EXERCISE 01. JEE-Physics CHECK YOUR GRASP

Transcription

1 J-Physics XRIS 1 HK YOUR GRSP Select the correct alternatie (only one correct answer) 1. On account of the earth rotating about its axis :- the linear elocity of objects at equator is greater than at other places the angular elocity of objects at equator is more than that of objects at poles the linear elocity of objects at all places at the earth is equal, but angular elocity is different at all places the angular elocity and linear elocity are uniform. fly wheel originally at rest is to reach an angular elocity of 6 radian/s in 6 second. The total angle it turns through in the 6 second is 5 radian 18 radian 6 radian 16 radian. The rotating rod starts from rest and acquires a rotational speed n = 6 reolution/minute in seconds with constant angular acceleration. The angular acceleration of the rod is 1 rad/s 5 rad/s 15 rad/s None of these. The number of reolutions must the 6 cm diameter wheel of a car turn as the car traels.5 km is 8 reolution 1 reolution 1 reolution 5 reolution 5. Two gear wheels which are meshed together hae radii of.5 cm and.15 cm. The number of reolutions does the smaller turns when the larger turns through reolution is 5 reolution reolution 1 reolution 1 reolution 6. The radius of a wheel of a car is.m. The car is accelerated from rest by an angular acceleration of 1.5 rad/s for s. The linear elocity of the wheel is 1 m/s m/s 1 m/s m/s 7. In the adjoining figure along which axis the moment of inertia of the triangular lamina will be maximum- [Gien that < < ] For all axis 8. Three particles, each of mass m are situated at the ertices of an equilateral triangle of side cm (as shown in the figure). The moment of inertia of the system about a line X perpendicular to and in the plane of, in gram cm units will be : m 5 m m m X m m m Y NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65 9. circular disc is to be made by using iron and aluminium so that it acquired maximum moment of inertia about geometrical axis. It is possible with : aluminium at interior and iron surrounded to it. iron at interior and aluminium surrounded to it. using iron and aluminium layers in alternate order. sheet of iron is used at both external surface and aluminium sheet as internal layer. 1. We hae a rectangular slab of same thickness., F, G, H are the middle point of,, D and D respectiely then which of the following axis the moment of inertia will be minimum : D G D G D HF 1 1. Two disc one of density 7. g/cm and the other of density 8.9 g/cm are of same mass and thickness. Their moments of inertia are in the ratio : (8.9 7.) : 1 1 : (8.9 7.) H F

2 J-Physics 1. Off two eggs which hae identical sizes, shapes and weights, one is raw and the other is half-boiled. The ratio between the moment of inertia of the raw egg and that of the half-boiled egg about a central axis is : one greater than one less than one incomparable 1. The moment of inertia of a thin uniform rod of mass M and length about an axis perpendicular 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 point is : I/ I/ I I 1 1. The moment of inertia of a rod about an axis through its centre and perpendicular to it is 1 ML (where M is the mass and L is the length of the rod). The rod is bent in the middle so that the two half make an angle of 6. The moment of inertia of the bent rod about the same axis would be : 1 8 ML 1 1 ML 1 ML ML Four similar point masses (each of mass m) are placed on the circumference of a disc of mass M and radius R. The M.I. of the system about the normal axis through the centre O will be:- MR + mr 1 MR + mr O MR mr None of these 1 6. Two rings of the same radius and mass are placed such that their centres are at a common point and their planes are perpendicular to each other. The moment of inertia of the system about an axis passing through the centre and perpendicular to the plane of one of the rings is (mass of the ring = m, radius = r) : 1 mr mr mr mr 1 7. Three point masses, each of m, are placed at the corners of an equilateral triangle of side. Then the moment of inertia of this system about an axis along one side of the triangle is : m m m m 1 8. Two rods each of mass m and length are joined at the centre to form a cross. The moment of inertia of this cross about an axis passing through the common centre of the rods and perpendicular to the plane formed by them, is : m m m m If the mass of hydrogen atom is g and interatomic distance in a molecule of hydrogen is 1 8 cm, then the moment of inertia [in kg-m ] of a molecule of hydrogen about the axis passing through the centre of mass and perpendicular to the line joining the atoms will be: If a body completes one reolution in sec then the moment of inertia would be: qual to rotational kinetic energy Double of rotational kinetic energy Half of rotational kinetic energy Four times of the rotational kinetic energy 1. For the same total mass which of the following will hae the largest moment of inertia about an axis passing through its centre of mass and perpendicular to the plane of the body a disc of radius a a ring of radius a a square lamina of side a four rods forming a square of side a. Two rods of equal mass m and length lie along the x axis and y axis with their centres origin. What is the moment of inertia of both about the line x=y : m m 68 m 1 m 6 NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65

3 J-Physics. 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 gien by I = x x The x-coordinate of centre of mass is : x= x= x=1 x=. The axis X and Z in the plane of a disc are mutually perpendicular and Y-axis is perpendicular to the plane of the disc. If the moment of inertia of the body about X and Y axes is respectiely kg m and kgm then M.I. about Z-axis in kg m will be: Zero 5. wheel is rotating about an axis through its centre at 7 rpm. It is acted on by a constant torque opposing its motion for 8 second to bring it to rest finally. The alue of torque in Nm is : (gien I kg m ) rod of mass M and length L is placed in a horizontal plane with one end hinged about the ertical axis. horizontal force of F= Mg rod will be :- g 5L is applied at a distance 5L 6 5 g L from the hinged end. The angular acceleration of the g L g L 7. person supports a book between finger and thumb as shown (the point of grip is assumed to be at the corner of the book). If the book has a weight of W then the person is producing a torque on the book of b a W a anticlockwise W b anticlockwise Wa anticlockwise Wa clockwise NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65 8. string is wrapped around the rim of a wheel of moment of inertia. kg-m and radius cm. The wheel is free to rotate about its axis and initially the wheel is rest. The string is now pulled by a force of N. The angular elocity of the string after 5 seconds will be : 9 rad/s 7 rad/s 95 rad/s 1 rad/s 9. In the figure half of the meter scale is made of wood while the other half of steel. The wooden part is pioted at O. force F is applied at the end of steel part. In figure the steel part is pioted at O' and the same force is applied at the wooden end: More angular acceleration will be produced in wood steel steel wood More angular acceleration will be produced in O P O' P' Same angular acceleration will be produced in both conditions F F Information is incomplete. In the following figure r 1 and r are 5 cm and cm respectiely. If the moment of inertia of the wheel is 51 kg-m then its angular acceleration will be :- 1 rad/sec 1 rad/sec 1 rad/sec 1 1 rad/sec 1. non uniform rod O of liner mass density x co nst. 69 1N N r 1 r 1N is suspended from ceiling with hinge joint O & light string as shown in figure. Find the angular acceleration of rod just after the string is cut g L g L g L None of these O 9N g x

4 J-Physics. If the earth is a point mass of 6 1 kg reoling around the sun at a distance of km and in time T= second, then the angular momentum of the earth around the sun is : kg m /s kg m /s kg m /s.7 1 kg m /s. particle of mass m moes with a constant elocity. Which of the following statements is not correct about its angular momentum : it is zero when it is at and moing along O the same at all points along the line D of the same magnitude but oppositely directed at and D increases as it moes along the line. thin rod of mass M and length L is struck at one end by a ball of clay of mass m, moing with speed as shown in figure. The ball sticks to the rod. fter the collision, the angular momentum of the clay-rod system about, the midpoint of the rod, is Y D O X M m L M m 1 L 9 ml ml 5. If the earth were to suddenly contract to 1 th n of the new day will be nearly : of its present radius without any change in its mass then the duration n hour n hour n hour n hour 6. The angular elocity of a body changes from 1 to without applying torque. The ratio of initial radius of gyration to the final radius of gyration is : : 1 1 : : 1 1 : 7. 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 eaporation, the speed of rotation of the system : becomes zero remains constant at the same alue of increases to alue greater than decreases to a alue less than 8. thin circular ring of mass M and radius r is rotating about its axis with a constant angular elocity. Four objects each of mass m, are kept gently to the opposite ends of two perpendicular diameters of the ring. The new angular elocity of the ring will be : M m M M m 7 (M m) M (M m) M m 9. person is standing on the edge of a circular platform, which is moing with constant angular speed about an axis passing through its centre and perpendicular to the plane of platform. If person is moing along any radius towards axis of rotation then the angular elocity will : decrease remain unchanged increase data is insufficient. n ant is sitting at the edge of a rotating disc. If the ant reaches the other end, after moing along the diameter, the angular elocity of the disc will :- remain constant first decreases and then increases first increases, then decrease Increase continuously NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65

5 J-Physics 1. boy stands oer the centre of a horizontal platform which is rotating freely with a speed of reolutions/s about a ertical axis through the centre of the platform and straight up through the boy. He holds kg masses in each of his hands close to his body. The combined moment of inertia of the system is 1 kg-m.. The boy now stretches his arms so as to hold the masses far from his body. In this situation the moment of inertia of the system increases to kg-m.. The kinetic energy of the system in the latter case as compared with that in the preious case will- Remain unchanged Decrease Increase Remain uncertain. horizontal platform is rotating with uniform angular elocity around the ertical axis passing through its centre. t some instant of time a iscous fluid of mass m is dropped at the centre and is allowed to spread out and finally fall. The angular elocity during this period : Decreases continuously Remains unaltered Decreases initially and increases again Increases continuously. particle starts from the point (m, 8m) and moes with uniform elocity of i ˆ m/s. fter 5 seconds, the angular elocity of the particle about the origin will be y 8 89 rad/s 8 rad/s 89 rad/s 8 17 rad/s. Two rotating bodies hae same angular momentum but their moments of inertia are I 1 and I respectiely (I 1 >I ). Which body will hae higher kinetic energy of rotation: First oth will hae same kinetic energy Second Not possible to predict 5. thin rod of length L is suspended from one end and rotated with n rotations per second. The rotational kinetic energy of the rod will be: 8m O m/s x ml n 1 ml n ml n 1 6 ml n 6. rigid body of mass m rotates with angular elocity about an axis at a distance d from the centre of mass G. The radius of gyration about a parallel axis through G is K. The kinetic energy of rotation of the body is : 1 mk 1 md 1 m(d k ) 1 m(d k) NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65 7. weightless rod is acted on by upward parallel forces of N and N at ends and respectiely. The total length of the rod is = m. To keep the rod in equilibrium a force of 6N should act in the following manner: Downwards at any point between and 71 Downwards at mid point of Downwards at a point such that =1m Downwards at a point D such that D =1m 8. In an experiment with a beam balance an unknown mass m is balanced by two known masses of 16kg and kg as shown in figure. The alue of the unknown mass m is : 16kg 1 m m 1 1 kg 6 kg 8 kg 1 kg kg

6 J-Physics 9. rod is hinged at its centre and rotated by applying a constant torque starting from rest. The power deeloped by the external torque as a function of time is : P ext P ext P ext P ext time time 5. If a ring, a disc, a solid sphere and a cylinder of same radius rolls down on inclined plane, the first one to reach the bottom will be : disc ring solid sphere cylinder 5 1. body is rolling without slipping on a horizontal surface and its rotational kinetic energy is equal to the translational kinetic energy. The body is : disc sphere cylinder ring 5. solid cylinder of mass M and radius R rolls without slipping down an inclined plane of length L and height h. What is the speed of its centre of mass when the cylinder reaches its bottom : gh gh 5. 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 elocity of the disc is, the height to which the disc will rise will be : g g g g 5. solid sphere is rolling on a frictionless surface, shown in figure with a translational elocity m/s. If it is to climb the inclined surface then should be: 1 7 gh > gh gh 1/7gh gh h time gh time h 5 5. rod hinged at one end is released from the horizontal position as shown in the figure. When it becomes ertical its lower half separates without exerting any reaction at the breaking point. Then the maximum angle ' ' made by the hinged upper half with the ertical is There is rod of length. The elocities of its two ends are 1 and in opposite directions normal to the rod. The distance of the instantaneous axis of rotation from 1 is : Zero HK YOUR GRSP NSWR KY XRIS -1 Q u e n s. D D D Q u e n s. D D D D D Q u e n s. D D NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65

7 J-Physics XRIS RIN TSRS Select the correct alternaties (one or more than one correct answers) 1. thin rod of length, mass m is bent at the points as shown in the fig. What is the moment of inertia of the rod about the axis passing point O & perpendicular to the plane of the paper O 9 9 m 1m m 1 m. smooth tube of certain mass is rotated in graity free space and released. The two balls shown in the figure moe towards ends of the tube. For the whole system which of the following quantity is not consered :- ngular momentum Kinetic energy Linear momentum ngular speed. uniform rod of mass m and length at rest on a smooth horizontal surface. n impulse P is applied to the end. The time taken by the rod to turn through a right angle is :- m P m 1P m P m P. n equilateral prism of mass m rests on a rough horizontal surface with coefficient of friction µ. horizontal force F is applied on the prism as shown in the figure. If the coefficient of friction is sufficiently high so that the prism does not slide before toppling, then the minimum force required to topple the prism is- mg mg µmg µmg a 5. uniform rod of mass M and length L lies radially on a disc rotating with angular speed in a horizontal plane about its axis. The rod does not slip on the disc and the centre of the rod is at a distance R from the centre of the disc. Then the kinetic energy of the rod is- F a P a 1 L m R 1 1 m R L R NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65 1 m L 7 None of these 6. particle of mass m is projected with a elocity making an angle of 5 with the horizontal. The magnitude of the angular momentum of the projectile about the point of projection when the particle is at its maximum height h is :- zero m ( g) m g m gh 7. tube of length L is filled completely with an incompressible liquid of mass M and closed at both the ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular elocity. The force exerted by the liquid at the other end is :- M L M L M L M L

8 J-Physics 8. Two point masses of. kg and.7 kg are fixed at the ends of a rod of length 1. m and of negligible mass. The rod is set rotating about an axis perpendicular to its length with a uniform angular speed. The point on the rod through which the axis should pass in order that the work required for rotation of the rod is minimum, is located at a distance of :-. m from mass of. kg.98 m from mass of. kg.7 m from mass of.7 kg.98 m from mass of.7 kg 9. sphere S rolls without slipping, moing with a constant speed on a plank P. The friction between the upper surface of P and the sphere is sufficient to preent slipping, while the lower surface of P is smooth and rests on the ground. Initially, P is fixed to the ground by a pin T. If T is suddenly remoed- S S will begin to slip on P. = r T P will begin to moe backwards. r P the speed of S will decrease and its angular elocity will increase. there will be no change in the motion of S and P will still be at rest. 1. disc of mass M and radius R is rolling with angular speed on a horizontal y plane as shown. The magnitude of angular momentum of the disc about the origin O is : O M x 1 MR MR MR 1 1. Two spheres each of mass M and radius R/ are connected with a mass less rod of length R as shown in the-figure. What will be the moment of inertia of the system about an axis passing through the centre of one of the spheres and perpendicular to the rod MR P R/ M Y M Q Y 1 M R 5 M R 5 5 M R 5 M R 1 1. cord is wound oer a cylinder of radius r and moment of inertia I. mass m is attached to the free end of the cord. The cylinder is free to rotate about its own horizontal axis. If mass m is released from rest, then the elocity of the mass after it had fallen through a distance h will be- (gh) 1/ mghr² I 1 / mghr² I mr² 1 / mghr² I mr² 1 / 1. solid sphere of radius R is placed on smooth horizontal surface. horizontal force F is applied at height h from the lowest point. For the maximum acceleration of centre of mass, which is correct- h = R h = R h = No relation between h and R 7 F h R Lowest Point 1. solid sphere is placed on a horizontal plane. horizontal impulse I is applied at a distance h aboe the central line as shown in the figure. Soon after giing the impulse the sphere starts rolling. The ratio h/r would be- I h R NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65

9 J-Physics 1 5. ring of radius a is fixed rigidly on a table. small ring whose mass is m and radius a, rolls without slipping inside it as shown in the figure. The small ring is released from position. When it reaches at the lowest point, the speed m a a of the centre of the ring at that time would be- ga ga 6ga ga 1 6. The moment of inertia of semicircular plate of radius R and mass M about axis ' in its plane passing through its centre is MR MR sin MR cos MR 1 7. 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 an ellipse a parabola a circle a straight line y o ' x 1 8. man can moe on a horizontal plank supported symmetrically as shown. x= The ariation of normal reaction on support with distance x of the man from the end of the plank is best represented by 1m 1m N N N N x 1 9. Find minimum height of obstacle so that the sphere can stay in equilibrium R 1 cos R 1 sin R 1 sin R 1 cos x x R m h x. sphere is placed rotating with its centre initially at rest in a corner as shown in figure &. oefficient of friction between all surfaces NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65 and the sphere is 1. Find the ratio of the frictional force fa f in situations & 75 b by ground 1 9/1 1/9 None 1. hinged construction consists of three rhombus with the ratio of sides 5::. Vertex moes in the horizontal direction at a elocity. Velocity of is

10 J-Physics. disc of radius R is rolling purely on a flat horizontal surface, with a constant P angular elocity. The angle between the elocity and acceleration ectors of point P is Zero 5 15 tan -1 (1/). Portion of the wedge shown in figure is rough and is smooth. solid cylinder rolls without slipping from to. The ratio of translational kinetic energy to rotational kinetic energy, when the cylinder reaches point is / 5 7/5 8/ D \\\\\\\\\\\\\\\\\\\\\\\ =. ring of mass m and radius R has three particles attached to the ring m as shown in the figure. The centre of the ring has a speed. The kinetic energy m of the system is : (slipping is absent) 6 m 1 m m 8 m 5. slender uniform rod of length is balanced ertically at a point P on a horizontal surface haing some friction. If the top of the rod is displaced slightly to the right, the position of its centre of mass at the time when the rod becomes horizontal lies at some point to the right of P lies at some point to the left of P must be to the right of P lies at P 6. solid sphere with a elocity (of centre of mass) and angular elocity is gently placed on a rough horizontal surface. The frictional force on the sphere must be forward (in direction of ) must be backward (opposite to ) cannot be zero none of the aboe 7. uniform circular disc placed on a rough horizontal surface has initially elocity and an angular elocity as shown in the figure. The disc comes m to rest after moing some distance in the direction of motion. Then r is 1/ 1 / 8. body is in equilibrium under the influence of a number of forces. ach force has a different line of action. The minimum number of forces required is, if their lines of action pass through the centre of mass of the body, if their lines of action are not parallel, if their lines of action are parallel, if their lines of action are parallel and all the forces hae the same magnitude 9. particle falls freely near the surface of the earth. onsider a fixed point O (not ertically below the particle) on the ground ngular momentum of the particle about O is increasing Torque of the graitational force on the particle about O is decreasing The moment of inertia of the particle about O is decreasing The angular elocity of the particle about O is increasing. 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 oer the plank cceleration of centre of sphere is less than that of the plank cceleration of centre of sphere is greater than the plank because friction acts rightward on the sphere cceleration of the centre of sphere may be towards left cceleration of the centre of sphere relatie to plank may be greater than that of the plank relatie to floor 76 F NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65

11 J-Physics 1. In the figure shown, the plank is being pulled to the right with a constant speed. If the cylinder does not slip then R the speed of the centre of mass of the cylinder is the speed of the centre of mass of the cylinder is zero the angular elocity of the cylinder is /R the angular elocity of the cylinder is zero. uniform disc is rolling on a horizontal surface. t a certain instant is the point of contact and is at height R from ground, where R is radius of disc The magnitude of the angular momentum of the disc about is thrice that about The angular momentum of the disc about is anticlockwise The angular momentum of the disc about is clockwise The angular momentum of the disc about is equal to that of about. If a cylinder is rolling down the incline with sliding after some time it may start pure rolling after some time it will start pure rolling it may be possible that it will neer start pure rolling None of these m. uniform bar of length 6a and mass 8m lies on a smooth horizontal table. Two point masses m and m moing in the same horizontal plane with speed and respectiely, strike the bar (as shown in the fig.) and stick to the bar after collision. Denoting angular elocity (about the centre of mass), total energy and centre of mass elocity by, and respectiely, we hae after collision : c a a m = = 5a = 5a = m 5 5. The moment of inertia of a thin square plate D, of uniform thickness about an axis passing through the centre O and perpendicular to the plane of the plate is ( where I 1, I, I and I are respectiely moments of inertia about axis 1,, and which are in the plane of the plate) I 1 + I I + I O 1 I 1 + I I 1 + I + I + I D NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65 RIN TSRS NSWR KY XRIS - Q u e n s. D D D Q u e n s. D D D D,,D,,D Q u e. 1 5 n s.,,,,,,d,, 77

12 J-Physics XRIS Tr ue/false 1. If more mass is concentrated near the axis of rotation, the moment of inertia will be less and the angular acceleration produced by a gien torque will be more than if the masses were uniformly distributed.. wheel is reoling about a fixed axis through its centre and perpendicular to the plane of wheel. onsider a point on the rim. When the wheel rotates with constant angular elocity, the point has only a radial acceleration and zero tangential acceleration.. If the torque is not zero, rotational equilibrium will not be there.. triangular plate of uniform thickness and density is made to rotate about an MISLLNOUS TYP QUSTIONS axis perpendicular to the plane of the paper and (a) passing through, (b) passing through, by the application of the same force, F at (mid-point of ) as shown in the figure. The angular acceleration in both the cases will be the same. 5. thin uniform circular disc of mass M and radius R is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with an angular elocity. nother disc of the same dimensions but of mass M is placed gently on the first disc coaxially. The angular elocity of the system now is 6. ring of mass. kg and radius.1 m and a solid cylinder of mass. kg and of the same radius are gien the same kinetic energy and released simultaneously on a flat horizontal surface such that they begin to roll as soon as released towards a wall which is at the same distance from the ring and the cylinder. The rolling friction in both cases is negligible the cylinder will reach the wall first. Fill in the blanks 1. cylinder of mass M and radius R is resting on a horizontal platform (which is parallel to the x-y plane) with it axis fixed along the y-axis and free to rotate about its axis. The platform is gien a motion in the x-direction gien by x = cos(t). There is no slipping between the cylinder and platform. The maximum torque acting on the cylinder during its motion is.... stone of mass m, tied to the end of a string, is whirled around in a horizontal circle. (Neglect the force due to graity). The length of the string is reduced gradually keeping the angular momentum of the stone about the centre of the circle constant. Then, the tension in the string is gien by T = r n where is a constant, r is the instantaneous radius of the circle and n =.... uniform disc of mass m and radius R is rolling up a rough inclined plane which makes an angle of with the horizontal. If the coefficient of static and kinetic friction are each equal to µ and the only forces acting are graitational and frictional, then the magnitude of the frictional force acting on the disc is... and its direction is... (up or down) the inclined plane.. rod of weight w is supported by two parallel knife edges and and is in equilibrium in a horizontal position. The knifes are at a distance d from each other. The centre of mass of the rod is at distance x from. The normal reaction on is... and on is The length of the minute hand of watch is 1 cm. The linear speed of tip of minute hand will be... cm/sec. 6. The translational kinetic energy is the... percent of total energy for a rolling hollow sphere. 7. uniform cube of side a and mass m rests on a rough horizontal table. horizontal force F is applied normal to one of the faces at a point that is directly aboe the centre of the face, at a height a aboe the base. The minimum alue of F for which the cube begins to tip about the edge is...(ssume that the cube does not slide). 8. smooth uniform rod of length L and mass M has two identical beads 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 an angular elocity about an axis L/ L/ perpendicular to the rod and passing through the mid-point of the rod (see fig). There are no external forces. When the beads reach the ends of the rod, the angular elocity of the system is F 5. NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65

13 J-Physics 9. symmetric lamina of mass M consists of a square shape with a semicircular section oer of the edge of the square as shown in figure. The side of the square is a. The moment of inertia of the lamina about an axis through its centre of a mass and perpendicular to the plane is 1.6 Ma. The moment of inertia of the lamina about the tangent in the plane of the lamina is... M TH TH OLUMN 1. In each situation of column-i, a uniform disc of mass m and radius R rolls on a rough fixed horizontal surface as shown. t t = (initially) the angular elocity of disc is and elocity of centre of mass of disc is (in horizontal direction). The relation between and for each situation and also initial sense of rotation is gien for each situation in column-i. Then match the column the Statement in column-i with the corresponding results in column-ii. olumn I olumn II O ( > R ) (p) The angular momentum of disc about point (as shown in figure) remains consered ( > R ) (q) The kinetic energy of disc after it starts rolling without slipping is less than its initial kinetic energy ( < R ) (r) In the duration disc rolls with slipping, the friction acts on disc towards left. ( < R ) (s) In the duration disc rolls with slipping, the friction acts on disc for some time to right and for some time to left.. Four rods of equal length l and mass m each form a square as shown in figure. Moment of inertia about four axes 1,, and are say I 1, I, I and I. olumn I olumn II NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65 I 1 (p) I (q) 79 m m I (r) 1 m I (s) None. olumn I olumn II In pure rolling work done (p) is always zero by friction (q) may be zero In forward slipping work done (r) is negatie by friction (s) is positie In backward slipping work (t) may be negatie done by friction (u) may be positie 1

14 J-Physics. disc with linear elocity and angular elocity is placed on rough ground. Suppose a and be the magnitudes of linear and angular acceleration due to friction. Then :- olumn I olumn II When = R (p) a = R (a ) When = R (q) a > R When = R (r) a < R (s) 5. solid sphere is rotating about an axis as shown in figure. n insect follows the dotted path on the circumference of sphere as shown. olumn I 8 None olumn II Moment of inertia (p) will remain constant ngular elocity (q) will first increase ngular momentum then decrease Rotational kinetic energy (r) will first decrease (s) (t) (u) then increase will continuously decrease will continuously increase data is insufficient 6. In the adjacent figure a uniform rigid body of mass m and radius R is kept at rest on a rough horizontal surface. constant horizontal force F is applied at the top most point of the body. The body starts rolling without slipping. Different shapes of bodies are gien in the column I and based on this problem some physical quantities related to them are gien in column II. olumn I olumn II Solid sphere (p) Friction force is zero Ring (q) Magnitude of friction force is Hollow sphere maximum Disc (r) cceleration of. O. M. SSRTION - RSON is F/ m (s) Magnitude of friction force is F/5 In each of the following questions, a Statement of ssertion is gien followed by a corresponding Statement of Reason (R) just below it. Of the Statements mark the correct answer as 1. Statement 1 : rigid disc rolls without slipping on a fixed rough horizontal surface with uniform angular elocity. Then the acceleration of lowest point on the disc is zero. a n d Statement : For a rigid disc rolling without slipping on a fixed rough horizontal surface, the elocity of the lowest point on the disc is always zero. Statement 1 is True, Statement is True ; Statement is a correct explanation for Statement 1 Statement 1 is True, Statement is True ; Statement is not a correct explanation for Statement 1 Statement 1 is True, Statement is False. Statement 1 is False, Statement is True. R Rough Insect F a cm NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65

15 J-Physics. Statement 1 : The torque can be applied only about two points. (i) centre of mass and (ii) point about which the body is rolling. a n d Statement : The equation a = r can always be applied in case of rolling. Statement 1 is True, Statement is True ; Statement is a correct explanation for Statement 1 Statement 1 is True, Statement is True ; Statement is not a correct explanation for Statement 1 Statement 1 is True, Statement is False. Statement 1 is False, Statement is True.. Statement 1 : In case of rolling friction force can in forward and backward direction both. a n d Statement : The angular momentum of a system will be consered only about that point about which external angular impulse is zero. Statement 1 is True, Statement is True ; Statement is a correct explanation for Statement 1 Statement 1 is True, Statement is True ; Statement is not a correct explanation for Statement 1 Statement 1 is True, Statement is False. Statement 1 is False, Statement is True.. Statement 1 : For the purpose of calculation of moment of inertia, a body s mass can be thought to be concentrated at its centre of mass. a n d Statement : Moment of inertia is a measure of how the mass is distributed about a certain axis. Statement 1 is True, Statement is True ; Statement is a correct explanation for Statement 1 Statement 1 is True, Statement is True ; Statement is not a correct explanation for Statement 1 Statement 1 is True, Statement is False. Statement 1 is False, Statement is True. 5. Statement 1 : If a body (ball) is rolling on a surface without slipping, no frictional force acts on it. a n d Statement : In the case of rolling without slipping point of contacts are relatiely at rest. Statement 1 is True, Statement is True ; Statement is a correct explanation for Statement 1 Statement 1 is True, Statement is True ; Statement is not a correct explanation for Statement 1 Statement 1 is True, Statement is False. Statement 1 is False, Statement is True. 6. Statement 1 : Torque a n d acting on a rigid body is defined as L, is a constant ector and L is the angular momentum of the body. The magnitude of the angular momentum of the body remains same. NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65 Statement : is perpendicular to L and also perpendicular to, hence torque does not delier any power to the body. Statement 1 is True, Statement is True ; Statement is a correct explanation for Statement 1 Statement 1 is True, Statement is True ; Statement is not a correct explanation for Statement 1 Statement 1 is True, Statement is False. Statement 1 is False, Statement is True. 7. Statement 1 : The moment of inertia of a rigid body is not unique, about a gien axis. a n d Statement : The moment of inertia of a rigid body depends on axis about which it has to be calculated. Statement 1 is True, Statement is True ; Statement is a correct explanation for Statement 1 Statement 1 is True, Statement is True ; Statement is not a correct explanation for Statement 1 Statement 1 is True, Statement is False. Statement 1 is False, Statement is True. 81

16 J-Physics 8. Statement 1 : sphere rolling on a rough horizontal surface with constant elocity then it start going up on a smooth inclined plane. Rotational K of sphere decreases continuously on horizontal and inclined surface. a n d Statement : Rotational K decreases if torque due to friction opposes angular elocity of sphere. Statement 1 is True, Statement is True ; Statement is a correct explanation for Statement 1 Statement 1 is True, Statement is True ; Statement is not a correct explanation for Statement 1 Statement 1 is True, Statement is False. Statement 1 is False, Statement is True. 9. Statement 1 : disc is rolling on an inclined plane without slipping. The elocity of centre of mass is. These others points on the disc lies on a circular arc haing same speed as centre of mass. a n d Statement : When a disc is rolling on an inclined plane. The magnitude of elocities of all the point from the contact point is same, haing distance equal to radius r. Statement 1 is True, Statement is True ; Statement is a correct explanation for Statement 1 Statement 1 is True, Statement is True ; Statement is not a correct explanation for Statement 1 Statement 1 is True, Statement is False. Statement 1 is False, Statement is True. 1. Statement 1 : non-uniform sphere is placed such that its centre is origin of coordinate system. If I x and I y be moment of inertia about x axis and y axis respectiely then moment of inertia about z axis is I x + I y. a n d Statement : ccording to perpendicular axis theory I z = I x + I y when object is lying in x-y plane. Statement 1 is True, Statement is True ; Statement is a correct explanation for Statement 1 Statement 1 is True, Statement is True ; Statement is not a correct explanation for Statement 1 Statement 1 is True, Statement is False. Statement 1 is False, Statement is True Statement 1 : sphere is performing pure rolling on a rough horizontal surface with constant angular elocity. Frictional force acting on the sphere is zero. a n d Statement : Velocity of contact point is zero. OMPRHNSION TYP QUSTIONS omprehension # 1 solid sphere is kept oer a smooth surface as shown is figure. It is hit by a cue at height h aboe the centre. 1. In case 1, h = R and in case, h = R. Suppose in case 1 the sphere acquires a total kinetic energy K 1 and in case total kinetic energy is K. Then :- (Note: That in both the cases, sphere is hit by the same impulse) K 1 = K K 1 > K K 1 < K data is insufficient. If the surface is rough, then after hitting the sphere, in which case the force of friction is in forward direction:- in case 1 in case in both the cases in none of the case 8 h NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65

17 J-Physics omprehension # In rotational motion if angular acceleration (or retardation) is constant we can apply equations of motion = ± t etc. Here = I. 1. solid sphere of mass 5 kg and radius 1 m after rotating with angular speed = rad/s is placed between two smooth walls on a rough ground. Distance between the walls is slightly greater than the diameter of the sphere. oefficient of friction between the sphere and the ground is µ =.1. Sphere will stop rotating after time t =... s : omprehension # solid sphere is rolling without slipping on rough ground as shown in figure. It collides elastically with an identical another sphere at rest. There is no friction between the two spheres. Radius of each sphere is R and mass is m. 1. Linear elocity of first sphere after it again starts rolling without slipping is :- 5 R 7 R 7 1 R 7 5 R. What is the net angular impulse imparted to second sphere by the external forces? 7 mr 5 7 mr 5 mr 7 1 mr omprehension # small sphere of mass 1 kg is rolling without slipping on a stationary base with linear speed = m/s. It leaes the inclined plane at point Find its linear speed at point :- 1m 1 7 m/s 5 7 m/s 1 5 m/s 5 m/s. Find ratio of rotational and translational kinetic energy of the sphere when it strikes the ground after leaing from point : omprehension # 5 NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65 solid sphere has linear elocity = m/s and angular elocity =9 rad/s as shown. Ground on which it is moing, is smooth. It collides elastically with a rough wall of coefficient of friction µ. Radius of the sphere is 1 m and mass is kg. 1. If the sphere after colliding with the wall roll without slipping in opposite direction, then coefficient of friction µ is : What is net linear impulse imparted by the wall on the sphere during impact :- N-s 17 N-s 5 N-s 15 N-s 8

18 J-Physics omprehension # 6 rod of length m and mass kg is lying on smooth horizontal x- y plane with its centre at origin O as shown figure. n impulse J of magnitude 1 N-s is applied perpendicular to at. J y O x 1. The distance of point P from centre of the rod which is at rest just after the impact is :- m 1 m 1 m 1 m. o-ordinates of point of the rod after time t = s will be :- 5 1 m, m 9 m, m 1 1 m, m m, m omprehension # 7 rod of mass m and length in placed on a smooth table. n another particle of same mass m strikes the rod with elocity in a direction perpendicular to the rod at distance x from its centre. Particle sticks to the rod. Let be the angular speed of system after collision, then : 1. s x is increased from to, the angular speed :- will continuously increase will first increase and then decrease will continuously decrease will first decrease and then increase. Find maximum possible alue of impulse (by arying x) that can be imparted to the particle during collision. Particle still sticks to the rod :- m omprehension # 8 m disc of mass m and radius R is placed oer a plank of same mass m. There is sufficient friction between disc and plank to preent slipping. force F is applied at the centre of the disc. m m 5 F 1. cceleration of the plank is :- F m F m. Force of friction between the disc and the plank is :- F F 8 F F m F m F NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65

19 omprehension # 9 J-Physics When a force F is applied on a block of mass m resting on a horizontal surface then there are two possibilities, either block moes by translation or it moes by toppling. If the surface is smooth then the block always translates but on a rough surface it topples only when the torque of the applied force F is greater than the torque of mg about a point in contact with the ground. F m a h When the force F is applied the body may topple about or it may translate. 1. When the block topples about, the normal force :- passes through centre of mass is zero shifts to the right and passes through rightmost edge containing is zero if the surface is smooth. If the block be a cube of edge a and µ =. then :- the body will translate the body will topple the body may translate or topple none of the aboe. If the block is a cube of edge a and µ =.6 then :- the body will translate the body will topple the body first translates and then topples none of the aboe omprehensi on # 1 In figure, the winch is mounted on an axle, and the 6-sided nut is welded to the winch. y turning the nut with a wrench, a person can rotate the winch. For instance, turning the nut clockwise lifts the block off the ground, because more and more rope gets wrapped around the winch. nut winch lock person grips wrench here Wrench turns winch clockwise NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65 Three students agree that using a longer wrench makes it easier to turn the winch. ut they disagree about why. ll three students are talking about the case where the winch is used, oer a 1 s time interal, to lift the block one metre off the ground. Student 1 y using a longer wrench, the person decreases the aerage force he must exert on the wrench, in order to lift the block one metre in 1 s. Student : Using a longer wrench reduces the work done by the person as he uses the winch to lift the block 1m in 1s. Student : Using a longer wrench reduces the power that the person must exert to lift the block 1m in 1s. 85

20 J-Physics 1. Student 1 is :- correct, because the torque that the wrench must exert to lift the block doesn't depend on the wrench's length correct, because using a longer wrench decreases the torque it must exert on the winch incorrect, because the torque that the wrench must exert to lift the block doesn't depend on the wrench's length Incorrect, because using a longer wrench decreases the torque it must exert on the winch.. Which of the following is true about student and :- Student and are both correct Student is correct, but student is incorrect Student is correct, but student is incorrect Student and are both incorrect. If seeral wrenches all apply the same torque to a nut, which graph best expresses the relationship between the force the person must apply to the wrench, and the length of the wrench :- force force force force (1) () () () length length length 1 length omprehensi on # 11 In the figure shown a plank of mass m is lying at rest on a smooth horizontal surface. disc of same mass m and radius r is rotated to an angular speed and then gently placed on the plank. If we consider the plank and the disc as a system then frictional force between them is an internal force. Momentum of the system changes due to external force only. It is found that finally slipping cease, and 5% of total kinetic energy of the system is lost. ssume that plank is long enough. is coefficient of friction between disc and plank. m, r m 1. Final elocity of the plank is r r 1 r r 1. Time when slipping ceases r g r 1g 86 r g. Magnitude of the change in angular momentum of disc about centre of mass of disc mr 1 mr zero r 1g. Distance moed by the plank from the placing of disc on the plank till the slipping ceases between disc and plank r 16g r 8g r g 1 mr r g NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65

21 J-Physics omprehensi on # 1 cylinder and a ring of same mass M and radius R are placed on the top of a rough inclined plane of inclination. oth are released simultaneously from the same height h 1. hoose the correct statement(s) related to the motion of each body The friction force acting on each body opposes the motion of its centre of mass The friction force proides the necessary torque to rotate the body about its centre of mass Without friction none of the two bodies can roll The friction force ensures that the point of contact must remain stationary. Identify the correct statement(s) The friction force acting on the cylinder may be more than that acting on the ring The friction force acting on the ring may be more than that acting on the cylinder The elocity of centre of mass of the ring is gh The elocity of centre of mass of each body is gh omprehensi on # 1 ring of mass M and radius R sliding with a elocity suddenly enters into of friction is, as shown in figure. rough surface where the coefficient Rough( ) 1. hoose the correct statement(s) s the ring enters on the rough surface, the limiting friction force acts on it The direction of friction is opposite to the direction of motion The friction force accelerates the ring in the clockwise sense about its centre of mass s the ring enters on the rough surface it starts rolling NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65. hoose the correct statement(s) The momentum of the ring is consered The angular momentum of the ring is consered about its centre of mass The angular momentum of the ring consered about any point on the horizontal surface The mechanical energy of the ring is consered. hoose the correct statement(s) : The ring starts its rolling motion when the centre of mass stationary The ring starts rolling motion when the point of contact becomes stationary The time after which the ring starts rolling is The rolling elocity is g 87

22 J-Physics. hoose the correct alternatie(s) The linear distance moed by the centre of mass before the ring starts rolling is 8g The net work done by friction force is 8 m The loss in kinetic energy of the ring is m The gain in rotational kinetic energy is + m 8 NSWR KY MISLLNOUS TYP QUSTION XRIS - Tr ue / False 1. T. T. T. F 5. F 6. F Fill in the lanks 1. 1 MR.. mg 6, up. d x w d, xw d mg 8. M M 6m 9..8 Ma Match the olumn 1. p,q,r p,q,r p,q p,q,r. q, s, q, q. q,t,u q,t,u q,t,u. s, r, r 5. q, r, p r 6. q, p, s r sserti on - Reason Questions 1.. D... D 5. D D 8. D D 11. omprehension ased Questions omprehension #1 : 1.. omprehension # : 1. D omprehension #: 1.. omprehension # : 1.. omprehension #5 : 1. D. omprehension #6 : 1.. omprehension #7 : 1.. omprehension #8 : 1.. omprehension #9 : 1... omprehension #1 : 1.. D. D omprehension #11 : omprehension #1 : 1.,,,D., omprehension #1 : 1.,,..,,D.,,D 88 NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65

23 J-Physics XRIS [] ONPTUL SUJTIV XRIS 1. Find out the moment of inertia of the following structure (written as PHYSIS) about axis and made of thin uniform rods of the mass per unit length.. disc of certain radius is cut from a disk of mass 9M and radius R. Find its moment of inertia about an axis passing through its centre and perpendicular R R/ R/ to its plane.. alculate the moment of inertia of a wheel about its axis which haing rim of mass M and twenty four spokes each of mass M and length.. carpet of mass M made of inextensible material is rolled along its length in the form of a cylinder of radius R and is kept on a rough floor. The carpet starts unrolling without sliding on the floor when a negligibly small push is gien to it. alculate the horizontal elocity of the axis of the cylindrical part of the carpet when its radius reduces to R. S 5. homogeneous rod of length L = 1.8 m and mass M is pioted at the centre O in such a way that it can rotate freely in the ertical plane (fig.). The rod is initially in the horizontal position. n insect S of the same mass M falls ertically with speed on the point, midway between the points O and. Immediately after falling, the insect moes towards the end such O that the rod rotates with a constant angular elocity. (i) Determine the angular elocity in terms of and L. L/ L/ L/ (ii) If the insect reaches the end when the rod has turned through an angle of 9, determine. 6. Two uniform rods and of length.6 m each and of masses.1 kg and. kg respectiely are rigidly joined end to end. The combination is pioted at the lighter end, P P as shown in figure. Such that it can freely rotate about point P in a ertical plane. small object of mass.5 kg, moing horizontally, hits the lower end of the combination and sticks to it. What should be the elocity of the object, so that the system could just be raised to the horizontal position. NOD6 : \Data\1\Kota\J-danced\SMP\Phy\Unit No-\Rotation\ng\xercise.p65 7. Determine the minimum co-efficient of friction between a thin rod and a floor at which a person can slowly lift the rod from the floor without slipping, to the ertical position applying to its end a force always perpendicular to its length. 8. block X of mass.5 kg is held by a long massless string on a frictionless inclined plane of inclination to the horizontal. The string is wound on a uniform solid cylindrical drum Y of mass kg and of radius. m as shown in figure. X The drum is gien an initial angular elocity such that the block X starts moing up the plane. (i) Find the tension in the string during the motion. (ii) t a certain instant of time the magnitude of the angular elocity of Y is 1 rad s 1. alculate the distance traelled by X from that instant of time until it comes to rest. 89 Y