Rotation Questions and Answers

27 The initial angular acceleration of the uniform rod of length just after the string A is cut as shown in the figure is hickA 1 3 4g 3L 12g 71 14 L76 2 E 3g 4L 7g 12L

m and radius R is made to move on a horizontal floor with a horizontal velocity vo in such a way that at the time t 0 its motion is purely sliding C t 0 Fig 2 37 If the disc acquires a pure rotational motion after a time to i what is the velocity of the centre of mass of the disc in time to ii If the coefficient of friction is find the value of to Express the work done by frictional force as a function of time and also determine the total work done in time t which is much more than the time 1 mi to 2 Vo O t to Fig 2 37 Vo ii to Vo

Moment of inertia of a wheel is 0 04 kg m If its number of revolutions is increased from 60 rpm to 150 rpm in 3 s determine the magnitude of the torque acting on it 3704 0 1256 N m

A uniform rod of mass M and length is dropped on horizontal smooth surface from the position shown Immediately after elastic collision with horizontal surface velocity of centre of mass of rod is found to be n m s Find the value of n g 10 m s h 9 8 m 60

Four spheres each of mass M and diameter 2a are placed at one corner of square of side b as shown below Calculate the moment of inertia w r t given a axis BB b axis AA Sol a Axis BB passes through the centre and perpendicular to the plane 4 IBB 4 1CM 4M JAA 41CM 2M 2 B B 4 Ma 4M x 4M Ma 5 4x 2 5 2 2 b Axis AA passes through the centre and spheres 1 and 3 b 2 A Q

3 24 N A string wrapped on a pulley of moment T Other end of the string is connected to block of mass m as shown If m is released from rest then kinetic energy of block when it descends by h is 1 mgh m 3 Less than mgh 2 Greater than mgh mgh 4

In both the figures all other factors are same except that in figure i AB is rough and BC is smooth while in figure ii AB is smooth and BC is rough Kinetic energy of the ball on reaching the bottom A h B A h 1 is same in both the cases 2 is greater in case i 3 is greater in case ii 4 Information insuffici B C

39 A wheel of radius 1 m rolls forward half of a revolution on a horizontal ground The magnitude of the displacement of the point P shown in the figure is Po 1 4 m 3 2 m 90 2 2 m 4 m

A force of FK acts at the origin of the co ordinate system The torque of the force about a point having co ordinate 1 1 is 1 F i 1 3 F i 1 j oxd encup 2 F i j 2 F i 4 F i j

3 If a rigid body is rotating about an axis passing through the point 21 1 k and parallel to 1 21 2k with an angular velocity 3 radians sec then find the velocity of the point of the rigid body whose position vector is 21 31 4k a 2 31 4 k b 2 31 4k c 2 3 A A 1 1 14

A bullet of mass 10 g and speed 500 m s is fired into a door and it gets embedded exactly at the centre of the door The door is 1 0 m wide and of mass 12 kg It is hinged at one end and it rotates about a vertical axis practically without friction Find the angular speed of the door just after the bullet gets embedded into it Hint The moment of inertia of the door about the vertical axis at one end is ML 3 INCERTO 625 rad 6 11

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 5 3 m along the incline What can be the possible shape s of the object K Ans R 1 Thus a ring or

Uniform circular ring of radius 0 5 m has a mass 10 kg and a uniform circular disc of the same radius has a mass 10 kg Calculate their moment of inertia about an axis passing through the centre and perpendicular to the plane of a ring or a dise 2 5 kg m 1 25 kg m

44 Two loops P and Q are made from a uniform wire The radii of P and Q are R and R respectively and their moment of inertia about axes normally through centre are 1 and 12 respectively If 1 then find 2 1 43 3 2 43 R R 1 2 43 12 1 4 4 3 4

23 A rod of length and radius ris joined to a rod of length and radius end of small rod is fixed to a rigid base and the free end of larger rod is given a twist of 0 the twist angle at the joint will be 1 3 0 50 6 2 of same material The free 2 4 2 80

A uniform straight rod of mass 50 kg and length 4 m is to be supported on two supports equidistant from the respective ends What can be the maximum distance of the supports so that a 30 kg counterpoising weight can be kept in equilibrium at any point on the rod 1 25 m

A rope remains wrapped over the circumference of a wheel of radius r and of moment of inertia I about its axis If a mass m is suspended from the lower end of the rope what will be the angular acceleration of the wheel mgr T mr

P A solid cylinder has mass M length L and radius R The moment of inertia of this cylinder about generator is An axis tangential to surface and parallel to axis of cylinder

A uniform circular disc has radius R and mass m A particle also of mass m is fixed at point A on the edge of the disc as shown in the Fig 2 38 The disc can rotate freely about a fixed horizontal R chord PQ which is at a distance 4 from the centre C of the disc The line AC is perpendicular to PQ Initially the disc is held vertical with point A as its highest point Find the linear speed of point A as it reaches its lowest position when the disc starts rotating about PQ P A m C R Fig 2 38 5gR

A plank of mass M is placed over smooth triangular wedge with angle of inclination 30 Block of mass m is connected to the plank by massless string A cylinder of mass M and radius R is now placed over plank such that one end of the string wrapped over cylinder is fixed and parallel to incline Coefficient of friction between cylinder and plank is sufficient for pure rolling of cylinder If system is released from rest with condition that M 4m find the acceleration of plank and angular acceleration of cylinder m M B

A semicircular wire of radius r 2m is supported in its vertical plane by a hinge at O and a smooth peg P starts moving from O with constant acceleration a 2 m s along the horizontal axis passing through O The tangential acceleration of the point Q when the ring J 2 17 has rotated by 37 isK m s Then find the value of K Initially 16 the diameter OQ is vertical P

9 A disc of radius 0 2 m rolls without slipping on a horizontal surface with a velocity of 6 m s Then it ascends a smooth track as shown in the figure Calculate the height up to which it will ascend a 2 4 m b 1 8 m d 0 6 m c 3 2 m

The moment of inertia of a wheel of radius 0 2 m about an axis passing through its centre is 0 1 kg m It is now allowed from rest to roll down a plane inclined at 30 What will be its angular velocity after it rolls down a distance of 2 m along the inclined plane Given mass of the disc 5 kg 18 07 rad s

A uniform solid cylindrical roller of mass m is being pulled on a horizontal surface with force F parallel to the surface and applied at its centre If the acceleration of the cylinder is a and it is rolling without slipping then the value of F is 1 ma 2 2 ma 3 3 12 1 ma

4 A constant torque of 3 14 N m is exerted on a pivoted wheel If the angular acceleration of the wheel is 4 rad s then the moment of inertia of the wheel is b 2 5 kg m d 25 kg m a 0 25 kg m c 4 5 kg m

A beam has a mass m 3 kg and a length L 6 m The beam is pivoted to the ground at point P and the beam makes an angle of 45 with the horizontal A mass M4 is hanging from the beam a distance of 4 m from point P The beam is supported by a rope that is attached to the beam at a distance L from point P The rope makes an angle of 30 with the horizontal and the rope has a tension T The beam is in equilibrium Hint The torque from the tension about point P is Tr LT sin 15 a What is T the tension on the rope b What is N the x component of the normal force acting at the pivot c What is Ny the y component of the normal force acting at the pivot 30 T P 45 M

Pulley A Spring Scale Figure 1 ii Is your answer to part Bi consistent with your answer to part a why Pulley B Spring Scale B i Derive an equation for the time tA it takes Pulley A to complete one revolution Let la be the pulley s rotational inertia Express your answer in terms of Fs la MR Rp and physical constants as appropriate

A uniform ladder of mass 30kg and length 5 m rests against a smooth vertical wall with its lower end on rough horizontal ground The ladder is inclined at 60 to the floor The coefficient of friction between the ladder and the ground is 0 4 Find how far a man of mass 80 kg can climb up the ladder without it slipping

Problem 3 A wheel radius 12 cm rotates without friction about an axle through its center A constant force with magnitude 80 N is applied tangent to the rim of the wheel After 2 0s the angular speed of the wheel is 12 rev s What is the moment of inertia of the wheel

The seesaw in the figure below is in equilibrium Assume that the seesaw is uniform and it has symmetric shape The pivot is placed right below its midpoint Which option below shows correct calculation of how far from the center of the seesaw the boy the child on the right is sitting 4m W 300 N O 4 400 300 O4 300 400 O 300 400 4 400 300 m W 400N

55 II A merry go round has a mass of 1440 kg and a radius of 7 50 m How much net work is required to accelerate it from rest to a rotation rate of 1 00 revolution per 7 00 s Assume it is a solid cylinder

A 42 g bug crawls from the center to the outside edge of a 170 g disk of radius 20 0cm The disk was rotating at 15 0 rad s What will be its final angular velocity Treat the bug as point mass Hint 1 The moment of inertia of a disk Idisk MR 2 Moment of inertia of a a point mass particle Iparticle bug R Winitial mr bug R fimal

4 Let the sun to be at the origin of a coordinate system The earth is orbiting around the sun The earth is also self rotating Let the distance of the earth from the sun D At time t the earth is at angle Et from the x axis The self rotation angle of the earth is St Give the transformation matrix of the earth You do not have to simplify the matrix multiplications 4 marks

In cycling the torque generated about the crank axis is determined by the magnitude of the force from a foot pushing the pedal as well as the angle between the crank arm and the force vector The diagram shows the 17 5 cm crank pedal at two positions 90 degrees and 135 degrees What is the torque at each of the points if the foot applies a force of 800 N Which position creates the larger torque Note Use the angle between the force application and crank arm in your calculations 17 5 cm 90 0 180 270 90 0 180 270

37 II A softball player swings a bat accelerating it from rest to 2 6 rev s in a time of 0 20 s Approximate the bat as a 0 90 kg uniform rod of length 0 95 m and compute the torque the player applies to one end of it

Foy Fe mg 3 0 m 21m Fw 1 40 m 2 8 m ME FM 15 12 cm 24 cm Problem 5 Consider a ladder with a painter climbing up the ladder as shown in the figure above on the left The length of the ladder is 5 0 m the mass of the ladder is m 12 kg and the mass of the painter is M 55 kg The ladder begins to slip at its base when her feet are a distance of 3 5 m up the length of the ladder Since the ladder begins to slip at its base the maximum friction force Ffric Nground is acting on the ladder where Nground is the normal force acting on the ladder from the ground There is no friction between the wall and the ladder What is the static friction coefficient between the ladder and the ground Answer 0 50

A uniform horizontal rod of length L and mass M is in complete equilibrium and has a frictionless pivot at the left end and is held in place by a wire at the right end as shown The wire makes an angle with the rod What is the magnitude of the total torque on the horizontal rod OMg sin 0 m N Mg cos 0 O Mg 2 sin 0 2Mg sin 0 m N O Mg m N m N O 4Mg m N Mg m N m N Mgm N O None of the above M L T

Two thin sheets on the surfaces of a slope of angle of elevation of 0 30 form a pair of rails The distance between the sheets is d 1 6 cm A solid ball of diameter of 2 cm is placed to the rails and its rolls down without slipping Assume that the ball does not touch the surface of slope Calculate the acceleration of the center of the ball

A 14 6 N force is applied to a cord wrapped around a pulley of mass M 3 22 kg and radius R 28 9 cm The pulley accelerates uniformly from rest to an angular speed of 26 7 rad s in 2 56 s If there is a frictional torque T 1 33 mN at the axle a determine the moment of inertia of the pulley b determine the rough estimate of the moment of inertia The pulley rotates about its center What is the difference be a and b 2

seesaw at an angle of 27 degrees at a distance of 7 5 m from the fulcrum e F input R How much force will their friend

A rotating device consists of a uniform bar mass M length L with a pivot at the center O and a small point mass m attached a distance L 3 away from the the left end L 3 m L What is the moment of inertia of the system Hint Moment of Inertia of the bar about an axis through one end Lend M L Moment of inertia of the bar about an axis through its center O L 2

Two identical disks rotate about fixed axes with negligible mass Identical masses are attached to strings wrapped around the axis and the outside rim respectively of the two disks The masses are then released from rest at the same height Just before each mass hits the ground which mass has more translational kinetic energy O The mass in system I The mass in system II 1 11

m 0 R m 1983M2 A uniform solid cylinder of mass m and radius R is mounted on frictionless bearings about a fixed axis through O The moment of inertia of the cylinder about the axis is I m R A block of mass m suspended by a cord wrapped around the cylinder as shown above is released at time t 0 a On the diagram below draw and identify all of the forces acting on the cylinder and on the block b In terms of m m R and g determine each of the following i The acceleration of the block ii The tension in the cord

Problem 2 A rod of mass M 0 2 kg and length 1m is initially at rest and it is pivoted about one of its ends A mass m 0 05 kg is moving with a speed vo 20 m sec towards the rod at a distance d 0 5 m from the pivot The mass m then gets embedded in the mass M and the system begins to rotate about the pivot after the collision The moment of inertia of a rod about one of its ends is I M and the moment of inertia of a particle is I mr a What is the angular speed w of the m M system after the collision Hint Angular Momentum is conserved about the pivot b What is the kinetic energy of the system before the collision c What is the kinetic energy of the system after the collision m vo Initial pivot M l Final pivot M d 8 me W

164 OR m Bullet Rod Block Problem 7 The puck in the figure above on the left has a mass of 0 120 kg The distance of the puck from the center of rotation is originally 0 4 m and the puck is sliding with a speed of u 0 8 m sec The string is pulled downward through the hole in the frictionless table and the new distance of the puck from the center of rotation is 0 25 m What is the new speed of the puck Answer vy 1 28 m sec Problem 8 In the figure above on the right a 0 001 kg bullet with initial speed 1800 m sec is fired into a 0 5 kg block attached to the end of a 0 60 m rod of mass 0 50 kg Point A is the pivot point for the rod and the bullet stays inside the block after the collision What is the angular speed w of the system after the collision The moment of inertia of the rod about point A is I Me and the moment of a particle is Imr Treat the block as a particle and also treat the bullet as a particle Answer 4 5 rad sec

Solid sphere of mass m and radius r R R 13 Before After Problem 3 A solid sphere of mass m and radius r rolls without slipping v rw along the track shown in the figure above on the left It starts from rest with the lowest point of the sphere at height h above the bottom of the loop of radius R much larger than r What is the minimum value of h in terms of R such that the sphere completes the loop The moment of inertia of a sphere is I 2 5 mr2 Hint You will need to draw a free body diagram of the ball at the top of the loop and you will need to find the minimum speed the ball needs to complete the loop Recall that the centriperal acceleration is a 2 Answer h 27 10 R 2 7R Problem 4 A disk with moment of inertia I rotates about a frictionless vertical axle with angular speed wi A second disk this one having moment of inertia 12 and initially not rotating drops onto the first disk see figure above on the right Because of friction between the surfaces the two eventually reach the same angular speed wf a Calculate wj Answer 1 wi h Colorleto KE the ratio of the final to the initial rotational energy Answer

Bullet Rod Block Problem The puck in the figure above on the left has a mass of 0 120 kg The distance of the pu from the center of rotation is originally 0 4 m and the puck is sliding with a speed of n 0 8m s The string is pulled downward through the hole in the frictionless table and the new distance of the puck from the center of rotation is 0 25 m 11 dat is the new speed of the puck Answer y 1 28m so

A 6 0 kg particle moves to the right at 4 0 m s as shown The magnitude of its angular momentum about the point O is 4 m s 6 kg O 288kg m s zero O 144kg m s O 24kg m s 249ka m s 12m 30 O

A disk rotates about an axis through its center Point A is located on its rim and point B is located exactly one fourth of the way from the center toward the rim What is the ratio of the angular velocity A to that of wg and the tangential velocity VA to that of VB HINT a the angular velocity WA WA WB 1 VB to that of w B b the tangential velocity VA to that of VB VA 0 25 Enter a number ASK YOUR TEACHE

If global warming continues it s likely that some ice from the polar ice caps of the Earth will melt and the water will be distributed closer to the Equator If this occurs the length of the day one rotation would do which of the following O remain the same O increase O decrease