Rotation Questions and Answers

x u u OF m B 1 m 24 are based on the Two identical dumbbells are fabricated by welding small identical p each of mass m at the ends of light identical rods each of length dumbbells are moving in free space with equal speeds u towards ea without rotation as shown in the figure Collisions between the part perfectly elastic 3 Angular velocity of each dumbbell after the first collision is u 21 2u 1 a L u b 77 d 4u 1 24 How many times will the dumbbells collide a One b Two ned

A particle is projected at an angle of 45 with the horizontal with a speed u in a uniform gravitational field as shown in the figure The instantaneous angular speed of the particle w r t point of projection at the moment it falls at the same horizontal level equals u A C 2g 8Q g 2 2u 45 g g 2u D none of the above B

If a sphere is rolling the ratio of translational energy to total kinetic energy is given by a 2 5 b 10 7 c 7 10 d 5 7 A wheel has angular acceleration of 3 0 rad sec2 and an initial angular speed of 2 00 rad sec In a time of 2 sec it has rotated through an angle in radian of ajo b 4 d 10 c 12

A wire of mass M length L and cross section A is shaped into a ring It is placed on smooth horizontal surface and is rotated about its axis with constant angular speed was shown If Young s modulus of material of wire is Y then 0

At the instant shown the disk rolls without sliding such that it has an angular velocity of 2 rad CW and angular acceleration of 13 rad CW Determine the x component of the acceleration vector of point A in at this instant Your Answer 0 15 m 0 4 m y 5 X B

Pulley is M radius is R and radius of gyration is k An inextensible string is wrapped tightly on it and a ball of mass M is attached to the free end After falling a height h the string becomes tight The speed of ball just after string becomes tight is Assume no friction anywhere TAB A 2gh B TIRUN step 2gh 1 R k H TUTES ETC 2gh IT 1 k R D none of these

A disc of mass 2 5 kg and moment of inertia 10 kg m about an axis perpendicular to plane of disc and passing through centre rotates with angular velocity of 10 rad sec An owl of mass 1 kg jumps and sit on periphery of disc under no external torque condition Final angular velocity of system of disc and owl is 1 10 rad s 2 15 rad s 3 5 56 rad s 44 12 5 rad s Iw mr w

An 85 kg man steps off a high board from rest If the high board is 3 2 m above the surface of the water and the man comes to rest 0 57 s after reaching the water what is the impulse net force Challenge What force does the water exert on the man

Two identical meter sticks are connected as shown in the diagram The origin of the coordinate system is located where the two meter sticks are joined I m 1 m X What is the moment of inertia of the two sticks about an axis perpendicular to meter sticks and passing through the center of mass of the whole system Assume each meter stick has a mass m and length L

Six spheres each of mass m and radius r are placed with their centres on the six corners of a regular hexagon of side the moment of inertia of the system about any one side will be A 12mr 15 ma B 5 f R 1 2 MR 5 C 12mr 15 7 2 12mr 15 5 3 me me D 12mr 15 LO me 2 5 me 4

Figure shows the variation of the moment of inertia of a uniform rod about an axis passing through its centre and inclined at an angle to the length The moment of inertia in kg m of the rod about an axis passing through one of its ends and making an angle will be 0 TU 3 1 I kg m TT 0 rad

a A particle moves from A 6 2 3 to B 8 12 6 when a force of F 5i 8j 9k Newton acts on it The coordinates are in metres Find the work done b A force of 6 newtons acts through the point P 2 1 3 in the direction of a 3i 2j8k Find the moment of the force about A 5 2 1

A rod of mass m and length is placed vertically on a smooth floor as shown in the figure The lower end of the rod is given a negligible slight push towards left due to which the lower end of the rod moves towards left and collides perfectly inelastically with the fixed point P and then the rod rotates about the point P 2 3 m Q18 The angular velocity of the rod just after its lowest point collides with P is A g 4 B g 2 C D 21 8 21 32

A circular disc M R can freely rotate in vertical x y plane about a fixed axis perpendicular to the plane of the disc and passing through a point A on periphery of the disc as shown in figure A point mass m moving with velocity vo collides with stationary disc at its central line elastically and particle stops after collision Disc M R m Vo Axes A If w is the angular speed of disc after collision then v Rw If w is the angular speed of disc after collision then v 2wR Linear momentum of system is conserved just before and just after collision Angular momentum of system about hinge point is conserved just before and just after collision

AB is a cylinder of length 1 0 m filled with a thin flexible adibatic membrane C see figure at the middle and two other thin flexible adiabatic membranes A and B at the ends The portions AC and BC contain hydrogen and oxygen gasses respectively at same pressure Standing waves are setup in two columns such that membrane A acts as antinode and B as node speed of sound in oxygen is 330 m s The frequency of these vibrations for which the membrane C is a node is A 330 Hz B 550 Hz C 1485 Hz D 660 Hz B

A T shaped object with dimensions shown in the figure is lying on a smooth floor A force F is applied at the point P parallel to AB such that the object has only the translational motion without rotation Find the location of P with respect to C 1 28 A F P C 20 2 C 36 B

A thin plank of mass m is placed over a solid cylinder of same mass m and radius R A horizontal force F mg is applied on the plank as shown in the figure There is no slipping anywhere then F C R Rough Frictional force on the plank is Frictional force on cylinder by ground is mg 11 Acceleration of plank is 3mg 11 6g 11 Acceleration of centre of cylinder is 3g

A rope of length L and mass per unit length A passes over a disc shaped pulley of mass M and radius R The rope hangs on both sides of the pulley and the length of larger hanging part is 1 The pulley can rotate about a horizontal axis passing through its centre The system is released from rest and it begins to move The pulley has no friction at its axle and the rope has large enough friction to prevent it from slipping on the pulley a Find the acceleration of the rope immediately after it is released b Find the horizontal component of the force applied by the axle on the pulley immediately

A bamboo stick of mass 1kg is leaning against a frictionless wall The other end is resting on a rough horizontal surface making an angle 37 but the contact forces between the horizontal surface and the stick Friction and Normal reaction Take g 10m s A B f 100N N 65N f 10N 67N

A thin uniform disc of mass M and radius R is in combined translation and rotation as shown The magnitude of angular momentum of the disc about the origin is MVOR where p and q are integers in simplest 9 forms Find the value of p q Y V 000 Disc M R 000 4V R

A thin uniform spherical shell of mass M and radius R rolls without slipping on a horizontal surface with angular velocity w The kinetic energy of the upper part of the 2 shell above the horizontal line AB is MR where p 9 and q are co prime The value of p q is A R 9 B

b The disc in Figure 3 2 has diameter 1 m and is rolling on the level ground with angular velocity 4 rad s Bar AB has length 1 m and is attached to the disc at A while B slides on the ground Determine the angular velocity of bar AB and the velocity of point B 1m 4 rad s Figure 3 2 1m B 9 Marks

A thin uniform rod of mass M and length L is free to rotate about a horizontal axis passing through its end A and perpendicular to its length A sharp angular impulse Jis imparted to the rod about the hinge at A such that at maximum deflection the rod becomes horizontal Given M 10 kg L 0 3 m g 10 m s2 the value of Jin S I units is

45 A rod of length I and mass m hinged at one end is released from rest from vertical position as The angular velocity of the rod in the position shown is 9 3 3g cose 2 sine 4 6g sin 0 2 1

As seen in the figure given a pendulum like tray having a uniform homogenous cube of side length a placed at the centre with a small stopper at along its side The centre of rotation of the tray lies at a distance 10 a from the geo metrical centre of cube There is a massive wall as shown in figure Now the whole system is deflected like a pendulum at an angle The pendulum arrives back and collides with the wall Q What should be the angle of deflection so that the cube overturned Masses of the tray and the strings are negligible and the MOI of cube through any axis can be assumed ma to be Collision is perfectly inelastic 6 a a m CHAMA 10 a

47 A light rod carries three equal masses P Q and R as shown in figure The angular acceleration of Q in vertical position of rod If it is released from horizontal position Uch ercht vs TP Q R fr El for safe e P e Q o gl 2 2gl 3 l R 3 gl 4 0

29 A rod PQ of mass M and length L is hinged at 1000 end P The rod is kept horizontal by a massless string tied to point Q as shown in figure When string is cut the initial angular acceleration of the rod is 2013 I sitient 10 ovat to aixs To To P mom niedt ovisi esbod owT uods vlovitosges 12 bus ni od liiw 1200 a g L 2g 20 b 2g L d 3g 1 0 6 21

1 Two bodies have their moments of inertia I and 21 respectively about their axis of rotation If their kinetic energies of rotation are equal their angular momenta will be in the ratio 3 2005 a 2 1 c 2 1 b 1 2 d 1 2

19 A ballet dancer spins about a vertical axis at 2 5 radians per second with his both arms outstretched With the arms folded the moment of inertia about the same axis of rotation changes by 25 percentage Calculate the new frequency of rotation in RPM

5 A uniform sphere of mass 500 g rolls without slipping on a plane horizontal surface with its centre moving at a speed of 5 00 cm s Its kinetic energy is 1 8 75 x 10 J 3 6 25 x 104 J 2 8 75 x 10 3 J 4 1 13 x 10 3 J

ML A circular disk of moment of inertia I is rotating in a horizontal plane about its symmetry axis with a constant angular speed Another disk of moment of inertia I is dropped coaxially onto the rotating disk Initially the second disk has zero angular speed Eventually both the disks rotate with a constant angular speed f The energy lost by initially rotating disc to friction is 2010 a 1 1 2 1 1 o 18 12 b 1 1 2 1 1 1 1 1 o 0 2

A metre stick is held vertically with one end on the floor and is then allowed to fall If the end touching the floor is not allowed to slip the other end will hit the ground with a velocity of g 9 8m s sulaul YOU A 3 2 m s m 5 4 m a

A rod of length has length e and mass per unit length linearly increases from A to 2 as shown in figure Rod rotates with constant angular velocity in a gravity free space Find the tension in Newton in the rod at its middle point take wf 2 m sec and 3 kg m 2

25N 30cm 25N a Calculate the torque acting on the wheel due to these forces 4 marks b Explain whether the steering wheel is in equilibrium when subjected to these two forces only 3 marks

ve 202 rotating about their centre of mass under their mutual gravitational attraction Let us consider such a double star which has two stars of masses m Reduce each at separation If T is the time period of rotation about their centre of mass then find T 41 Im Toy 113 m Basix 4 men 1 T 2 330 fr 3 T 2T 1 mG 1 3mG 1 2 T 2 2 4 T 2 1 2mG 1 4mG Gmxm mu 12 W 24m 3 30 1 29 w 9 F F F r2 4 F ifr F NOTAN NNNN if r of the follow tional field centre of a spheric

Consider the arrangement shown in the figure The pulley is light and the spool has mass M Table is sufficiently rough for pure rolling String is light and inextensible 2R R Aa zey adda uz furz AR Az4 5ch AU FREC WHA M I u Y Aay qufta gez 31 34 zeh am f m 2R B 5 1 MR given 2 C FRAROT 2 4F 9M 5 MR PT Acceleration of the spool is FRAROT Acceleration of the spool is 2 4F 13M Correct Answer 4F 9M 9F 13Mg y a fag si zyada 913 Correct Answer 4F 13M F Minimum value of u for pure rolling is 9F 13 Mg Minimum value of u for pure rolling is

A uniform solid cylinder of mass M 3 kg and radius R 10 cm is connected about an axis through the centre of the cylinder to a horizontal spring with spring constant 32N m The cylinder is pulled back stretching the spring 1m from equilibrium When released the cylinder rolls without slipping What is the speed of the center of the cylinder when it returns to equilibrium wwww

2 MR material in K then rise in temperature of the sphere is M I of hollow sphere 3 kg hollow sphere of mass M and radius R is rotating with angular frequency o It suddenly tops rotating and 75 of kinetic energy is converted to heat If S is the specific heat of the A Ro 48 B R 4S C Ro 2S R 2S D

DA 2 6 points A small steel block of mass M is attached to a rod of mass m and length L whose other end is connected to a hinge on the ceiling The block and the rod can rotate in the vertical plane but initially they are at rest A rubber bullet of mass me is fired at the block horizontally and bounces back while the block and the rod start rotating The initial velocity of the bullet is v and the final velocity of the bullet is v The moment of inertia of the rod with respect to the hinge is mL 3 and the moment of inertia of the block is ML a Find the angular velocity co with which the rod and the block start rotating after the collision with the bullet Hint Use the conservation of angular momentum b Find the energy lost ME E Erin this collision

Consider the child s playground merry go round wheel If the steel disk has mass of 200 kg a radius of 2 meters and a smooth low friction bearing at its center you can make it spin by apply force to the rim This torque increases the angular momentum of the disk Suppose the force is 100 newtons What mass could you pick up with a force of 100 newtons That will give you a sense of how much force this really is How long would you have to apply this force it to get the wheel spinning 2 times a minute What would happen to the rate of spin if you then jumped on the rim of the wheel with your mass of 75 kg Explain your answers

A uniform disc of mass m and diameter 2 R moves forward towards another uniform disc of mass 2 m diameter 2 R on a frictionless surface as shown in figure M 1 0000 2M M 2M When the first disc contacts the second they stick to each other and move as a single object the velocity of combined disc after the collision is S the angular velocity of combined disc after the collision is if we the combined disc will not rotate if combined disc does not rotate the energy loss is MV

A thin hollow cylinder is free to rotate about its geometrical axis It has a mass of 8 kg and a radius of 20 cm A rope is wrapped around the cylinder What force must be exerted along the rope to produce an angular acceleration of 3 rad sec 0 84 N 0 5 8 N 048N O None of these Marks 4 1

A hollow straight tube of length 1 and mass m can turn freely about its centre fixed on a smooth horizontal table Another smooth uniform rod of same length and mass is fitted into the tube so that their centres coincide The system is set in motion with an initial angular velocity The angular velocity of the rod at an instant when the rod slips out of the tube is 1 3 4 w 7 2 w 2 3 w 4

A particle of mass m moves with a constant 44 velocity Which of the following statements is not correct about its angular momentum about point O E IC B 1 It is zero when it is at A and moving along OA 2 It is same at all points along the line DE 3 It is of the same magnitude but oppositely directed at B and D 4 It increases as it moves along the line BC 47 1 mm to and fra fan fo after a FACE 1 f E IC Im Im B fazia 2 DE 3f en farg 3 feam faudra t 4 BC and an an fer

4 0 53 mA A cylinder of mass M and radius r is mounted on 29 a frictionless axle over a well A rope of negligible mass is wrapped around the solid cylinder and a bucket of mass m is suspended from the rope The linear acceleration of the bucket will be M 1 a 3 a bucket 2mg M 2m mg M 2m TA 2 a 4 a mg M m Mg M 2m 4 0 53 mA M r v i i 347 B V I 30H TH T f alled fanger as the PER 1 a 3 a M bucket m 2mg M 2m mg M 2m 2 a 4 a mg M m Mg M 2m

A metal cylinder of radius R is placed on a wooden plank BD The plank is kept horizontal suspended with the help of two identical string AB and CD each of length L The temperature coefficient of linear expansion of the cylinder and the strings are a and a respectively Angle e shown in the figure is 30 It was found that with change in temperature the centre of the cylinder did not move Find the ratio 1 if it is know that L 4R Assume that change in 2 value of 0 is negligible for small temperature changes L B B R L

A small mass attached to a string rotates on a frictionless table top as shown If the tension in the string is increased by pulling the string causing the radius of the circular motion to decrease by a factor of 2 the kinetic energy of the mass will

A disc of mass m and radius R rests on a horizontal smooth as shown in figure Two particle of mass m and 2m moving with velocity V collides in elastically e 0 with the ring at diametrically opposite points Find the angular velocity rad s with which the system rotates after the particle strikes the ring V 26 m s R 11 m m m

A uniform solid cylinder of radius R 15 cm rolls over a horizontal plane passing in to an inclined plane forming an angle a 30 with the horizontal Find the maximum value of the velocity vo which still permits the cylinder to roll onto the inclined plane section without a jump The sliding is assumed to be absent T Vo Ja

A solid sphere is rolling on a surface as shown in figure with a translation velocity v m s If it is to climb the inclined surface continuing to roll without slipping then minimum v for this to happen is AIEEE 2012 Online 1 10 7 gh 2 7 2 gh A h 3 7 gh 4 2gh