Rotation Questions and Answers

Three point masses each of mass m are placed at the corners of an equilateral triangle of side 1 Then the moment of inertia of this system about an axis passing through one corner and perpendicular to the plane of the triangle 3 1 2 m 3 ml 2 2 ml 4 m

g is uniform rod of length 200 cm and mass balanced on a wedge placed at 40 cm mark A mass of 2 kg is suspended from the rod at 20 cm and another unknown mass m is suspended from the rod at 160 cm mark as shown in the figure Find the value of m such that the rod is in equilibrium g 10 m s 0 20 cm 40 cm e a 3 2 kg 1 kg 12 kg kg A kg 160 cm m

Create a bucket by rotating around the y axis the curve y 3 ln z 3 from y 0 to y 7 If this bucket contains a liquid with density 800 kg m filled to a height of 6 meters find the work required to pump the liquid out of this bucket over the top edge Use 9 8 m s for gravity Work Joules

Through a smooth fixed cylinder of radius R the axis of which is horizontal a weightless rope is thrown to the ends of which a load and a thin uniform rod is attached as shown in the figure In the position of equilibrium rod makes an angle a 30 with the horizontal The distance from the end of rod to which the rope is attached to the point of contact of the rod and the cylinder is If mass M of rod is 6 kg 2 R then find the value of mass min kg m RAZ M

2 The whole set up shown in the figure where 23 particles m and m are rotating with constant angular velocity w on a horizontal frictionless table If T and T are tension in given strings then ratio of tensions T T is 2 1 M M 3 M M T m T m 1 2 M 2M 2M 4 M M M

A wheel of radius r rolls without slipping with a speed v on a horizontal road When it is at a point A on the road a small blob of mud separates from t wheel at its highest point and lands point B on the road The distance AB AVVE A 4v C W B 2v D LLI CO RD OPT E

This section contains 35 SINGLE COR A uniform rod of length 3 m and mass 1 kg is free to rotate vertically about a fixed point A The rod is released from rest in the horizontal position The initial angular acceleration of the rod will be AC B O 9 Q 3g2 TYPE questions Each question has choices 1 2 3 and g2

The angle of a prism is A One of its refracting surfaces is silvered Light rays falling at an angle of incidence 24 on the first surface retums back through the same path after suffering reflection at the silvered surface The refractive index of the prism is 1 2 sin4 1 3 cos A co 2 2 cos 4 4 Tan A

A rope is wound round a hollow cylinder of mass 2 kg and radius 0 5 m If the rope is pulled with a force of 80 N the angular acceleration of the cylinder will be 1 20 rad s 3 60 rad s 2 40 rad s 4 80 rad s

If the origin of co ordinate sy of mass the sum of the moments of the masses of the system about the centre of mass 1 May be greater than zero 2 May be less than zero 3 May be equal to zero 4 Is always zero

4 Is always zero In the diagram given below a solid sphere of radius r moving on horizontal surface with velocity of 2v0 and angular velocity wo The angular momentum about an axis passing through horizontal surface as shown is voor 1 2mvor 101 8 myr wo r m 2Vo axis I to page 2 2 mvor 5 12

A cubical block is on a rough fixed incline plane of angle of inclination 0 What is the shift in normal reaction from the center of the cubical block if block is at just rest 1 6 2 a 3 tane 411 0 m 2 a tane a 4 tane

A man stands on a rotating platform with his arms stretched holding a 5 kg weight in each hand The angular speed of the platform is 1 2 rev s The moment of inertia of the man together with the platform may be taken to be constant and equal to 6 kg m If the man brings his arms close to his chest with the distance of each weight from the axis changing from 100 cm to 20 cm The new angular speed of the platform is 1 2 rev s 2 3 rev s

2 A particle of mass m moving with velocity v hit a of identical mass free to rotate about one end an stick to it Find angular velocity of rod just afte I want collision length of rod 1 2 V 3 4 V How 2l 3v 4l 6v 71 5v 6l That s 0 I have applied arg know 31 Qm l AN K

82 A solid sphere a hollow sphere and a dis having same mass and radius are placed at the top of an incline and released The friction coefficients between the objects and the incline are same and sufficient to allow pure rolling Least time will be taken in reaching the bottom by 1 All will take same time 2 The hollow sphere 3 The disc

A disc of radius 20 cm and mass 1 kg is rolling with slipping on a flat horizontal surface At a certain instant the velocity of its center is 4 m s and its angular velocity is 10 rad s The lowest contact point is O The velocity of point O is 2 m s 10 rad sec P G 0 Angular momentum of disc about ICOR is 1 8 kgm sec 4 m s The distance of instantaneous center of rotation ICOR from the point O is 0 2 m 27

g Three identical rods each of length x are joined form a rigid equilateral triangle Its radius of gyration about an axis passing through a corner and perpendicular to the plane of triangle is X 1 3 3 3 X 2 X N 2 XA 1 2

A thin uniform bar lies on a frictionless surface and is free to move in any way on the surface Its mass is 0 16 kg and length 3 meters Two particles each of mass 0 08 kg are moving on the same surface and towards the bar in a direction perpendicular to the bar one with a velocity 10 m s and the other with 6m s as shown in figure The first particle strikes the bar at A and the other at the point B The points A and B are at distances 0 5 m from centre of the bar The particles strike the bar at the same instant of time and stick to the bar on collision Calculate the loss of kinetic energy of the system on the above collision process in Joule 10 m s 0 5 m 3 6 m s 0 5 m 10 5 m 3m

65 3 5 One quarter sector is cut from a uniform circular disc of radius R This sector has mass M It is made to rotate about a line perpendicular to its plane and passing through the centre of the original disc Its moment of inertia about the axis of rotation is LOS MR 2 1 2 MR 4

A small solid cylinder of radius r is coaxially from point A inside the fixed large cylindrical bowl of radius R as shown in figure If the friction between the small and large cylinder is sufficient enough to prevent any slipping then what fraction of total kinetic energy will be rotational Kinetic energy 2 3 A R LIST 3

37 A particle of mass m is projected with a velocity at an angle 60 with horizontal When the partic is at its maximum height the magnitude of angular momentum about the point of projection 1 Zero 3mv 16g

A solid sphere of mass M and radius R is placed on a rough horizontal surface It is pulled by a horizontal force F acting through its centre of mass as a result of which it begins to roll without slipping Angular acceleration of the sphere can be expressed as 3F 1 4MR 3 7F 11MR 425 DJ 5F 7MR 5F 2MR

3 10 m A cubical block of mass m and edge length a slides with a down a rough inclined plane of inclination uniform speed The torque of the normal force acting on the block about its centre is 1 ma sine 2 m 0 u 2 1 mag sine 2

2 A wire of mass m and length bent in the form of an arc of a circle subtends an angle 0 at it centre Its moment of inertia 1 about an axi through centre of circle and normal to the plane c circle varies as 1 1 8 3 0 2 0 2 2 10 4 10

FA 2 3 45 0 3m ds 30 1 B F F2 F 7 KN F 4 KN dg 1 5m d 1 8m a About O 1 Moment of RHS F the one LHS F 1 e 2 F2 A 6 Moment 7 d 1 RHS F 1 LHS F L 1 4 5 Resultant couple Moment about 0 b About A ANSWER PLEASE 1 Acting at A about 0 B about o of LHS F the one LHS F 8 Resultant couple moment about A 17 about B about D acting at B about A B about A

JA horizontal disc rotating freely about a vertical axis through its centre makes 90 revolutions per minute A small piece of wax of mass m falls vertically on the disc and sticks to it at a distance r from the axis If the number of revolutions per minute reduce to 60 then moment of inertia of disc is 1 mr 3 2 mr 72 mr 2 4 3 mr

45 0 3 m B A 30 F2 F1 O For the beam shown in the figure forces are acting at A and B Determine the resultant couple moment of the given four forces Solve the problem in two ways a sum moments about point O and b sum moments about point A Given F 7 kN F 4 kN d 1 5 m d 1 8 m Note that the points Band O are lying on the same horizontal line a About O 1 Moment of the RHS F the one acting at A about O 2 Moment of the LHS F the one acting a t B about 0 3 Moment of the RHS F the one acting at A about O Blahou

I thin rod each of mass m and length e are joined to form an equilateral triangle find the moment of inertia about an axis xx parallel to plane in the triangle as shown in figure is 1 2 3 2m 3 me 3 me 4 l e 2 2

A hollow sphere released at the top of a smooth incline of length and inclination 0 reaches the bottom in time to A solid sphere released at the top of a rough incline of length and inclination a rolls down to the bottom in time 2t Choose the correct relation 1 13 sinx 2 sine 3 10 sina 9 sine 220 2 20 sina 7 sine 4 sina 2 sine

A recording disc rotates steadily at 45 rev minute on a table When a small mass of 0 02 kg is dropped gently on the disc at a distance of 0 04 m from its axis and stuck to the disc the rate of revolution falls to 36 rev min The moment of inertia of the disc about its centre is 45 UA CH E 0 04 m un tas at fald 36 Options 1 3x10 kgxm 1 3x10 5 kgxm 1 3x10 kgxm 13x10 2koym 0 02 de 34 34 i m

Anon uniform bar of weight W and length L is suspended by two strings of negligible weight as shown in figure The angles made by the strings with the vertical are a L c L 0 and 0 respectively The distance d of the centre of gravity of the bar from its left end is tan 0 tan 0 tan 0 tan 0 tan 0 tan0 2 tan 0 tan 0 tan 0 b L L d L Vw dk 0 tan 0 tan 0 tan 0

A solid sphere is rolling on a frictionless horizontal surface shown in figure with a translational velocity v m s If it is to climb the rough inclined surface then v should be 1 10 gh S 2 2gh

The figure shows a metallic plate of uniform density The value of in terms of L so that the centre of mass of the system lies at the common interface of the triangular and rectangular portion is 1 LIM L 9 d b 2 L 1 3

A thin uniform rod of mass m length L is rotating about an axis passing through one end of the rod At t 0 a wave pulse is generated at one end of the rod which is just close to the axis Angular velocity of the rod is w The time taken by the wave pulse to reach other end is of the form The value of A will be M L 00

A roller is made by joining together two cones at their vertices O It is kept on two rails AB and CD which are placed asymmetrically see figure with its axis perpendicular to CD and its centre O at the centre of line joining AB and CD see figure It is given a light push so that it starts rolling with its centre O moving parallel to CD in the direction shown As it moves the roller will tend to 1 turn right 3 turn left and right alternately go straight 4 turn left A B D C

A cubical block of side L and mass m rests on a rough horizontal surface with coefficient of friction u A horizontal force F is applied on the block as shown in figure If the coefficient of friction is sufficiently high so that the block does not slide before toppling then the minimum force required to topple the block is 1 3 2mg 3 mg 4 F 3L 4 2 4 mg 3 3mg 4

From a disc of radius R and mass M a circular hole of diameter R whose rim passes through the centre is cut What is the moment of inertia of the remaining part of the disc about a perpendicular axis passing through the centre 1 15 MR2 32 2 13 MR2 32 3 11 MR2 32 4 9 MR2 32

A square plate of side a has been removed from a uniform big metal square plate having mass M and side 2a as shown in figure The moment of inertia of remaining part about its centre of mass and the given axis is M mass of original plate 15Ma 1 24 5Ma 24 Ma 18 17002 2 3 0 w ka 2a K

If a street light of mass M is suspended from the end of a uniform rod of length L in different possible patterns as shown in figure then V C A B 1 Pattern A is more sturdy 2 Pattern B is more sturdy 3 Pattern C is more sturdy 4 All will have same sturdiness

A slender rod of mass M and length L 1 m is lying on the frictionless surface A particle of mass moving M 8 perpendicular to the length of the rod with speed v and strikes the rod at a distance from the centre of 14 14 mass of the rod Find the y coordinate in m of the end 4 L B in time t 4 Assume that after collision particle comes to rest take 3 14 YA A L 2 L 4 MO B X

Consider the arrangement shown m 7 5 kg m 10 kg and mass per unit length of the string is 1 5 x 10 2kg m At t 0 the system is released from rest and at the same instant of time a transverse pulse is produced at A The pulley B is at a distance 10 m from A at t 0 Assuming all surfaces to be smooth and pulleys to be light what is the time taken by the pulse to reach B approximately g 10 m s m 8 m

TEST FEEDBACK A non conducting ring of mass m and radius R is charged as shown The linear charge density for corresponding region shown is A It is placed on rough horizontal surface A uniform electric field Ei is switched on and the ring starts rolling without sliding Angular velocity acquired by the ring when it turns through an angle ofis 16 rad s XEo Value of in SI unit is m 321 m e u Ri ngch 3dlch t Tefy Gyuz m fka fhu ga 1 zenf na ga fag Zwu 3nde qara 21 gzza afau aab 92 20 qm 21 55 94x9 faya da 0 ald fahul Jill i au fa i thi c MRIH GR 1 G adu a 23 g u guy Gal da zah ga ura fchu nu cu 16 rad s SI XEO ZI m Correct Answer 61

Two particle of mass 2m and m attached to a light string as shown Complete system is rotated in a horizontal circle with constant angular velocity 2w about an axis passing through O point and perpendicular to plane of circle Find out TOA TAB is TOA TAB IS tension in OA and AB string respectively 200 1 uestion Type Single Correct Type 2 4 1 r 1 1 A 2m B m

r at a Let I be the moment of inertia of a uniform square plate about an axis AB that passes though its centre and is parallel to two of its sides CD is a line in the plane of the plate that passes through the centre of the plate and makes an angle 0 with AB The moment of inertia of the plate about the axis CD is then equal to ntal tion ce r ven tP A I C I sin 0 Figure 5 125 B I cos 0 D I cos 0 2

From a uniform square plate of side a and mass m square portion DEFG of side is removed Then the a moment of inertia of the remaining portion about the axis AB is 1 2 3 4 7 ma 16 2 2 3 ma 16 2 3 ma 4 16 ma a 2 A E D a F G B C

A uniform thin rod of length and mass m is hinged at one end O and released from horizontal position as shown in figure The angular acceleration of the rod about the hinge point when rod becomes vertical is 1 1 2 l 3 Zero 2 g 3 l 3g 2 l

39 A rod of mass mand length 1 is hinged at one of its end A as shown in figure A force F is applied at a distance x from A The acceleration of centre of mass a F varies with x as a c at at old seam to bosqe alug a soc X b d a A 0 X X X

30 A horizontal turn table in the form of a disc of radiu r carries a gun at G and rotates with angular velocity o about a vertical axis passing through the centre O The increase in angular velocity of the system if the gun fires a bullet of mass m with a tangential velocity v with respect to the gun is moment of inertia of gun table about O is Io a mvr 10 mr c G 000 b d 2mvr lo mvr 2la

A belt of mass 2 kg is wrapped over two cylindrical wheels of small radii rolling on a horizontal rough surface Each wheel has velocity 3 m s see in the figure and assume that the belt does not slip The kinetic energy of the belt is 9K joule Find the value of K Belt o 3m s 3m s

A disc of mass M and radius R is rolling with angular speed on a horizontal plane as shown The magnitude of the angular momentum of the disc about the origin O is YA 1 3 2 MR 00 MR 0 MO X 2 MR w 4 2 MR2 B