.9 B Q.0 C Q. A Q. C Q. C Q.4 B Q.5 B Q.6 B Q.7 D Q.8 A Q.9 D Q.0 B Q. C Q. A Q. A Q.4 D Q.5 B Q.6 D Q.7 C Q.8 B Q.9 D Q.40 A Q.4 D Q.4 B Q.4 C Q.44 C Q.45 B Q.46 A Q.47 C Q.48 C Q.49 C Q.50 C Q.5 A Q.5 A Q.5 D Q.54 C Q.55 C Q.56 B Q.57 D Q.58 A Q.59 D Q.60 B Q.6 D Q.6 D Q.6 C Q.64 B Q.65 B Q.66 B Q.67 A Q.68 C Q.69 D Q.70 D Q.7 B Q.7 C Q.7 C Q.74 B Q.75 C Q.76 A Q.77 C Q.78 A Q.79 C Q.80 B Q.8 B Q.8 D Q.8 D Q.84 C Q.85 A Q.86 C Q.87 C Q.88 D Q.89 D Q.90 A Q.9 A Q.9 D Q.9 C Q.94 A Q.95 A Q.96 D Q.97 A Q.98 A Q.99 D Q.00 B

PHYSICS LOCUS / / d dt. ( vi) mass, m moment of inertia, I. ( ix) linear momentum, p Angular momentum, l p mv l I

PHYSICS LOCUS / / d dt. ( vi) mass, m moment of inertia, I. ( ix) linear momentum, p Angular momentum, l p mv l I 6 n terms of moment of inertia, equation (7.8) can be written as The vector form of the above equation is...(7.9 a)...(7.9 b) The anguar acceeration produced is aong the direction of appied externa torque.