Rotation Questions and Answers

34 Four masses are joined to a light circular frames in the figure The radius of gyration of this system about an axis passing through the centre of the circular frame and perpendicular to its plane would be where a is the radius of the circle A 1 7 2 3 a 2m0 R 3m m A 2 4 R 7 2 a 20 92m

solid sphere of mass M and radius R The sphere is on rough horizontal surface and there is no slipping What is acceleration of centre of mass of sphere X M R 5 F 2M 5 F 7M 5 F 3 M F

79 A uniform rod having length is hinged at one end and is free to rotate in vertical plane The rod is released from the position making angle 0 with the vertical The acceleration of the free end of the rod at the instant it was released is 1 3g sin 0 4 3a sing 0 VN AX 3g cos 0 2 short to inspM C 2 ng

A uniform rod of mass 2 kg and length rotate in a vertical plane about a fixed horizontal axis through one end The rod begins rotating from rest from its unstable equilibrium position Its angular acceleration when it has turned through an angle of 30 is 1 g rad s2 010 2g 3 3 rad s2 3g 2 2 MINTONO 3g 4 rad s2 rad s2

X3 ew from Let s now see what our symmetric top looks like from a fixed frame Euler s equations won t help much here because they deal with the components of w in the body frame But fortunately we can solve for the motion from scratch In terms of the changing principal axes 1 2 3 we have W W x1 W 2 W3x3 L I x1 2x2 13w3x3 Eliminating the w1x1 2x2 term from these equations gives in terms of the 2 defined in Eq 9 47 L 1 2x3 s L Nx3 where L L and is the unit vector in the L direction The linear relationship 9 52 LA 9 53

Column I a Ratio of change in time period of a simple pendulum with temperature to its original time period b Ratio of the value of a length to its scale reading c Reciprocal of coefficient of volume expansion for an ideal gas of constant pressure d F YA Column II i aAT ii T iii 1 AT iv 2 AT

Two thin discs each of mass M and radius r are attached to form a rigid body The moment of inertia of the system about an axis passing through the diameter of the disc and in the plane of the disc as shown in figure will be 1 7Mr 2 014 2 B X A 2 Mr2 2 3Mr

A uniform rod having length is hinged at one end and is free to rotate in the vertical plane The rod is released from the position making an angle with the vertical The acceleration of the free end of the rod at the instant it is released is 1 2 3 4 I I I I I 3gsin 4 3gcos 2 3gsin 2 3gcos 4 C

A particle moves in a circle with a uniform speed when it goes from a point A to a diametrically opposite point B the momentum of the particle changes by PA PB 2Kg m s j and the centripetal force acting on it changes by FA FB 8N 1 where i j are unit vectors Then the angular velocity in rad s of the particle is A 1 B 2 C 4 D 8

Q 24 A metal coin of mass 5g and radius of 1 cm is fixed to a thin stick AB of negligible mass as shown in the figure The system is initially at rest the constant torque that will make the system rotate about AB at 25 rotations per second in 5s is closed to A 3 Your Answer 2 0 x 10 5 Nm Option 1 4 0 x 10 6 Nm Option 4 Option 2 1 6 x 10 5 Nm Option 3 7 9 x 10 6 Nm 10 5 N

A mass m supported by a massless string wound around a uniform solid cylinder of mass m and radius R If the string does not slip on the cylinder with what acceleration will the mass fall after release m g 4g 3 R m

3 Three identical uniform rods each of length L are joined to form a rigid equilateral triangle as shown in the figure Radius of gyration of this system about an axis passing along the any rod in the plane of the triangle is 3 6 L 2 4 L ZLI 3 4

The moment of inertia of a uniform semicircular wire of mass mand radius r about an axis passing through its centre of mass and perpendicular to its plane is 1 mr 2 3 mr 1 4 2 2 mr 4 mr 1 4

Q 3 The circular disk shown rotates in horizontal plane about point O at uniform angular velocity w while O remains fixed Two points A and B are located on the line OZ at distances A and B from O respectively zeu qau aga ghaha h ufu w u fau an qufa Ged Gafch far far B 0 Tkut 1 am 7 A Z A The velocity of point B with respect to point A is w rB rA in a direction opposite to the direction of motion of point B fang A i ngang B i a fang B T n i farm w rB A a Your Answer B The velocity of point B with respect to point A is w rB A in a direction same as the direction of motion of point B far A fa B chi fa Ba rB A Correct Answer C The acceleration of point B with respect to point A is w B A in a direction from O to Z far A B w rB TA D The acceleration of point B with respect to point A is w rB rA in a direction from Z to O far A fa B chi carui Z 0 H 32 PR w r

vi On the flat surface of a disc of radius a a small circular hole of radius b is made with its centre at a distance c from the centre of the disc If mass of the whole uncut disc is M calculate the moment of inertia of the holed disc about the axis of the circular hole 1 M 2 22 b 9

The angular speed of a circular hoop is constant and is given by The hoop has a mass M and radius a and is rolling without slipping along the horizontal axis in the xy plane Calculate the total angular momentum of the hoop about the origin when the hoop s center is at a distance d 3a from the origin a Maw b 2Ma a c 2Ma o d 3Ma Answer OA B C

A cubical block of side a moving with velocity a horizontal smooth the plane as shown in figure It hits a ridge at point O and starts rotating about the edge in contact with O The angular speed of the block after it hits O is 1 3 MA 3v 4a 3 v 181 V 2 3v 2a DA mod beda 4 Zero O

A uniform cylinder of mass 1 kg and radius 2 cm is rotating about its main axis at the rate of 4 rpm The constant torque required to stop it after 2 revolution is 4 44 x 10 7 N m 1 x 10 7 Nm 5 x 10 6 Nm 1 78 x 106 Nm

51 A vehicle with a horn of frequency n is moving with a velocity of 30 m s in a direction perpendicular to the straight line joining the observer and the vehicle The observer perceives the sound to have a frequency n n Then if the sound velocity in air is 300 m s 1 n 0 1 n 2 n 0 3 n 10 n 4 n 0 1 n

ng of mass m 1 kg and radius R 1 25 m is kept on a rough horizontal ground A small body of mass m is struck to the top of the ring When it was given a slight push forward the ring started rolling purely on the ground What is the maximum speed of the centre of the ring in m s MECL

whose constant is 1 N m The middle offers a damping force that is numerically equal to twice the speed snapshot of the block The block is initially released from a point that is one meter above the equilibrium position with a downward velocity of 1 m sec Determine the equation of motion of the block VANAKA E

3 2 M I a I b 21 3 Four small spheres each of radius r and mass m are placed with their centres at the four com a square of side L The M I of the system about a diagonal of the square is 8mr a mL mL 5 b 8mr 5 2mL c b I sin 0 5mr 8 d c I cos 0 5mr 2 8 Let I be the M I of uniform square plate about an axis AB that passes through its centre parallel to two of its sides CD is a line in the plane of the plate that passes through the centre d plate and makes an angle with AB The M I of the plate about the axis CD is equal to a I 2mL d I cos 0

mass Example 7 15 A cord of negligible mass is wound round the rim of a fly wheel of 20 kg and radius 20 cm A steady pull of 25 N is applied on the cord as shown in Fig 7 35 The flywheel is mounted on a horizontal axle with frictionless bearings a Compute the angular acceleration of the wheel b Find the work done by the pull when 2m of the cord is unwound c Find also the kinetic energy of the wheel at this point Assume that the wheel starts from rest d Compare answers to parts b and c

A rod is bent into the L shape and hinged at O so that it can be rotated about z axis in x y plane as shown in the figure At the position shown in the figure the angular velocity is 2rad sec and angular velocity is decreasing at the rate of 4rad s Acceleration of end A will be A a 41 281 4m B 3m 90 B a 281 41 C a 281 41 D a 281 41

A rod of mass M and length L is hinged about 1 one end and a particle of mass M is attached to its other end If it is kept horizontal with the help of a spring of spring constant K as shown in fig find the extension in spring 1 D 3Mg K 1 2 Mg K 2 3 Rod M L 2Mg M 4Mg

12 A solid paraboloid shaped object is formed by rotating on parabola y 2x about y axis as shown in figure If the height of the body is h then the position of center of mass from origin Assume density to be uniform throughout A h 2 B h 3 C 2h 3 Y D 3h 5

A solid sphere of mass M and radius r slips on a rough horizontal plane At some instant it has translational velocity vo and rotati velocity about the centre vo 2r The translational velocity after the sphere starts pure rolling 0 V 2r Vo O 6vo 7 in forward direction O 6v 7 in backward direction O 7v 6 in forward direction O6vo 6 in backward direction

A boy is standing on a horizontal massless rotating wheel with his hands stretched Kinetic energy of boy is 120 J If he now folded his hands then M I is reduced to 80 Then a new K E of boy 1 Increases by 30 J 2 Decreases by 30 J 3 Increases by 20 J 4 Remains constant 8 Pusa Road New Delhi 110005 Ph 011 47623456

A uniform rod of length 12 metre and mass m is hinged at midpoint and lies horizontally at rest Rod is free to rotate in vertical plane about hinge O A particle of mass m is released from height h as shown in figure Particle collides with end A and sticks to it Calculate minimum height h in meter so that system can complete vertical circular motion

A basketball rolls without slipping around a basketball rim in such a way hat the contact points trace out a great circle on the ball and the CM moves around in a horizontal circle with frequency 2 The radii of the ball and rim are r and R respectively and the ball s radius to the contact point makes an angle 0 with the horizontal see Fig 9 52 Assume that he ball s moment of inertia around its center is I 2 3 mr2 Find 22 Fig 9 52 R 6

Maps 4 Four bodies a ring a solid cylinder a hollow sphere and a solid sphere of same mass are allowed to roll down a rough inclined plane without slipping from same level The body with greatest rotational kinetic energy at bottom is INCERT Pg 178 1 Ring 2 Solid cylinder 3 Hollow sphere 4 Solid sphere

A force Facts tangentially at highest point of a solid sphere of mass M and radius R The sphere is on rough horizontal surface and there is no slipping What is acceleration of centre of mass of sphere F M R 5 F 2 M 5 F 7 M 5 F 21 M

To one end of an uniform rod of length L a point mass is attached whose mass is twice the mass of the rod To what distance from the loaded end should a point mass system be struck in a direction perpendicular to length of rod so that it performs pure translational motion A B L 2 D L 3 C L 6 L 4

3 30 pts A wheel that is the shape of a solid disk has a radius of 35 5 cm The wheel is used to shape pieces of metal The wheel starts from rest and makes 75 3 revolutions in 25 8 s A What is the angular acceleration of the wheel B What is the angular velocity of the wheel at that time C When the metal piece is pushed against the wheel the metal piece exerts a tangential force to the wheel of 12 5 N to slow the wheel down How many revolutions does the wheel make to come to a stop Pictorial Representation Mathematical Representation Mass of disk 50 kg

equal to that of a lamina 27 Inside a fixed frictionless spherical cavity of radius r a dumbbell is touching the bottom of the cavity at its lower end The dumbbell cons of two identical small balls each of mass m affixed at the ends of a rod of length r 2 What will the force of interaction between the lo ball and the cavity be immediately after the dumbbell is released

When a carpenter shuts off his circular saw the 10 0 inch diameter blade slows from 3984 rpm to zero in 3 25 S A What is the magnitude of the angular acceleration of the blade Express your answer in radians per second squared to three significant figures B What is the distance traveled by a point on the rim of the blade during the deceleration Express your answer in feet to three significant figures C What is the magnitude of the net displacement of a point on the rim of the blade during the deceleration Express your answer in inches to three significant figures

3 M 0 1 1 m s 4 13 A uniform disc of radius 20 cm is perfom rolling on a moving surface if it s angula is 5 rad s then find velocity of it s centre V 2 m s 2 2 m s

A weight w rests on the bar PQ as shown in the figure The wire connecting and Q passes over two ideal pulleys Bar PQ has negligible mass The vertical component of reaction at P is 5 T Choose only 1 answer W W ba b 2a

A solid cylinder of mass M and radius R rolls without slipping down an inclined plane of length L and height h What is the speed of its centre of mass when the cylinder reaches its bottom A 2gh C 13 gh 3 B gh D 4gh

78 Pre Medical Physics A constant force of 80N is applied tangentially the rim of a wheel of radius 10cm The wheel h moment of inertia 40 kg m Then the work da on the wheel in 4 seconds 1 Zero 10cm 80N 2 10 J

A billiard ball of mass m radius R is given an impulse passing through its centre of mass so that centre of mass starts moving with a velocity Vo The coefficient of friction between ball horizontal floor is u After how long ball starts pure rolling 7Vo 1 2 g 2V0 3 7 g 2 g 2 7Vo 7 g 4 2Vo

A rod of mass m 2kg and length 1 3m rotates about an axis passing through one end O and perpendicular to its length The other end of rod slides without friction on a circular conducting rail negligible resistance Point O is the center of circular rail A resistance R connected to the rod and rail as shown in fig whose set up lies in horizontal plane with uniform magnetic field B directed in to plane of ring The rod is given angular speed wo and released The total heat generate in R is 27 joule Find value of wo in rad sec 3 X rod

A rod of length is pivoted at origin The moment of inertia of the rod about this axis if linear mass density of rod varies as ax b kg m 3 al4 bl 4 3 al bl 2 4 als bl 5 2 al5 bl L

Find the frequency with which the solid sphere radius r oscillates in the hallow sphere radius R The solid sphere rolls back and forth along the circumference of the hallow sphere when disturbed from equilibrium position R

help me Tigure the problem Determine the moment of inertia in4 IY of the area shown with respect to the vertical line that passes to the centroid of the composite area if b 5 78 in h 0 76 in L 7 93 in and W 0 72 in Round off only on the final answer to expressed in 3 decimal places b W h

85 min 10 secs 100 A solid disc of radius 5 cm rolls without slipping on a horizontal floor as shown with an angular velocity of 3 rad s and angular acceleration of 2 rad The magnitude of acceleration of point P on the disc at the given instant is in cm s 30 82

In the figure i a disc of mass M kg and radius R m is rotating smoothly about a fixed vertical R 2 axis AB with angular speed 26 rad s A rod CD of length m and mass M kg is hinged at one end at point D on the disc The rod remains in vertical position and rotates along with the disc about axis AB At some moment the rod CD gets a very small impulse at point C due to air due to which the rod falls on the disc along one radius and sticks to the disc as shown in figure ii Now find the angular velocity of the disc in rad s Figure i B Figure ii R B 3B 1

Two particles A and B of mass m each are joined by a rigid massless rod of length 7 A particle Po mass m moving with a speed u normal to AB strikes A and sticks to it The centre of mass of the A B P system is C P m 11 The velocity of C before impact is u 3 The velocity of C after impact is u 3 B m The velocity of A P immediately after impact is u 2 The velocity of B immediately after impact is zero

Two blocks of mass m and 2m connected by a weightless spring of stiffness k rest on a smooth horizontal plane Block of mass m is shifted to a small distance x to the left and the released If the duration between releasing mass m and breaking off of mass 2m is At then average force in mk n Find the value of n Newton exerted by wall on mass 2m is given as At 12mb 0000000 Im

performing pure translational motion is falling with speed vo just before hitting a smooth ground as shown in the figure If collision is inelastic e and 60 then 3 Option A CM g 0 Rate this question speed of centre of mass will becomes 3 Vo just after collision