Rotational Level 1 QuestionsQ1. A uniform solid cylinder of mass m and radius a freely rotates about its axis of

symmetry under the action of constant torque 4mga. Find the angular velocity of the cylinder after 4 seconds if it started from rest.

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Q2. A uniform disc of mass 20 kg and radius 0.5m can turn about a smooth axis through its centre and perpendicular to the disc. A constant torque is applied to the disc for 3

seconds from rest and the angular velocity at the end of that time is revolutions per

minute. Find the magnitude of the torque. If the torque is then removed and the disc is brought to rest in t seconds by a constant force of 10 N applied tangentially at a point on the rim of the disc , find t.

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Q3. A flywheel whose moment of inertia about its axis of rotation is 16 kg-m2 is rotating freely in its own plane about a smooth axis through its centre. Its angular velocity is 9 rad s-1 when a torque is applied to bring it to rest in t0 seconds. Find t0 if (a) the torque is constant and of magnitude 4Nm.(b) the magnitude of the torque after t seconds is given by kt.

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Q4. A horizontal force F is applied to the sphere shown in the accompanying figure. What value of F is needed to hold the sphere in equilibrium? What is the frictional force of the incline on the sphere?

Ans: F = mg tan θ, f = 0Q5. A wheel is attached to a fixed shaft, and the system is free to

rotate without friction. To measure the moment of inertia of the wheel-shaft-system, a tape of negligible mass wrapped around the shaft is pulled with a known constant force F. When a length L of tape has unwound, the system is rotating with angular speed ωo. Find the moment of inertia of the system.

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Q6. A grindstone in the form of a cylinder has a radius of 0.2m and a mass of 30 kg. (a) What constant torque will bring it from rest to an angular velocity of 250 rev/min in 10 sec?(b)Through what angle has it turned during that time? (c)Calculate the work done by the torque? Ans : (a)1.57 N-m, (b)130.9 rad, (c)206 J

Q7. Figure shows three identical yo-yos initially at rest on a horizontal surface. For each yoyo the string is pulled in the direction shown. In each case there is sufficient friction for the yo-yo to roll without slipping. Draw the free body diagram for each yo-yo. In what direction will each yo-yo rotate?Ans: In each case the yo-yo rotates clockwise

Q8. A uniform circular disc of mass m and radius a is rotating with constant angular velocity ω in a horizontal plane about a vertical axis through its centre A. A particle P of mass

2m is gently placed on the disc at a point distant a from A. If the particle does not slip

on the disc find the new angular velocity of the rotating system.

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Q9. A man stands at the centre of a circular platform holding his arms extended horizontally with a 4 kg block in each hand. He is set rotating about a vertical axis at 0.5 rev/sec. The moment of inertia of the man plus platform is 1.6 kg-m2, assumed constant. The blocks are 90 cm from the axis of rotation. He now pulls the blocks until they are 15 cm from the axis of rotation. Find:(a) his new angular velocity (b) the initial and final kinetic energy of the man and the platform (c) How much work should the man do to pull in the blocks?Ans: (a)14.3 rad/s = 2.27 rev/s (b) Ei = 39.9 J, Ef = 181 J (c) 141 J

Q10. A horizontally oriented disc of mass M and radius R rotates freely about a stationary vertical axis passing through its centre. The disc has a radial guide along which can slide without friction a small body of mass m. A light thread running through the hollow axle of the disc is tied to the body. Initially the body was located at the edge of the disc and the whole system was rotating with an angular velocity ωo. Then by means of a force F applied to the lower end of the thread the body was slowly pulled to the rotation axis. Find:(a) the angular velocity of the system in the final state. (b) the work performed by the force F.

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Q11. The polar ice caps contain about 2.3 ×1019 kg of ice. This contributes essentially nothing to the moment of inertia of the earth because it is located near the axis of rotation. Estimate the change in the length of the day to be expected if the polar ice caps melt distributing the mass uniformly over the surface of earth. Mass of earth = 5.98 ×10 24kg and the radius of earth = 6.37 ×106m.Ans: 0.6 sec slower

Q12. A uniform rod of length 2l and mass m is free to rotate in a vertical plane about a smooth fixed horizontal axis through one end of the rod. The rod is held in a horizontal position and then released. Find the maximum angular velocity of the rod in the subsequent motion.

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Q13. A uniform circular disc of mass m, radius r and centre O is free to rotate about a smooth horizontal axis which is tangential to the disc at point A. The disc is held in a vertical plane with A below O and is then slightly displace from this position. Find the angular velocity of the disc when its plane is next vertical.

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Q14. A uniform disc of radius R and mass M is pivoted about a horizontal axis through its centre C. A point mass m is glued to the disc as shown in the figure. If the system is released from rest find the angular velocity of the disc when m reaches the bottom point B.

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Q15. A uniform ring of mass m and radius a is free to rotate in a vertical plane about a fixed smooth axis which is perpendicular to the plane of the ring and passes through a point A on the ring. A particle of mass m is attached to the ring at point B, where AB is the diameter. When the ring is hanging in the position of equilibrium the particle is struck a

blow which gives it a velocity . Find the height above A to which the particle

rises.Ans: 2.5 a

Q16. A uniform disc of mass M and radius R is pivoted so that it can rotate freely about a horizontal axis passing through its centre and perpendicular to its plane. A small particle of mass m is attached to the rim of the disc directly above the pivot. The system is given a gentle start and the disc begins to rotate. (a ) what is the angular velocity of the disc when the particle is at the lowest point? (b) At this point what force must be exerted by the disc on the particle so as to keep it on the disc?

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Q17. ABC is a triangular framework of three rods each of mass m and length 2l. It is free to rotate in its own plane about a horizontal axis passing through A which is perpendicular to ABC. If it is released from rest with AB being horizontal and C above AB, find the maximum velocity of C during the subsequent motion

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Q18. A solid disc is rolling without slipping on a horizontal surface with constant speed of 2m/s. How far can it roll up a 30o ramp before it comes to rest? Ans: 0.612 m

Q19. A disc of mass M and radius R has a spring of constant k attached to its centre, the other end of the spring being fixed to a vertical wall. If the disc rolls without slipping on a horizontal floor, how far to the right does the centre of mass move, if initially the spring was unstretched and the disc had an angular speed ωo?

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Q20. A cord is wrapped around a solid cylinder of radius 0.25 m, and a constant force of 40 N is exerted on the cord. (see the accompanying figure.) The cylinder is mounted on frictionless bearings and its moment of inertia is 6.0 kg m2.(a) Use the work energy principle to calculate the angular velocity of the cylinder after 5.0 m of cord have been unwound. (b) If the 40 N force is replaced by a 40 N weight, what will be the angular velocity of the cylinder after 5m of cord have unwound.Ans: (a)8.16 rad, (b) 8.0 rad/s

Q21. A uniform ball of radius r rolls without slipping along the loop-the-loop track in figure. It starts at rest at a height h above the bottom of the loop. If the ball is not to leave the track at the top of the loop, what is the least value h can have(in terms of radius R >>r of the loop)? What would h have been if the ball were to slide along a frictionless track instead of rolling?Ans: 2.7R ; 2.5 R

Q22. A uniform rod AB of mass 3m and length 2l is free to rotate in a vertical plane about a smooth horizontal axis through A. A particle of mass m is attached to the rod at B. When

the rod is hanging in equilibrium it is set moving with an angular velocity .

(a) If k = 2 find the height of B above the level of A when the rod first comes to instantaneous rest.(b) Find the range of values of k for which the particle may execute complete circle about A.

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Q23. A uniform circular disc of mass M and radius a is pivoted at point O on its circumference so that it can rotate about tangent at O, which is horizontal, the centre of disc describes vertical circle of centre O in a plane perpendicular to the tangent. The point diametrically opposite O is A, and the disc is just displaced from A when A is vertically above O. Find the angular velocity of the disc when A is vertically below O. At this instant a particle of mass M traveling with a velocity u in the opposite direction of the motion of the centre of the disc hits the disc at its centre and sticks to it. Find the angular velocity of the system immediately before and after the impact. If the

disc just reaches its initial position then show that

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Q24. A sphere starts from rest at the upper end of the track shown in the accompanying figure. It rolls until it leaves the track at the right end. If the track is horizontal at this point then determine the distance d to the right of this point at which the sphere strikes the lower surface.Ans: 23.9 m

Q25. A ball of mass m and radius r rolls along a circular path of radius R. Its speed at the bottom of the path is vo. Find the force exerted by the path on the ball as a function of θ.

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Q26. A lawn roller in form of a thin-walled hollow cylinder of mass M is pulled horizontally with a constant horizontal force F applied by the handle to the axle. If it rolls without slipping find the acceleration and the frictional force.

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Q27. A solid cylindrical wheel of Mass M and radius R is pulled by a force F applied to the centre of the wheel at 37o to the horizontal. If the wheel is to roll without slipping, what is the maximum value of |F|? The coefficients of kinetic and static friction are μs=0.4 and μk=0.3.Ans: 0.79 mg

Q28. A uniform solid cylinder of mass M and radius 2R rests on the horizontal table top. A string is attached by a yoke to a frictionless axle of the cylinder so that the cylinder can rotate about the axle. The string runs over a pulley in shape of a disc of mass M and radius R that is mounted on a frictionless axle. A block of mass M is suspended from the free end of the string. The string does not slip over the pulley surface and the cylinder rolls without slipping. After the system is released from rest what is the downward acceleration of the block.

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Q29. A block of mass m = 1kg slides down the surface of a smooth inclined plane as shown in the figure. The block is tied to a string which is wrapped around a disc capable of rotating about a horizontal axis. The disc has a mass M = 5 Kg and a radius R = 0.2m. Initially the string is taut. If the mass is released calculate its acceleration.Ans: 1.4 m/s2

Q30. A yo-yo of mass M has an axle of radius b and a spool of

radius R. Its moment of inertia can be taken to be .

The yo-yo is placed upright on a table and the string is pulled with a horizontal force F as shown. The coefficient of friction between the table and the yo-yo is μ. What is the maximum value of F for which the yo-yo will roll without slipping?

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Q31. A heavy homogeneous cylinder has a mass m and radius R. It is accelerated by a Force F, which is applied through a rope wound around a light drum of radius r attached to the cylinder. The coefficient of static friction is sufficient for pure rolling. (a) Find the friction force (b)Find the acceleration a of the centre of the cylinder. Is it possible to choose r so that a is

greater than ? How? (d) What is the direction of friction

force in the circumstances of part (c)?

Ans: assuming f opposite to F

(c) yes if r is greater than R/2. (d) f in same direction as F

Q32. Two blocks are attached to ropes attached to wheels on a common axle as shown in the figure. The total moment of inertia of the two wheels is 40 Kg m2. The radii are R1=1.2m and R2=0.4m. (a) If m1=24 kg then find m2 such that the system is in equilibrium. (b) If 12 kg is gently added to the top if m1, find the angular acceleration of the wheels and the tension in the ropes.Ans: (a) 72 kg, (b) 3.53 rad/s2, T1=201N, T2=808 N

Q33. The system in the figure is released from rest. The 30 kg body is 2m above the floor. The pulley is a uniform disc with a radius of 10cm and mass 5kg. Find (a ) the speed of 30 kg body just before it hits the floor and the angular speed of the pulley at that time. (b)the tension in the strings (c)the time it takes the 30 kg body to reach the floor.Ans: (a)2.73 m/s, 27.3 rad/s, (b) TL = 233.5 N , TR=238.2 N (c) 1.47 sec

Q34. A uniform cylinder of mass m1 and radius R is pivoted on frictionless bearings. A string wrapped around the cylinder is tied to a mass m2, which is on a frictionless incline of angle θ. This system is released with m2 a height h above the bottom. (a) what is the acceleration of m2 (b)What is the tension in the string.(c)What is the total energy of the system when m2 is at a height h? (d) What is the total energy of the system when m2 is at the bottom of the incline and has a speed v? (e) What is the speed v?

Ans: (a) , (b) , (c)m2gh, (d) m2gh

(e)

Q35. Consider a cylinder of mass M and radius R lying on a rough horizontal plane. It has a plank lying on its top as shown in the figure. A force F is applied on the plank such that the plank moves and causes the cylinder to roll. The plank always remains horizontal. There is no slipping at any point of contact. Calculate the acceleration of the cylinder and the frictional forces at the two contacts.

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Q36. Find the acceleration of the cylinder of mass m and radius R and the plank of mass M placed on smooth surface if pulled with a force F as shown in the figure. Given that friction is sufficient to prevent slipping of the cylinder.

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Q37. The disc of radius R is confined to roll without slipping at A and B. If the plates have the velocities as shown, determine the angular velocity of the disc.

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Q38. In the figure the cylinder of mass 10kg and radius 10cm has a tape wrapped round it. The pulley weighs 100N and has a radius 5cm. When the system is released the 5kg mass comes down and rolls without slipping. Calculate the velocity and acceleration of the mass as a function of time.

Ans:3.6 m/s2,

Q39. A cylinder is sandwiched between two planks. Two constant horizontal forces F and 2F are applied on the planks as

shown in the figure. Determine the acceleration of the center of the cylinder and the top plank if there is no slipping at the top and bottom of the cylinder.

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Q40. Two identical uniform discs each of mass m and radius R are held as shown in the figure with help of a long massless string which is wrapped around discs in opposite directions. Disc A is attached to the ceiling in such a way that it can rotate freely about its axis. Disc B initially at same height as A , is then released to fall so that the string unwinds from both discs. Find the angular and linear acceleration of the falling disc and the tension in the string. Assume that the disc does not slip and motion is confined in the same vertical plane.

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Q41. A small steel sphere of mass m and radius r rolls without slipping on the surface of large hemisphere of radius R (>>r) whose axis of symmetry is vertical. It starts at the top from rest (a) What is the kinetic energy at the bottom? (b)What fraction is the rotational kinetic energy of the total kinetic energy? (c)What fraction is the translational kinetic energy of the total energy? (d) Calculate the normal force that the small sphere will exert at the bottom of the hemisphere. How will the results be affected if r is not very small as compared to R?

Ans: (a) mg (R-r) (b) (c) (d)

Q42. A uniform rod AB of length 2l and mass 2m is rotating in a horizontal plane about a vertical axis through A, with an angular velocity ω, when the mid point of the rod strikes a nail and is brought immediately to rest. Find the impulse exerted by nail.

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Q43. A uniform rod of length L rests on a frictionless horizontal surface. The rod is pivoted about a fixed frictionless axis on one end. The rod is initially at rest. A bullet moving parallel to the horizontal surface and perpendicular to the rod with speed v strikes the rod at its centre and gets embedded. The mass of the bullet is one-sixth the mass of the rod. (a) What is the final angular velocity of the rod. (b)What is the ratio of kinetic energy of the system after the collision to the kinetic energy before the collision?

Ans: (a) (b)

Q44. A solid wood door 1.0m wide and 2.00m high is hinged along one side and has total mass of 50 kg. Initially open and at rest, the door is stuck at its centre by a handful of sticky mud of mass 0.5kg, traveling at 12.0m/s just before the impact. Find the final angular velocity of the door. Is the moment of Inertia of the mud significant?Ans: 0.179 rad/s, yes

Q45. A uniform stick of length L and mass M hinged at one end is released from and angle θo

with the vertical. Show that when the angle with the vertical is θ, the hinge exerts a force Fr along the stick and Ft perpendicular to the stick given by

and

Q46. A ring of mass m and radius r has a particle of mass m attached to it at point A. The ring can rotate about a smooth horizontal axis which is tangential to the ring at point B diametrically opposite to A. The ring is released from rest when AB is horizontal. Find the angular velocity and angular acceleration of the body when AB has turned

through an angle .

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Q47. A uniform rod AB of length 2l and mass 2m has a particle of mass m attached at B. The rod is free to rotate in a vertical plane about a horizontal axis through A . When the rod

is hanging at rest with B below A, it is given an angular velocity . Find the

reaction at the axis when the rod first becomes horizontal.Ans:

Q48. A pulley in the form of a uniform disc of mass 2m and radius r, is free to rotate in a vertical plane about a fixed horizontal axis through its center. A light inextensible string has its one end fastened to a point on the rim of the pulley and is wrapped several time around the rim. The portion of the string not wrapped on the pulley is of length 8r and carries a particle of mass m at its free end. The particle is held close to the rim of the pulley and level with the centre. If the particle is released from this position find the initial angular velocity of the pulley and the impulse of sudden tension in the string when it becomes taut.

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Q49. A uniform solid cylinder of mass m and radius a is free to rotate about its axis which is smooth and vertical. A light inextensible string is wound around the cylinder and its free end is pulled horizontally with a constant force 2mg. Find the angular velocity of the cylinder when the free end of the string has moved through a distance 4a.

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Q50. A uniform rod AB of mass 3m and length 4l, which is free to turn in a vertical plane about a smooth horizontal axis passing through A, is released from rest when horizontal. When the rod first becomes vertical, a point C of the of the rod where AC = 3l, strikes a fixed peg. Find the impulse exerted by the peg on the rod if

(a) the rod is brought to rest by the peg(b)the rod rebounds and next comes to an instantaneous rest inclined to the downward

vertical at an angle radians.

Ans: (a)

Q51. A uniform rod AB of mass 3m and length 2l is lying at rest on a smooth horizontal table with a smooth vertical axis through end A. A particle of mass 2m moves with a speed 2u across the table and strikes the rod at its mid point C. If the impact is perfectly elastic find the speed of the particle after the impact if(a) it strikes the rod normally(b) its path before the impact was inclined at 60o to AC.

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Q52. A light string hangs over a pulley which is a uniform disc of mass 4m and radius a, and carries a pan of mass m at each end. If a particle of mass m is dropped in to one of the scale pans from a height 10a above it, find the initial angular velocity of the pulley assuming that the string does not slip on it and the particle does not rebound from the pan.

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Q53. A cylinder r=0.1m and mass M = 2kg is placed such that it is in contact with a vertical wall and a horizontal surface as

shown in the figure. The coefficient of static friction μ=

for both the surfaces. Find the distance d from the centre of the cylinder at which a force F = 40 N should be applied so that the cylinder just starts rotating in the anticlockwise direction.Ans: d = 0.06m

Q54. A heavy roll of wrapping paper in the form of a solid cylinder is resting on a table top. Its mass is M and radius R. If a horizontal force F is applied evenly to the paper as shown in the figure, what are the linear and angular acceleration a around the centre of the roll? The coefficient of kinetic friction between the table and the paper is μk.

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Q55. A thin horizontal uniform rod AB of mass M and length l can freely rotate about a vertical axis passing through its end A. At a certain moment the end B starts experiencing a constant force F which is always perpendicular to the original position of the stationary rod and directed in the horizontal plane. Find the angular velocity of the rod as a function of its rotation angle passing through its end A. At a certain moment the end B starts experiencing a constant force F which is always perpendicular to the

original position of the stationary rod and directed in the horizontal plane. Find the angular velocity of the rod as a function of its rotation angle φ counted relative to the initial position.

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Q56. A cylinder of mass M and radius R is rotated in a uniform V groove with a constant angular velocity ω. The coefficient of friction between the cylinder and each surface is μ. What torque must be applied to the cylinder to keep it rotating?

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Q57. A uniform cylinder of mass M and radius R has a string wrapped around it. The string is held fixed, and the cylinder falls vertically, as in the figure.(a) Show that the cylinder

falls with a vertically downward acceleration of (b)

Find the tension in the string.

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