Rotation Questions and Answers

The moment of inertia of semicircular ring about an axis which is perpendicular to the plane of the ring and passes through the centre AIEEE 2002 a MR2 c MR2 4 b MR 2 d None of these

A wheel has a constant angular acceleration of 3 0 rad s2 During a certain 4 0 s interval it turns through an angle of 120 rad Assuming that at t 0 angular speed wo 3 rad s how long is motion at the start of this 4 0 second interval 1 7 sec 2 9 sec 3 4 sec 4 10 sec

Two thin circular discs of mass 2 kg and radius 10 cm each are joined by a rigid massless rod of length 20 cm The axis of the rod is passing through the centres of the discs Fig 2 35 The whole arrangement is kept on a moving truck in such a way that the axis of the arrangement is horizontal and perpendicular to the direction of the motion of the truck The friction of the floor is such that the arrangement can roll on the floor of the truck without slipping Take x axis along the direction of motion of the truck and z axis along the vertically upward direction If the truck has an acceler ation of 9 m s 2 calculate i the force of friction on each disc ii the magnitude and the direction of the frictional torque acting on each disc about the centre of mass O of the arrangement Express the torque in the vector form in terms of unit vectors i j and k along the x y and z directions respectively i f 6 N i ii 7 0 6 N m j 0 6 N m k 20 cm Fig 2 35

A disc of mass M and radius R has massless string wrapped over it with one end fixed on disc and other end connected to block of same mass M Initially system is held at rest Now system is released from rest Immediately after system is released assume there is no slipping at any contact surface 2 A Acceleration of block in ground frame is gi gi B Acceleration of block in the frame of centre of disc is 5 2 C Acceleration of block in ground frame is 5 D Tension in string is 3mg 5 figure pulleys and strings are ideal au 1 m s M AL

Calculate the radius of gyration of a uniform thin rod of mass M and length L about an axis of rotation perpen dicular to its length and passing through its centre k L

A stick of mass density 8 kg m rests on a disc of radius R 20 cm as shown in the figure The stick makes an angle 0 37 with the horizontal and is tangent to the disc at its upper end Friction exists at all points of contact and assume that it is large enough to keep the system at rest Find the friction force in Newton between the ground and the disc take g 10 m s 37 HAN R

A uniform rod AB bent in the shape of an arc of circle is free to rotate in a vertical plane about a fixed horizontal axis passing through the mid point of the wire The situation is shown in the figure The angular acceleration of the rod immediately after it is released from rest from Calculate p the position shown is a pg 4b 0 B

2 A bullet of mass 0 02 kg is travelling with speed 100 m s at an angle of 30 with respect to the horizontal The bullet collides with and sticks to the end of a vertical thin rod of length 2 0 m and mass 3 kg as shown in the figure The rod then rotates about its end which is hinged to the floor Find a the angular velocity of the rod immediately after the collision and b the angular velocity of the rod just before when it hits F the floor Treat the bullet as a point mass 23 m 0 02 kg 100 m s 30 M 3 kg 505 1 M B 2m 2 3 B 2 2 x4 w 12

s 69 3 hrs number of to t 11 gases itself mass en its K AIIMS 2019 A rod starts slipping from rest on the smooth surface with its initial position vertical If its mass is 100g length 20 cm then find its rotational kinetic energy at the instant it makes 30 with the vertical and the centre of mass has velocity 2 cm s CM 30 V 2cm s 300 1 2 66 x 10 J 3 2 66 x 10 5J 2 4 60 10 J 4 4 60 x 10 5 J

A uniform wire frame of linear mass density having of length 2a is kept on a smooth horizontal surface An impulse J is applied at one end as shown in the figure P is the mid point of AB Now answer the following question Read the above paragraph and given the answer of following question The angular velocity of system just after the impulse 3J A B 221a 2J 221a 4 5 C The velocity of point P just after the impulse is A C a ha 1 71 11 6 D B 220a 4J 221a 6 a yt a 11 J AL 1

4 A wheel of radius R and mass M is placed at the bottom of a fixed step of heig shown in the figure A constant force is continuously applied on the surface of the wheel so that it just climbs the step without slipping Consider the torque T about an axis normal to the plane of the paper passing through the point Q Which of the following options is are correct S P X R Q A If the force is applied at point P tangentially then 7 decreases continuously as the wheel climbs Spoo B If the force is applied normal to the circumference at point X then T is constant C If the force is applied normal to the circumference at point P then 7 is zero D If the force is applied tangentially at point S then T 0 but the wheel never climbs the step

clinder of height h is placed on an inclined x 10 x 1 5 x 10 t plane the angle of inclination of which is slowly increased It begins to slip when the angle of inclination is 45 What is the radius of the cylinder 2 2 h 1 h 3 h 4 h

about A rod lying on a frictionless horizontal is initially given an angular velocity vertical axis which passes through centre of mass The centre of mass is at rest but not fixed The mass of the rod is M and its length is L Subsequently end 4 of the rod collides with nail P which is near to such that end A becomes stationary immediately after impact Velocity of end B just after collision will be B ol eck all that apply A 2 L 2 C A 3 L 4 4 L 3

Two particles A and B of mass m each are connected together by a rigid massless rod of length 20 cm Initially rod is vertical and particle A is given velocity V horizontally while particle B is at rest Consider the adjacent figure Find the minimum value of V in m s for which particle B loose contact with ground immediately after giving velocity V Take g 10 m s yomg B

A and B are two loops made from same wire The radii of A and B are r and r and their moment of inertia about same axes are 1 and 12 If 1 4 1 1 1 2 4 3 2 3 43 then the value of 2 is 1 2 1 2 2 3 2 4 23 23

A disc of mass M and radius R is placed on a rough horizontal surface A light rod of length 2R is fixed to the disc at point A as shown in the fig and force Mg is applied at the other end Find the 2 3 minimum value of 10 where is coefficient of friction upto one decimal place between disc and horizontal surface so that disc starts to roll without slipping R2 A R R 2R 32

10 In the setup shown the string is inextensible and light there is friction at the axles of the pulleys but friction between the string a the pulley wheels is sufficient to prevent slipping All the pulleys a light except the pulley P which has whole of its mass m concentrated its axle Initially the block B of mass M is on the ground and the bla A is held at rest How much minimum mass should the block A have that after releasing it the block B starts rising a 0 5M b 0 5M m c Whatever mass the block A has the block B will not rise d Pulley

145 A closed chain A of mass m 0 36 kg is attached to a vertical rotating shaft by means of a thread Fig 1 37 and rotates with a constant angular velocity 35 rads The thread forms an angle 0 45 with the vertical Find the distance between the chain s centre of gravity and the rotation axis and the tension of the thread 146 A round cone A of mass m 3 2 kg and half angle a 10 rolls uniformly and without slipping along a round conical surface B so that its apex O remains stationary Fig 1 38 The centre of gravity of the cone A is at the same level as the point O and at a distance I 17 cm from it The cone s axis moves with angular velocity on Find boog oooooo 2a m Fig 1 36 Fig 1 38

1 A 60 kg mass is spun in a circle of radius 10 m with a velocity of 100 m s a What is the frequency of rotation if the mass spins with a period of 0 625 s b What is its acceleration

A bicycle is moving with constant acceleration The frictional force on rear wheel is 2 In forward direction 1 Zero 3 Backward direction 4 All of these

31 Three point masses each of mass m are connected by 3 light rigid rod and system is rotating about one rod as shown in the given figure Moment of inertia of the system is 1 m 3 3 m1 2 m1 4 3m

A wire frame AOPQB lying in the horizontal plane is free to rotate about a vertical axis passing through center C of the same circle and I to plane of AOPQB The mass M of the frame is uniformly distributed over its whole length The moment of inertia of the frame about this axis is OA QB r and CP r the radius of semicircular part 14 3 A Mr2 3 6 C Mr2 3 4 TC B Mr D 2 T r 2r Mr C

Find radius gyration about axis shown in figure Solid Hemisphere IC 1 0 67R 3 1 4R 2 0 4R 4 1 33R

CRP 1819 JEEA RP PI PH 4 20 A uniform cylinder of mass m and radius r rolls on a fixed cylindrical surface of radius R At a certain instant the line OC has an angular velocity and angular acceleration a Find a the acceleration of point of contact P of the cylinder with respect to the surface b the kinetic energy of the cylinder col 2 O T w7 P110 1 2 8 Hora 27 R 28

A boy and a man carry a uniform rod of leng horizontally in such a way that the boy gets of the load If the boy is at one end of the ro distance of the man from the other end is 1 L 3 3 2L 3 2 L 4 4 3L 4

15 Moment of inertia of the sector of mass m and radius R about axis through centre of curvature and normal to plane is 1 2 3MR2 4 MR 1200 R 2 2MR2 5

Numericals 1 A nut can be opened by lever of length 20 cm If the minimum force required is 4 N calculate the moment of force to turn the nut Ans 0 8 Nm

46 When a body is rolling without slipping on a rough horizontal surface the work done by friction is 1 Always zero 2 May be zero 3 Always positive 4 Always negative

is zero An electron of kinetic energy of 7 2 x 10 18 J is revolving on circular path in magnetic field 9 x 10 5 Wb m 2 then radius of its circular path aluation is 1 1 25 cm 2 2 5 m 4 25 0 cm

12 The ratio of rotational kinetic energy to the to kinetic energy of a disc in pure rolling motion is 1 2 1 2 3 3 2 13 4 1

Moment of inertia of a disc about OO is of A 3 mr 2 B mr 2 m C 5 mr2 2 D 5 mr 4

JEFE A uniform wooden plank of mass 30 kg is kept on two supports plank is L g as shown The length of the Reaction force applied by Y as shown is 1 100 N 3 200 N akash Educational Services Con K Y 2 150 N 4 250 N

A rod of mass m length 100 cm is hanging as show in diagram A block of mass 2m is hanged from on end at distance 25 cm Find tension T and TB A mg 2 B mg 4 C 3mg 4 T 75cm 25cm

31 Find the acceleration of the body if a force F 8N pulls the string at P that passes over the body and it is 2 connected by another string to a rigid support at Q take radius of gyration k mass of the body m 3 Kg Q R P O F Ar 21 4 m R 2m r 1m and

9 8 A disc of moment of inertia kg m is rotating at 600 rpm If the frequency of rotation the work done changes from 600 rpm to 300 rpm then what A 1467 J B 1452 J C 1567 J D 1632 J

Two rings equal thickness of radius R and n made up of same material have the moment o inertia about an axis passing through centre as 1 8 The value of n is 1 2 3 4 2 2 2 4 N 2

A rod of mass 100 g and length 1 0 m makes an angle 30 with the axis passing through its centre Find i the moment of inertia of the rod about the given axis ii If the rod rotates with an angular velocity of 2 0 rad s about the given axis find the rotational kinetic energy of the rod i 2 08 x 10 3kg m ii 4 17 x 10

A circular ring of mass 20 g is rotating about its own axis with an angular velocity of 30 rpm Two bodies of equal masses are placed simultaneously at the two extreme points along its diameter Mass of each of these two bodies is 5 g In this situation what will be the angular velocity of the system 20 rpm

4 The magnitude of the resultant moment about point O in Nm of the two forces acting on the rod shown below is A 25 B 125 college 0 5 m C 175 D 225

By keeping moment of inertia of a body constant if we double the time period then angular momentum of body A remains constant C doubles B becomes half D quadruples

3 A metre stick is pivoted about its centre A piece of wax of mass 20 g travelling horizontally and perpendicular to it at 5 m s strikes and adheres to one end of the stick so that the stick starts to rotate in a horizontal circle Given the moment of inertia of the stick and wax about the pivot is 0 02 kg m the initial angular velocity of the stick is 1 1 58 rad s 2 2 24 rad s 3 2 50 rad s 4 5 00 rad s

The moment of inertia of a square lamina about the perpendicular axis through its centre of mass is 20 kg m Then its moment of inertia about an axis touching its side and in the plane of the lamina will be ma 20 2 30 kg m 6 4 25 kg m 1 10 kg m 3 40 kg m

are joined together by a Three identical particle thread as shown All the three particles are moving in a horizontal plane If the velocity of the outermost particle is vo then the ratio of tensions in the three sections of the string is G4 T O m 14 TI 21 m T 3L m

0 A couple produces 1 Linear and rotational motion 2 No motion 3 Purely linear motion LAX Ruroly rotational motion

A cubical block of mass m and edge a slides down a rough inclined plane of inclination e with uniform velocity Find the torque of the normal force acting on the block about its centre 1 1 mg asin 0 2 1 3 mga cos 2 mga tan0 4 mga sin 20

A ring of mass m 1 kg and radius R 125m is kept on a rough horizontal ground A small body of same mass m 1kg is struck to the top of the ring When it was given a slight push forward the ring started rolling purely on the ground What is the maximum speed of the centre of the ring in m s 1 3 4 5 2 in a

A small body of mass m is attached to a light thread which passes through a hole at the centre of a smooth table The body is set into rotation in a circle of radius r with a speed v The thread is pulled down so that the radius of the path is changed to r2 What will be the new linear speed and the new angular speed of the body Vi

A rope is wound on a circular disc of moment of inertia 0 1 kg m A tangential force of 20 N on the rope rotates the disc about its axis If the radius of the wheel is 0 1 m calculate its acceleration Now a mass of 2 kg is suspended at the lower end of the rope What will be the angular acceleration g 10 m s 2 Do 20 rad s 2 16 7 rad s 2 6

2 Two discs are rotating about their axes normal to the discs and passing through the centres of the discs Disc D has 2 kg mass and 0 2 m radius and initial angular velocity of 50 rad s Disc D has 4kg mass 0 1 m radius and initial angular velocity of 200 rad s 1 The two discs are brought in contact face to face with their axes of rotation coincident The final angular velocity in rad s of the system is 2 2 60 4 120 1 40 3 100

37 A thin wire of length and uniform linear mass density p is bent in the form of a square Its moment of inertia about an axis along one of its edge is 1 3 2p 3 4 50 4 5p 3 3 3 2 4 3p 2 4 1p 3 4 3 3