Center of mass and momentum Questions and Answers

7 A rigid square is made up of four identical light massless rods of length a and two light diagonal rods as shown in the figure Four small objects of mass m 2m 3m and 4m are attached to the four corners of the square as shown a Now the system is rotated with an angular velocity o about an axis through A the center of the system perpendicular to the plane of the square Find the rotational kinetic energy of the system b If the system is now rotated about an axis through the rod connecting the two masses 2m and 4m with angular velocity w find the new kinetic energy of the system 4m 3m a O a a m a 2m c Compare the results from a and b and make a statement as to why they are the same or different depending on your result

Consider the following two statements a Among a given set of axis for a rigid body the moment of inertia is always minimum about the one passing through the center of mass b Perpendicular axis theorem for moment of inertia would be applicable for a solid sphere The correct statement s is are 1 Only a 2 Only b 4 Neither a por b

B The figure shows the positions and velocities of two particles If the particles move under the mutual attraction of each other then the position of centre of mass at t 1 s is A x 5 m 1 kg 5 m s x 2 m B x 6 m 3 m s 1 kg x 8 m C x 3 m D x 2 m haight

9 A particle of mass m moving with horizontal speed 6 m s collides elastically with another particle of mass M m M moving in the same direction with speed 4 m s Find the speed of lighter particle after collision 1 2 m s in original direction 2 2 m s opposite to the original direction 3 4 m s opposite to the original direction 4 4 m s in original direction

1 A cubical block of side a moving with velocity v on 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 3v 4a 3 v 184 V 2 3v 2a 4 Zero

13 A table with smooth horizontal surface is turning at an angular speed w about its axis A groove is made on the surface along a radius and a particle is gently placed inside the groove at a distance a from the centre Find the speed of the particle with respect to the table as its from the centre becomes L

12 A ball of mass 1 kg is dropped from height 9 8 m strikes with ground and rebounds at height of 4 9 m if the time of contact between ball and ground is 0 1 s then find impulse and average froce acting on ball a 23 52 N s 235 2 N c 42 5 N s 525 N b 235 2 N s 23 53 N d 52 5 N s 525 N

A uniform rod of length 4 m and mass 2 2 kg revolves with constant angular velocity o about a vertical axis through a smooth joint A at one extremity of the rod so that it describes a cone of semi vertical angle 53 as shown in the figure Choose the correct statement s g 10 m s A The value of angular velocity is 2 5 rad s B The angle between hinge reaction at point A and vertical axis AC is 45 C The magnitude of hinge reaction at point A is 40 N D The angular momentum of the rod is constant 9 A 53 B

It is found that if a neutron suffers an elastic collinear collision with deuterium at rest fractional loss of its energy is p while for its similar collision with carbon nucleus at rest fractional loss of energy is p The values of Pa and P are respectively JEE Main 2018 1 28 89 2 0 0 3 0 1 In a collinear collision a particle with an initial cpood u otril 4 89 28

on A ball of mass m hits a floor with a speed v making an angle of incidence 0 45 with the normal rivom to the floor If the coefficient of restitution is e 1 2 of reflection 0 45 V find the speed of the reflected ball and the angle AIPMT Mains 2005 Since the floor exerts a force on the ball along the normal during the collision so horizontal component of velocity remains same and only the vertical component changes

In a step up transformer the turn ratio is 1 2 If voltage across primary is 1 5 V then voltage developed in the secondary would be 1 3 0 V 2 0 75 V 3 1 5 V 4 Zero

A ball of mass m moving at a speed v makes a head on collision with an identical ball at rest The kinetic energy of the ball after the collision is 4 of the original Find the coefficient of restitution 1 3 3 5 1 2 4 2 1 5 13

1 Determine the impedance seen by the source V 2420 V in the network shown in figure 202 11 0 ww ele 30 RO 4H ele 20

3 A body of 1000 kg moving with 0 3 m s in x direction A hole is present at the bottom of body such that mass is dropping at rate of 20 gm sec in y direction Find out force exerted by ejecting mass on body 1 0 4 N 3 0 6 N 2 0 N 4 None of these

Example 11 22 A ball of mass m moving at a speed v makes a head on inelastic collision with an identical ball at rest The kinetic energy of the balls after the collision is th of the original Find the coefficient of restitution 3 4 m V m V After collision V

3 A solid sphere of radius R is placed on a smooth horizontal surface A horizontal force F is applied at height h from the lowest point For the maximum acceleration of centre of mass which is correct 1 h R 2 h 2R 3 h 0 4 Centre of mass has same acceleration in

Three identical uniform spheres each of mass M and radius R are placed as shown in figure The moment of inertia of the system about the axis AB is A 3 E omil 1 MR 46 5 19 MR 2 4 J 36 5 17 MR MR B F

at height of from its base 4R MOTION OF CENTRE OF MASS 1 Motion of Centre of Mass 2R As for a system of particles position of centre of mass give by R m m CM m m m

Find the ratio of magnitude of angular momentum of a rolling disc about origin to that about the point P on the disc as shown in the figure A 2 1 C 1 3 B 1 1 D 4 3 YA M RYP R Ax

A wheel of mass 5 kg and radius 0 4 m is rolling on a road without sliding with angular velocity 10 rad s 1 The moment of inertia of the wheel about the axis of rotation is 0 65 kg m2 What is the percentage of the kinetic energy of rotation in the total kinetic energy of the wheel 44 8

1 Consider a gravity free hall in which a tray of mass M carrying a cubical block of ice of mass m and edge L is at rest in the middle If ice melts by what distance does the centre of mass of the tray and ice system descend 1 3 L L 3L 2 L 4 4 Zero 4 19

A diatomic molecule is formed by two atoms which may be treated as mass points m and m2 joined by a massless rod of length r Then the moment of inertia of molecule about an axis passing through centre of mass and perpendicular to the rod is C Zero m m r r Correct Answer m1 m2 m1m2 mi m

In which of the following cases the centre of mass of a rod may be at its centre 1 The linear mass density decreases continuously from left to right 2 The linear mass density increases continuously from left to right 3 The linear mass density decreases from left to right upto the centre and then increases 4 The linear mass density increases from left to right upto the centre and then decreases D 1 4 A 1 2 B 3 4 C 2 4

37 In a one dimensional collision between two identical particle A and B B is stationary and A has momentum P before impact During impact B gives an impulse J to A Then coefficient of restitution between the two is 2J P a 2J b 1 c 1 p P d J 1 P

18 A uniform thin rod AB of length L has linear mass density x a 7 124 the CM of the rod lies at a distance of bx L L from A then a and b are related as 1 2a b 2 3a 2b 3 a 2b 9 A large number n of identical beads each of mass m and radius raro strung where x is measured from A If JEE Main Online 2015 4 a b La

At a place true dip is 45 and apparent dip is 60 If plane of dip circle is rotated by 90 then dip shown by it will be 1 30 3 cot 1 INIM 2 2 cos 4 sin 1 3 2 23

dius r is joined to a uniform solid right circular cone of base radius r and height 3r If both have same density then find the position of centre of mass from centre of hemisphere 7638 O 3r 1 12 2 distance from O r 2 distance from O 3 r distance from O r NIM V 1x8 h 3 x x x x p 53

B A circular platform is mounted on a frictionless vertical axle Its radius R 2m and its moment of inertia about the axle is 200 kg m It is initially at rest A 50 kg man stands on the edge of the platform and begins to walk along the edge at the speed of 1 m s relative to the ground Time taken by the man to complete one revolution is 1 2n s 3 S 2 4 BIN S 3 2 S

12 The moment of inertia of a uniform circular disc is maximum about an axis perpendicular to the disc and passing through rbans 1 D 3 B B C A D 2 A 4 C

A large number n of identical beads each of mass m and radius r are strung on a thin smooth rigid horizontal rod of length L L r and are at rest at random positions The rod is mounted between two rigid supports see figure If one of the beads is now given a speed v the average force experienced by each support after a long time is assume all collisions are elastic JEE Main Online 2015 1 mv 2 L nr 2 my nr L 3 zero 4 T mv

5 A mass m moves in a circle on a smooth horizontal plane with velocity vo at a radius Ro The mass is attached to a string which passes through a smooth hole in the plane as shown 1 R The tension in the string is increased gradually and finally m moves in a circle of radius The final value of the kinetic energy is 1 mv m 2 2my 3 Ro 2 my 4 mir

and U2 1 Two identical mas M moving with velocity u collide perfectly inelastically The loss in kinetic energy is M 1 x 4 4 2 M 4 3 X U 4 2 1 2 4 4 Mx 4 1 1 X U 2 block of thickness

Consider the following two statements A Linear momentum of a system of particles is zero B Kinetic energy of a system of particles is zero Then 1 A does not imply B and B does not imply A 3 A does not imply B but B implies A AIE 2 A implies B but B does not imply A 4 A implies B and B implies A

0 In the figure shown ABC is a uniform wire If centre of mass of wire lies vertically below point A then AE is close to JEE Main Online 2016 1 1 85 2 3 B 60 2 1 C C LAY 1 37

A particle of mass m strikes elastically with a disc of same mass with speed v as shown in figure All the surface of contacts are smooth The velocity of disc just after collision is consider tangential line as x axis and radial toward C as y axis at point of impact m V RIZ C

A solid sphere is rotating freely about its symmetry axis in free space The radius of the sphere i increased keeping its mass same Which of th following physical quantities would remain constar for the sphere 1 Angular velocity 2 Moment of inertia 3 Rotational kinetic energy 4 Angular momentum

7 Consider the following two statements a body may have momentum without energy b A body may have energy without momentum The correct statements among the following are 1 a only 3 Both a and b 2 b only 4 Neither a por h

A railwagon moves at velocity v on a track toward three other railwagons at rest on the track All the wagons are identical and each stationary wagon is some distance from the others The moving wagon hits the first stationary wagon and locks onto it Then the two hit the second stationary wagon lock onto that one and so on When all four railwagons have locked onto one another what is the velocity vy of the four wagon train OAV 4V OBVES 1 2vJ OC 1 4 OD V 1 8V OE V 4 Reset Selection

Three identical rods are hinged at point A as shown The angle made by rod AB with vertical is a tan tan Q 3 4 b tan c tan 1 d 1 4 3 B 90 90 D

Four particles A B C and D with masses mA m mB 2m mc 3m and mp 4m are at the corners of a square They have accelerations of equal magnitude with directions as shown The acceleration of the centre of mass of the particles is to B t a A 1 i 1 AY C D to a X a 2 j 5 d

1 All the particles of a body are situated at a distance R from the origin The distance of centre of mass of body from origin is 1 2R 3 R 2 R 4 R

8 Pick three different trials that have a different mass ratio For each trial that you pick fill in the table below Trial A E H m m 0 5 1 0 1 0 2 0 1 50 1 0 VJ Vin 60 15 30 15 20 30 9 Analyze the data in the table above State a rule that describes how to predict the relat ve velocities of the two carts after the explosion

On a smooth level surface a red marble of mass 0 02 kg moving at 2 0 m s collides with a stationary yellow marble of equal mass After the collision the red marble stops completely and the yellow marble moves in the direction the red marble was moving a Calculate the momentum of both marbles before and after the collision Show your calculations and include units in your answer b If the velocity of the red marble doubles how will the velocity of the yellow marble change after the collision c If the red marble had more mass than the yellow marble how would the momentum of the yellow marble change after the collision

A slab of concrete of length L 1 6 m is supported by rollers as shown in Figure 1 If the slab is pushed to the right it will move off the supporting rollers one by one on the left side Figure L 1 of 1 Part A How far can the slab be moved before its center of gravity is to the right of the contact point with the rightmost roller at which point the slab begins to tip Express your answer with the appropriate units S 1 6 Submit m Previous Answers Request Answer X Incorrect Try Again 5 attempts remaining

Two balls of equal mass m undergo a collision One ball is initially stationary After the collision the velocities of the balls make angles of 25 5 and 45 9 relative to the original direction of motion of the moving ball a Draw and label a diagram to show the balls before and after the collision Label the angles and b Calculate the final speeds of the balls if the initial hall had a speed of 3 63 m s

spokes are massless and that F 1 N In order to impart identical angular accelerations how large must F be M 1 kg O 0 5 N 0 25 N 1 0 N 2 0 N 1 5 N 4 0 N 8 0 N F R 0 5 m M 1 kg F2 R 1 0 m

7 29 A 1000 kg car moving at 20 m s hits a 2000 kg parked car head on How much mechani cal energy is dissipated if the collision is a elas tic b completely inelastic

2 A 6 2 kg object heading north at 3 0 m s collides with an 8 0 kg object heading west at 3 5 m s If these two masses stick together upon collision what is their velocity after collision 2 4 m s 56 W of N

41 II A dad pushes tangentially on a small hand driven merry go round and is able to accelerate it from rest to a frequency of 15 rpm in 10 0 s Assume the merry go round is a uniform disk of radius 2 5 m and has a mass of 560 kg and two children each with a mass of 25 kg sit opposite each other on the edge Calculate the torque required to produce the acceleration neglecting frictional torque What force is required at the edge

large 10 kg sports ball is caught by a 70 kg student on the track team If the ball was moving at 4 0 and the student was stationary before catching the ball how fast will the student be moving after catching the ball P Initial Impulse P P Change in Momentum R O Final P Pute