Rotation Questions and Answers

A thin rod of mass M and length a is free to rotate in horizontal plane about a fixed vertical axis passing through point O A thin circular disc of mass M and of radius a 4 is pivoted on this rod with its center at a distance a 4 from the free end so that it can rotate freely about its vertical axis as shown in the figure Assume that both the rod and the disc have uniform density and they remain horizontal during the motion An outside stationary observer finds the rod rotating with an angular velocity and the disc rotating about its vertical axis with angular velocity 40 The total angular momentum of the system about the point O is Ma 2 n 48 The value of n is M a M 492 a 4

LE CHOICE CORRECT QUESTIONS Consider a uniform square plate of side a and mass m the moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is AIEEE 2008 5 1 ma 2 1 ma 12 1 1 7 3 ma 12 4 2 3 ma

A particle of mass m 5 units is moving with a 3 uniform speed v 3 2 m s in the XOY plane along the line Y X 4 The magnitude of the angular momentum of the particle about the origin is 1 zero 3 7 5 unit 2 60 unit 4 40 2 unit X 4 X 3 2 m s fafanufa c m TIT 1 3 7 5 unit 2 60 unit 4 40 2 unit

K shown in figure It is attached by an inextensible thread which is being pulled horizontal with constant velocity v Find tangential and normal acceleration as a function of 0 AN COS 7 2 v R 4 R DAN sin 7 2 v an R 4 ar v Rain 2 v 02 0 T 0

On the rotating disc A of mass M another disc 15 ME M of same dimension but of mass 4 gently with same axis New angular velocity of system M R is placed farsit ameft farg m M 4 M R

A conducting ring of mass 2 kg and radius 0 5 m is placed on a rough horizontal plane surface so that the ring does not slip on the surface The ring carries a current 4 A A horizontal magnetic field B 10 T is switched on at time t 0 The initial angular acceleration of the ring will be Options n 1 rad s

A thin spherical shell rolls down a curved track ABC without slipping At C it leaves the track at a 370 with horizontal The maximum height H reached by the shell 27 1 h 27h ne 7h 24

C B Two uniform rods each of mass m 1kg and length 1m are hinged at B and pivoted at C If a horizontal impulse FAt 7 N s is applied to the end A of the lower rod then find the angular velocity in rad sec of the upper rod just after the impulse At is very small and rods can rotate freely w r t hinge and pivots C e m B m

A thin spherical shell of mass M and radius R rolls down with out sliding on a rough inclined plane e the linear acceleration of the shell is 1 g sin e 2 3g sin 0 3 g sin 0 2 4 g sin 0

Which of the following statements is not correct about the moment of inertia 1 Moment of inertia of a body is not fixed as it changes with the axis of rotation 2 Numerically the moment of inertia is equal to the torque required to produce unit angular acceleration 3 If no external torque is applied on a system its moment of inertia remains the same 4 Its dimensions are M L To

Disc A and disc B with a hole have equal mass and radius and a string is wrapped around them as shown Equal force F is applied on the string of each body Frication is sufficient for rolling After time t velocity of centre of A is v and that of B is vand kinetic energy k k then F F VA VB Ok k B A

20 A cavity of radius R 2 is made inside a solid sphere of radius R The centre of the cavity is located a a distance R 2 from the centre of the sphere The gravitational force on a particle of mass m at a distance R 2 from the centre of the sphere on the line joining both the centres of sphere and cavity s opposite to the centre of cavity Here g GM R2 where M is the mass of the solid sphere D none of these A mg 2 B 3 mg 8 C mg 16

A hollow hemisphere rests on a truck as shown in figure The truck begins to accelerate with 2 m s2 Find the initial angular acceleration in rad s2 of the sphere Friction is sufficient to prevent slipping Take R 1m and the system is released from rest when 8 37

A massless rod of length b has one end pivoted and the other end joined perpendicular to the middle of a rod of mass m and length In case i two rods are held in horizontal plane and then released In case ii two rods are held in a vertical plane and then released Which of the following is true about the acceleration of the centre of mass The accelerations are same in both cases O The acceleration in case i is g 0 The acceleration in case ii is The acceleration in case ii is g g 1 12b b case i g b case ii g k

Question No 9 a x where x is the distance measured from A and po is a constant Mass A thin uniform rod AB of length a has variable mass per unit length p Po a rod is M The rod is freely pivoted at A and is hanging in equilibrium vertically when it is struck by a horizontal impulse J at B Rod complete vertica circular motion about A The moment of inertia of the rod about an axis through A perpendicular to AB is Ma 18 The moment of inertia of the rod about an axis through A perpendicular to AB is 7 12 0a J M 70ga B Velocity of centre of mass of rod just after impulse is given is 20J 21 poa

A wheel of radius r rolls without slipping on a horizontal surface with a constant velocity v The wheel starts its journey at time t 0 from the position shown in the figur The point P is fixed to the rim Which of the following statement is Correct y The point P touches the ground after time Y P C r V TITT777 X The coordinates of the point P at time t are x r 0 sin e and y r 1 cos 0 where 0 vt T The radius of curvature of the path followed by P is minimum when P is at its maximum height The coordinates of the wheel s centre when P touches the ground are x r and y r

A uniform rod of mass mand length is hinged at its one end so that it can rotate freely in a vertical plane If it is released from horizontal position then acceleration of centre of mass of rod just after release of rod equals to 3g 1 3 3g 2 4 3g 3g 5

5 If the perpendicular distance between fixed point and point of application of force increases the moment of force a Increases 6 The SI unit of pressure is a Newton metre b Decreases b Dyne centimetre c Remains same b 20 joule c 5 joule c Pascal d None of these d None of these 7 A force of 10 N acting on a body causes a displacement of 2m in its own direction The work done by the force is a 10 joule d 0 joule

9 A copper ball of mass 100 g is at a temperature T It is dropped in a copper calorimeter of mass 100 g filled with 170 g of water at room temperature Subsequently the temperature of the system is found to be 75 C T is Given room temperature 30 C specific heat of copper 0 1 cal g C 2017 Main a 885 C b 1250 C c 825 C d 800 C

Question No 33 The angular velocity of the wheel increases from 120 rpm to 300 rpm in 10 s The number of revolutions made during this time is O 25 O 35 O 30 O 40

Consider two masses each of size 2m at the ends of a light rod of length L with the axis of rotation through the center of the rod The rod is doubled in length and the masses are halved What happens to l A 2m JA IB IA IB IA IB L 2 L 2 2m m B L L m

a What is the tangential acceleration of a bug on the rim of a 13 0 in diameter disk if the disk accelerates uniformly from rest to an angular speed of 76 0 rev min in 4 50 s m s b When the disk is at its final speed what is the tangential velocity of the bug m s c One second after the bug starts from rest what is its tangential acceleration m s2 d One second after the bug starts from rest what is its centripetal acceleration m s e One second after the bug starts from rest what is its total acceleration m s2 from the radially inward direction

13 A student sits on a rotating stool holding two 3 5 kg masses When his arms are extended horizontally the masses are 1 m from the axis of rotation and he rotates with an angular speed of 0 85 rad s The moments of inertia of the students plus stool is 5 kg m and is assumed constant The student then pulls the masses to 0 35 m from the axis of rotation a Find the new angular speed of the system 5 points

The torque caused by a force F applied at a distance r from the axis of rotation is given by t rFsin 9 where qis the angle between F and F If the magnitudes r and F remain the same what other angle qe will produce the same torque as was produced at angle qif q was less than 90 O e 180 8 8 90 8 O None of the above 08 90 8

Suppose a wheel is initially rotating at 10 0 rad s while undergoing constant angular acceleration reaching a speed of 30 0 rad s after 20 0 seconds have elapsed How long after the initial time has the wheel undergone half of the angular displacement that it will have gone through during the entire 20 0 second interval O 15 0 s O 12 4 s O 14 2 s 10 0 s

A 0 30 m radius automobile tire accelerates from rest at a constant 2 0 rad s over a 5 0 s interval What is the tangential component of acceleration for a point on the outer edge of the tire during the 5 s interval 6 7 m s O 0 60 m s 33 m s 0 30 m s

A force of F 4 i 4j 7k lb acts at the point 12 6 5 ft Determine the moment about the point 3 4 1 ft

Check my wor A biologist is studying plant growth in outer space He wants to simulate Earth s gravitational field so he positions the plants on the inside rim of a rotating cylindrical shell inside the spaceship The distance of each plant from the central axis of rotation is r 0 212 m What angular speed is required rad s

10 37 Determine the moment of inertia I of the shaded area about the y axis 6 in y 6 in 3 in 6 in X

2 A system of forces is shown in the following figure F2 F Z 10 3 9 X 0 3 8 12 a Y a Express F and F2 in vector form b Find the angle between F and F c Find the resultant force of the system of forces d Find the resultant moment of the system of forces about the point B 4 1

A coat hanger of mass m 0 216 kg oscillates on a peg as a physical pendulum as shown in the figure below The distance from the pivot to the center of mass of the coat hanger is d 18 0 cm and the period of the motion is T 1 20 s Find the moment of inertia of the coat hanger about the pivot kg m 2 Pivot X CM

Babe Ruth steps to the plate and casually points to eft center field to indicate the location of his next ome run The mighty Babe holds his bat across is shoulder with one hand holding the small end of the bat The bat is horizontal and the distance rom the small end of the bat to the shoulder is 22 5 m The bat has a mass of 1 20 kg and has a enter of mass that is 69 0 cm from the small end of the bat Part A Part B J Part C Part D Find the direction of the force exerted by the force acts in the downward direction force acts in the upward direction

Problem 1 Using the perpendicular distance method to the line of action of the force Calculate the moments at A B and C 30 KN 1 75 10 K 75 125 3m 15 KN 3m 35

2 What is the moment of inertia of a uniform solid cylinder about its central axis of diameter d 0 190 m and total mass M 1 65 kg Show your work

ct Answer Trapeze Near the surface of the Earth a trapeze consists of a rod with length L and negligible mass supported at its ends by vertical ropes of equal length as shown in the figure Each rope can withstand a maximum tension of Tmax before breaking When hanging at rest a trapeze artist can hold on with each hand a distance of L 4 from the ends of the rod and neither rope breaks However if she takes one hand off and does not move the other hand she that one rope breaks causing her to fall What is her minimum possible mass Tmax 2g 4Tmax 3g 3 Tmax 4g 5Tmax 3g

C R r A thin rod of mass m and length is hinged to a ceiling and it is free to rotate in a vertical plane A particle of mass m moving with speed v strikes it as shown in the figure and gets stick with the rod The value of v for which the rod becomes horizontal after collision is A The value of v for which rod becomes horizontal after collision is 168 9 gl 60 gl B The value of v for which rod becomes horizontal after collision is 53 3 m l C Angular momentum of rod particle system will remain constant about hinge just before and after collision 2 m D Angular momentum of rod particle system will remain same about centre of mass just before and after collision

moo 192 A solid sphere is placed on a surface An impulse of short duration is applied horizontally at a distance y from centre as shown For pure rolling motion of sphere Impulse ou 1 y 3 y R 2 2R 5 TY 2 y R 2 4 y 0 ast es ex

Paragraph Question Nos 47 to 50 There is a uniform bar of mass m and length hinged at centre so that it can turn in a vertical plane about a horizontal axis Two point like masses m and 2m are rigidly attached to the ends and the arrangement is released from rest 7 Initial angular acceleration of the bar is 3g 51 1 m 3a 2 3g 21 2m

177 Particles A and B are separated by 10 m A is at rest and B started moving with a speed 20 m s as shown in figure Angular velocity of Bw r t A at this ensinstant is ELDS etilletsa ent nart synes and fade A 1 1 rad s 1 3 2 rad s 1 10 m B 30 20 m s 2 1 5 rad s 1 4 2 5 rad s 1

83 A hollow sphere having mass m and radius moving as shown in the figure If the speed of centre of mass of sphere is v and angular speed Vo is wo R about point O is 1 mv R 3 2 mv R 3 mv R 2 mv R The angular momentum of the sphere O F Wo

134 Moment of inertia of a thin uniform rod of length and mass m about an axis passing through a 1 from centre O and at an point at a distance of 4 angle of 60 to the rod is 1 3 7m1 64 m 4 15 4 60 2 4 m 8 m1 12 ML 2 M 3 21 Z 2 12 12992 4

Two uniform rods each of mass m 1kg and length 1m are hinged at B and pivoted at C If a horizontal impulse FAt 7 N s is applied to the end A of the lower rod then find the angular velocity in rad sec of the upper rod just after the impulse At is very small and rods can rotate freely w r t hinge and pivots D C e m B e m

11 A spool of mass M inner radius R and outer radius 2R is released from rest from the position as shown in figure If moment of inertia of spool 5 MR then at this about its central axis is 4 instant 1 T Ma 1 MP Q2R FR 1 Tension in the string is 2 Acceleration of point C is 3 Acceleration of point P is TOTRAM 4 Acceleration of point Qis 5Mg P 4g 9 4g 9 PEITA 5g 9 9 90 A 2RX 49

block of side L rests on a rough M A horizontal surface with coefficient of friction horizontal force F is applied on a block as shown If there is sufficient friction between the block and the ground then the torque due to normal reaction about its centre of mass is 1 Zero 3 FL F 7777 m t L 0 2 2 FL 4 N 3 FL 2 2

is r If there is no slipping of string over the pulley ia and its radius then acceleration of the block of mass m is 1 g mg 2 m mg 1 2 3 m r mg m 23

60 A disc is set into pure rolling Let us taken three points on the disc as shown in figure Then VB VC VD is VB VC and v are corresponding speeds DHE D 0 B A pia Em gx S 1 Sm ex 2X px 8 8 Emplar 4 1 1 1 1 stions in easi prived rake ond a prli 64 6

3 t 10 A particle of mass m is moving on x y plane with constant velocity in a circle of radius R as shown in figure Then its angular momentum is maximum about z axis for its position at YA 1 B 3 A 0 R D CAV B A R 0 VY V X The 2 C 4 Both 1 2

34 Four rods of mass M and length are joined at the point O Find the moment of inertia of the system about axis of rotation AA shown here 1 3 MI 3 4M1 3 M I 24 M I Oi A i 90 2 4 M I M I MI 12 15M X

3 4 2 4 A solid sphere a disc and a cylinder are kept on the top of a smooth inclined plane If they are allowed to fall along inclined plane then 1 Disc reach bottom first 2 Solid sphere reach bottom first 3 All reach bottom together 4 Only disc and cylinder reach bottom together

916 20 55 A solid sphere is rolling up on an incline If it is thrown in upward direction with some initial angular velocity and its motion is pure rolling then its angular momentum about the initial point of contact during its upward motion 1 Remains constant 2 Increases E 3 Decreases asoler ons bris A esibod owT ST 916 2eibod or to 2982sm BnT 4 Increases or decreases depend on the value of friction