Rotation Questions and Answers

A solid cylinder of mass M and radius 2R is rolled up on an incline with the help of a plank of mass 2M as shown in the figure A force constant F is acting on the plank parallel to the incline There is no slipping at any of the contact Solid cylinder M 2R Plank 2M mmmmmm numm Fixed incline Direction of friction force on cylinder due to fixed inclined plane is up the inclined B Magnitude of friction force between the plank and cylinder is given by 3F 2Mg sin 8 19 Magnitude of friction force between the plank and cylinder is given by 3F 2Mg sine 19

The moment of inertia of a uniform flat disc about its own axis is I The radius of the disc is a A section ABCD of the disc as shown in figure is cut off and separated The moment of inertia of the remaining part of the disc about the same axis will be 1 into 2 B C 30 A 3 D 45 Tin 4

A 0 25 kg meter stick hangs from its top by a peg A 0 12 kg ball is thrown at the meter stick from the side at a speed of 2 5 m s striking it 0 69 m from the top and bouncing back with a speed of 1 1 m s a What is the angular speed of the meter stick immediately after impact b To what maximum angle

A ring and a disc are initially at rest side by side at the top of an inclined plane which makes an angle 60 with the horizontal They start to roll without slipping at the same instant of time along the shortest path If the time difference between their reaching the ground is 2 3 10s the the height of the top of the inclined plane in meters is Take g 10 ms JEE Advanced 2018

A meter stick supported by two strings hangs horizontally with a block hanging from the meter stick The meter stick is in equilibrium The first string has a tension of 88 0 N The second string is attached at the 95 0 cm on the meter stick and has a tension of 51 0 N The block is attached at the 41 1 cm spot on the meter stick and has a weight of 139 N You can ignore the weight of the meter stick At what cm spot on the meter stick is the first string attached

Example 96 A homogeneous rod AB of length L and mass Mis pivoted at the center O in such a way that it can rotate freely in the vertical plane The rod is initially in the horizontal position An insect S of the same mass M falls vertically with speed v on the point C mid way between 0 and B Determine the initial angular velocity in terms of v and L A 1 2 O M V C L 24 2 B

A small mass m is attached to a massless string whose other end is fixed at P as shown in the figure The mass is undergoing circular motion in the x y plane with centre at O and constant angular speed If the angular momentum of the system calculated about O and P are denoted by Lo and Lp respectively then IIT JEE 2012 Z m a Lo and Lp do not vary with time b Lo varies with time while Lp remains constant c Lo remains constant while Lp varies with time

78 g 34 cm long rod hangs ertically on a frictionless orizontal axle passing through its enter A 13 g ball of clay traveling orizontally at 2 7 m s hits and ticks to the very bottom tip of the d Part A To what maximum angle measured from vertical does the rod with the attached bal of clay rotate Express your answer in degrees to two significant figures View Available Hint s Templates Symbols undo redo reset keyboard shortcuts help 0 7

Suppose you observe your friend mass 65 2 kg stand at the very edge of a motionless merry go round modelled as a thin disk with moment of inertia 9 kg m 2 At some point your friend walks around and along the edge at a speed 0 8 m s perfectly tangent to the edge at all times relative to the merry go round s surface Take the radius of the merry go round to be 1 8 m Calculate the speed in m s of your friend relative to the stationary ground O 0 04 m s O 0 06 m s 0 03 m s 1 59 m s

SolveLancer Test There is thin uniform rod of mass m and length I It is hinged at one end as shown The hinge makes it is free to rotate in the vertical plane The rod is slightly displaced from this position What is the hinge reaction given to rod when it is instantaneously in horizontal position A SolveLancer Test a b c 3mg 2 mg 37mg

Four identical rods each of mass m and length L are jonined to form a square Find the moment of inertia about the x axis 3ML mL 3 2 ML 12 23 ML 2 X Marks 4 1

radius 4R is rotating about fixed horizontal axis passing through point O with angular velocity A solid cylinder of radius R is rolling without slipping with respect to inner surface of hollow cylinder At the given instant the line OC has angular velocity of 2000 Point A and B are topmost and bottom most point of solid cylinder respectively and C is its centre Then at the given instant A Angular velocity of solid cylinder about its own axis is 10 B Magnitude of acceleration of point A of solid cylinder is 88 R C Magnitude of acceleration of point B of solid cylinder with respect to contact surface of hollow cylinder is 112 R D Radius of curvature of point A is 200 32R

A uniform rod of 60 kg is hanging in a horizontal position with the help of two threads It also supports of 40 kg mass as shown in the figure Find tension developed T and T2 respectively A 200 400 N 600 400 N 400 600 N 400 200 N T l 4 40 kg AT 2 B

A uniform sphere of mass M and radius R is placed on a rough horizontal surface Figure The sphere is struck horizontally at a height h from the floor Match Column I with Column II Column I R 2 A h B h R 3 C h R 2 7 D h R 5 h Column II p Sphere rolls without slipping with a constant velocity and no loss of energy q Sphere spins clockwise loses energy by friction r Sphere spins anti clockwise loses energy by friction s Sphere has only a translational motion loses energy by friction a A r B s C q D p b A s B p C r D q c A q B r C p D s D

SolveLancer Test Imaginary lines CD and AB are parallel Two uniform plates XYZ and XWZ are shown They have same and uniform surface area density and same thickness Moment of inertia of XYZ and XWZ is given by ly and Iw about AB Which of the following is true SolveLancer Test C A X a ly Iw b lw ly c Data insufficient d ly lu Z WD B

16 A uniform circular disc of mass 40kg and radius 0 2 m is pivoted at its centre point O and can freely rotate in a horizontal plane about a vertical axis YY Initially the disc is at rest The tangential force of 4 N is applied at a certain instant on a point P on its periphery as shown in the adjoining figure Assume that there is no frictionpresent a What is the moment of inertia of the disc about the given axis of rotation b What torque is produced by the applied force c What is the angular acceleration caused in the disc by the said torque y 0 d What will be the angular velocity of the dise 6s after start of its rotational motion I T anguc 1

A disc rotating with angular velocity wo Is placed on an inclined plane of inclination a with zero initial velocity as shown Coefficient of friction is u 2 tan 0 Find maximum displacement of the disc up the plane 00

T m 1 A mg cot 2 B mg cot 0 C mg tan 0 D 2mg tan 0 0 A uniform beam of mass m is pivoted at one end and held in equilibrium at an angle by a horizon tal wire as shown above The tension T in the wire is

Measuring Torque What is a possible reason why there is a difference percent difference 3 1 torque with force and lever arm perpendicular percent difference 4 1 torque and non perpendicular force

A uniform square plate has a small piece Q of an irregular shape removed and glued to the centre of the plate leaving a hole behind The moment of inertia about the z axis is then y Q a increased b decreased c the same x hole unpredicted manner

37 Which of the following statements is not correct a During rolling the instantaneous speed of the point of contact is zero b During rolling the instantaneous acceleration of the point of contact is zero c For perfect rolling motion work done against friction is zero R d A wheel moving down a perfectly frictionless inclined plane will slip but not roll on the plane

A turntable with mass m radius R and rotational mR2 inertia 2 initially rotates freely about an axis through its center at constant angular speed with negligible friction A piece of clay also of mass m falls vertically onto the turntable as shown above and sticks to it at point P a distance from the center of rotation R 2 2 A MR 7 What is the rotational inertia of the clay turntable system after the collision B 2 mR 2mR CO C A R D mR2 Single Person Usage Vansh R 1210 vanshraajin House No

5 A light rigid rod is placed on a frictionless horizontal floor and blocks of masses of m and 2m are attached to the ends of the rod with the help of light inextensible cords If a horizontal force F is applied perpendicular to the rod at its midpoint as shown in the figure what will acceleration of the midpoint of the rod be mC 3ml

horizontal square platform of mass m and side a is free to rotate about a vertical axis passing through s centre O The platform is stationary and a person of the same mass m as the platform is standing n it at point A The person now starts walking along the edge from A to B see figure The speed v of ne person with respect to the platform is constant Taking v 5 m s and a 1 m S A tion A 3 B Angular velocity of platform when the person is at A is 6 rad s Rate this question Angular velocity of platform increases first and then decreases again as it reaches B Angular velocity of platform decreases first and then increases again as it reaches B Angular velocity of platform remains constant as the person goes from A to B

A uniform ladder stands on a rough floor and rests against a frictionless wall as shown in the figure Since the floor is rough it exerts both a normal force N and a frictional force f on the ladder However since the wall is frictionless it exerts only a normal force N on the ladder The ladder has a length of L 4 00 m a weight of W 60 0 N and rests against the wall a distance d 3 75 m above the floor If a person with a mass of m 90 kg is standing on the ladder determine the following N N 1 a the forces exerted on the ladder when the person is halfway up the ladder Enter the magnitude only N N N b the forces exerted on the ladder when the person is three fourths of the way up the ladder Enter the magnitude only N N

A wheel of mass m which has a rotational inertia I about and a radius r rotates with an angular speed an axis through its center A retarding force F is applied tangentially to the rim of the wheel 20 Which of the following is equal to the magnitude of the angular acceleration of the wheel Fo A B Fr C T D la E or 21 The retarding force F finally stops the rotation of the wheel Which of the following best represents the total reduction in mechanical energy in the process of stopping the wheel A Fr B Io C r D Fr Single Person Usage Purch Vansh Raaj vanshraaj ind House No 180

released from rest in the vertical osition 0 0 on a rough Surface is sufficiently rough to prevent just sliding quare comer A as shown in figure then choose correct options D Rough If the rod just begins to slip when A 17 with vertical then the co efficient of static friction between the rod and the corner is 0 3 If the end of the rod is notched so that it cannot slip then the angle at which contact between the rod and the corner ceases is If the rod just begins to slip when A 17 then the coefficient of static friction between the rod and the corner is 4 If the end of the rod is notched so that it cannot slip then the angle at which contact between the rad

Two solid bodies rotate about stationary and mutually perpendicular intersecting axes with constant angular velocities 0 2 rad s and 2 4 5 rad s The angular acceleration of one body relative to the other body is 9 rad s A 2 rad s B 4 5 rad s C D 2 25 rad s A solid hemisphere of radius R has a variable volume density o 00where r is the distance from the center

tarting from rest an object rolls down along an incline that rises by 3 in every 5 along it The object gains a speed of 10 m s as it travels a distance of 5 malong the incline What can be the possible shape s of the object 6 Why is it useful to define radius of gyration Section C Attempt any FOUR Q 7 State and prove parallel axes theorem 8 Obtain an expression relating the torque with angular acceleration for a rigid body Discuss the inter link between translational rotational and total kinetic energies of a rigid object that rolls without slipping 14 x 3 M 12 MJ 1 A motorcyclist as a particle is undergoing vertical circles inside a sphere of death The speed of the motorcycle varies between 5 m s and 10 m s Calculate diameter of the sphere of death How much minimum values are possible for these two speeds Q11 Coefficient of static friction between a coin and a gramphone disc is 0 5 Radius of the disc is 8 cm Initially the centre of the coin is a cm away from the centre of the disc At what minimum frequency will it start slipping from there Section D Attempt any TWO 2 4 M 8M Q12 State and explain the principle of conservation of angular momentum Use a suitable illustration Do we use it in our daily life When Q 1 A rigid body is rolling down an inclined plane Derive expressions for the acceleration along the track and the speed after falling through a certain vertical distance Q 14 A flywheel used to prepare earthenware pots is set into rotation at 100 rpm It is in the form of a disc of mass 10 kg and radius 0 4 m A lump of clay to be taken equivalent to a particle of mass 1 o distance x from the centre Calculate x if the wheel now kg falls on it and adheres to it at a certai rotates at 80 rpm

ES OF RIGID BODIES 2 39 vertices of an equilateral triangle of side

A small solid ball mass 0 1 kg rolls without slipping along the track shown in figure The radius of the circular part of track is R If the ball starts from rest at a height 8R above the bottom what is the horizontal forced in N acting on it at P

A thin rod of length L and mass M is bent at the middle point o at an angle of 60 The moment of inertia of the rod about an axis passing through o and perpendicular to the plane of the rod will be L 2 60 L 2

AMICS I 2 In the following diagram pulley P is movable and pulley P is fixed The value of angle if system are in equilibrium 1 60 2 30 P A W P2 W B 3 45 4 15

A solid sphere of mass m rolls without slipping down on inclined plane with an acceleration a The frictional force o the solid sphere due to surface of the plane is 2ma 3 hr min 2ma

28 Starting from rest an object rolls down along an incline that rises by 3 in every 5 along it The object gains a speed of 10 m s as it travels a distance of 5353 can be the possible shape s of the object m along the incline What

The figure shows a uniform rod of length moving such that co 6 suddenly fixed The angular velocity of the rod will be A velocity of its centre of is v B 3m 3 4 E V If the end A of the rod is

Top View A particle of mass m moves counterclockwise around a horizontal circle of radius r as shown above The angular speed of the particle is given as a function of time by t bt where b is a positive constant and 0 31 Which of the following best describes the acceleration vector for the particle at the moment shown in the diagram A It is directed toward the left Co B It is directed toward the bottom of the page Bad C It is directed toward the top of the page D It has components directed toward the right and toward the bottom of the page It has components directed toward the left and toward the bottom of the page com SAT CO AP et What is the magnitude of the angular momentum of the particle about the center of the circle as a function of time A mbt r B mbrt C mbr D mb rt E mb 22

A hollow sphere of mass 12 kg rotates about its axis with angular speed 50 rad s The radius of the hollow sphere is 0 25 m The kinetic energy associated with rotation of the hollow sphere is 1250 J 700 J 625 J 1400

As shown in the figure a planet revolves around the sun in an elliptical orbit Determine the ratio of linear momentum at points P and Q Take R 10000 km and R 15000 km Vp P A M R 45 m R Q

54 Four point masses of value 2 5 6 and 3 kg are placed at the four corners of a square frame ABCD of side 0 4 meter having negligible mass Another point mass of 10 kg is placed at its centre Calculate the torque required to produce an acceleration of 20 sec 2 if the frame is to be rotated about a perpendicular axis passing through the centre Ans 0 446Nm Hint I Emr T Ia 1 Newton s between two poin masses and inver them thus

A hollow sphere of mass M and radius R is rolling on horizontal surface with speed v Find the angular momentum of sphere about point of contact on surface Question Type Single Correct Type 1 2 3MVR 5 2 3MVR

E Two identical discs each of negligible mass and radius r r l are connected by a uniform rod of mass m and length as shown in the figure The disc A is connected with an ideal spring of spring constant k at its centre The spring is unstretched when kl and 0 0 There is no slipping at any contact at any point of time during the motion Choose the correct statement for mg during the motion each disc remains in contact with the contact surface 00000 The value of 0 when the system is in stable equilibrium is 60 m The value of 0 when the system is in stable equilibrium is 30 0 kinetic energy of system in stable equilibrium is kl 8 Spring potential energy of the system in the stable equilibrium is kl 8

A thin rod of mass m and length 2L is made to rotate about an axis passing through its center and perpendicular to it If its angular velocity changes from 0 to w in time t the torque acting on it is Question Type Single Correct Type 1 2 3 mL 12t mL w 3t mL 2 t

44 180 Tark for Review A boy standing on a rotating table with heavy spheres in his hands suddenly brings his hands close to his body Then Angular momentum of the system remains same Angular velocity of system decreases hr min Angular momentum of system increases Both 2 and 3

A wooden bus of mass M 20 000 kg M represents the mass excluding the wheels is on a ramp with angle 0 Each of the four wheels is composed of a ring of mass and radius R 1 m and 6 evenly spaced spokes of mass and length R All components of the truck have a uniform density Find the acceleration of the us down the ramp assuming that it rolls without slipping Answer in m s R R

A boy standing on a rotating table with heavy spheres in his hands suddenly brings his hands close to his body Then Angular momentum of the system remains same Angular velocity of system decreases Angular momentum of system increases Both 2 and 3

4 Suppose that a spherical star spinning at an initial angular velocity a suddenly collapses to half of its original radius without any loss of mass Assume the star has uniform density before and after the collapse What will the angular velocity of the star be after the collapse A 4 1018 w 2 0 C D 20 E 40 son

A thread is wound around two discs on either sides The pulley and the two discs have the same mass and radius There is no slipping at the pulley and no friction at the hinge Find out the angular acceleration of the pulley

The figure shows a uniform rod of length moving such ol that velocity of its centre of is 6 the rod is suddenly fixed The angular velocity of the rod will be A B 8 m 4 3 N If the end A of V

7 A uniform sphere of mass m and radius r rolls without slipping down an inclined plane inclined at an angle 53 to the horizontal What is minimum value of coefficient of friction at which slipping is absent 21 hr 5 14 min