Center of mass and momentum Questions and Answers

Directio A rod of mass M and length L is suspended by a frictionless hinge at the point O as shown in figure A bullet of mass m moving with velocity v in a horizontal direction strikes the end of the rod and gets embedded in it M L 29 The angular momentum of the system about O befor collision is A mvL 1 C myL B MvL D MVL

P2 Block B of mass 3m on a frictionless horizontal surface attached to a spring with elastic constant k 200 N m The other end of the spring is fixed to a wall Block A of mass m 2 0 kg and moving initially with speed vo 4 0 m s collides elastically with the stationary block B Calculate 10 marks a Speed of block A immediately after the collision b Speed of block B immediately after collision c After collision what is the maximum compression of the spring 772 15 3m k

3 2 A particle of mass m moving with a velocity v a wall at an angle of incidence 0 and rebounds back as shown in the figure Calculate change in momentum y f L 1 2mv cose i 3 2mx sine i X 2 2mv cose 1 4 my sine i

Two astronauts with masses m and m are moving with the velocity u together They push each other off in the direction perpendicular to u spending the energy AE What is the angle between the velocities of the astronauts after the push off

In the figure shown each tiny ball has mass m and the string has length L One of the balls is imparted a velocity u in the position shown in which the initial distance between the balls is 3L 4 If the impulse of the tension in the string when it becomes taut is ma kg ms then find the value of a Given m 2kg u 15 m s L in 53 u 3L 4

gy In a torsional pendulum a uniform disc of radius 0 1m and mass 0 250kg is fixed at its center with a metal string and the other end of the string is fixed with a clamp If the time period of torsional oscillation is 0 2 sec find the torsional constant C of the system

Phy Section B This section contains NUMERICAL VALUE TYPE questions Enter the correct numerical value in Read More From a solid sphere of radius R a quarter of sphere is cut out as shown The centre of mass of remaining 2R part is at distance from centre of sphere Find n value of n

On a smooth horizontal surface there is a string of length 16 m and mass equal to that of man Now then man starts collecting the string Finally the man collects all the string and puts in his pocket What is displacement of man w r t earth in m in process of collecting 1111

A solid cone and a solid hemisphere of equal mass are stuck together to form a composite body as shown The distance of centre of mass of system from centre of base of cone is x R 16 R h R

Radius of bigger hollow hemisphere R 5 times radius r of sphere mass of hemisphere M 3 times mass m of sphere All surfaces are smooth Sphere is released from the position when all are at rest When sphere reaches bottom displacement of sphere is

A uniform rod released on a smooth horizontal surface from rest as shown in figure Rod makes an angle 8 60 with horizontal initially A Acceleration of centre of mass of rod just after releases in 3g 7 3g vertical B Acceleration of centre of mass of rod just after releases in C Acceleration of point of contact just after release is D Acceleration of point of contact just after release is 3 3g 3 3g Horizontal 0 60 rest

de decays to three particles One of them has energy 20 GeV and tum along the x axis of 10 GeV c Another has energy 15 GeV momentu ne y axis of 10 GeV c and momentum along the z axis of 5 GeV c The thir has energy of 15 GeV and momentum along the z axis of 15 GeV c All th omponents of the three particles momenta are zero What was the mass aying particle to the nearest integer in GeV c

A wheel of moment of inertia about its centre of mass 4kg m is spinning with an angular velocity of 15rad s If the wheel is brought to rest by a braking torque is 2 5 second then value of the torque is A 24 N m B 26 5 N m C 37 5 N m D 60 N m

4 Kinetic energy is 2 M m 2 A free body of mass 8 kg is travelling at 2 meter per second in a straight line At a certain instant the body splits into two equal parts due to internal explosion which releases 16 joules of energy Neither part leaves the original line of motion finally 1 Both parts continue to move in the same direction as that of the original body 2 One part comes to rest and the other moves in the same direction as that of the original body 3 One part comes to rest and the other moves in the direction opposite to that of the original body 4 One part moves in the same direction and the other in the direction opposite to that of th

8 The velocities of two particles A and B of same mass are A V ai and VB bj where a and b are constants The acceleration of particle A is 2ai 4bj and acceleration of particle B is ai 4bj in m s The centre of mass of two particle will move in 1 straight line 2 parabola 3 ellipse 4 circle

A thin uniform rod of mass 2m and length L hinged at its upper end hangs in equilibrium A small bullet of mass mis fired horizontally with velocity v The bullet hits rod very close to its bottom end and gets embedded in it 2 m L If after the collision the rod comes to rest in the horizontal position then the correct relation is 16gL 3 0 O220gL 3 O v 32gl 5 75gL C 2

A body of mass m is dropped from a height h Simultaneously another body of mass 2m is thrown up h vertically with velocity v such that they collide at height 2 If collision is perfectly inelastic then speed with which the combined mass system hits the ground is O 2gh O 3gh 7 gh 3

1 A bag mass M hangs by a long thread and a bullet mass m comes horizontally with velocity v and gets caught in the bag Then for the combined bag bullet system 1 Momentum is mvM M m 2 mv 2 2 Kinetic energy is 3 Momentum is mv M m M 4 Kinetic energy is m v

7 100 O Fig 6 b R 20 10 All dimensions are given in cm

The beam is subjected to applied loads as shown Determine the reactions at the supports A 20 kips 12 kips k 9 kips downward 96 kips upward 96 kips downward 2 kips downward 12 kips upward 27 kips downward 27 kips upward 12 kips downward 18 kips upward 20 kips upward 2 kips upward 18 kips downward 9 kips upward 26 kips downward 20 kips downward 6 ft 1 5 kips ft B D for t 101 8R 8 ft ft Lu E

18 A machinist starts with three identical square plates but cuts one corner from one of them two corners from the second and three corners from the third Rank the three according to the x coordinate of their centre of mass from smallest to largest y 1 1 3 1 2 3 3 2 1 2 1 1 3 2 1 3 2 4 1 and 3 tie then 2

21 A body of mass 5 kg explodes at rest into three fragments with masses in the ratio 1 1 3 The fragments with equal masses fly in mutually perpendicular directions with speeds of 21 m s What is the velocity in m s of the heaviest fragment

The carts in figure are sliding to the right at 2 m s on a smooth level ground The spring between them has a spring constant of 60 N m and is compressed 60 cm The carts slide past a flame that burns through the string holding them together The carts are not attached to the spring Afterward what is the speed in m s of 200 g cart after it loses contact with the spring 000000000 200g 2 m s 600g

rope is wound around a hollow cylinder of mass 3 kg and radius 40 40x cm What is the angular acceleration of the cylinder if the rope is pulled with a force 30 N a 0 25 rad s c 5ms 2 b 25 rad s 2 d 25 m s 2 d Ple

Consider a horizontal surface moving vertically upward with velocity 2 m s A small ball of mass kg is moving with velocity 21 23 If coefficient of restitution is 1 2 and friction coefficient is 1 3 then find horizontal component of velocity of the ball after collision y ay L 2 m s X

The wedge with hemi spherical cut of radius R has mass 5m It is placed on friction less horizontal surface A small sphere of radius r and mass m is released from rest from point A shown When m reaches B the speed of the wedge is x A R smooth g R r 60 2 find x

A small ball is thrown horizontally from a 20 m high window of a building with velocity 10 m s The ball collides with the smooth horizontal ground and immediately after the first rebound its velocity vector makes a 45 angle with the horizontal Let the maximum height reached by the ball between its first and second rebound be Hand let the angle that its velocity vector makes with the horizontal after the second rebound bee Which of these options is are correct g 10 m s 0 H 10 m H 5 m 8 tan 1 0 tan 9

Q 3 54 A window of a building with velocity 10 m s The ball collides with the smooth horizontal ground and immediately after the first rebound its velocity vector makes a angle with the horizontal Let the maximum height reached by the ball between its first and second rebound be Hand let the angle that its velocity vector makes with the horizontal after the second rebound be Which of these options is are correct g 10 m s H 10 m H 5 m 8 tan 52

7 Two bodies of masses 2 kg and 4 kg are m with velocities 2 m s and 10 m s towards each due to mutual gravitational attraction hat velocity of their centre of mass in magnituc 1 5 3 m s m v m v mit my 2 7 3 m s 3 Zero 2X2 4X 4 6 m s 20 If the momentum of a body increases by 20

9 a A shell of 80 kg mass is travelling with a velocity of 500 m s During its motion it splits into two equal portions which continue to travel in the same direction If an energy equivalent to 1 5 106 Nm is released at the instant of explosion determine the subsequent velocities of the two portions

Both the blocks oscillate with the same time period but their amplitude of oscill ation is different Find maximum elongation in the spring and the maximum compression in the s pring Also find time after which elonga tion will be maximum and compression will be maximum for the first time 3 K this 2m

A wedge is placed on ground All the surfaces are smooth The block rises to the highest possible point on the curved track and returns back The velocity of small block when it returns on to flat part is 3m V m

a Lightning bolt A hits a pole 200 m west of Albert Lightning bolt B hits a tree 800 m west of him Albert sees the flash from bolt A 1 0 s before he sees the flash from bolt B In Albert s rest frame which lightning strike occurred first b What was the time difference between the two strikes Enter O if the strikes were simultaneous otherwise enter a positive number Units us tolerance 0 1 s

A ball strikes a smooth horizontal ground at an angle a 45 with the vertical What cannot be the possible angl of its velocity with the vertical after the collisior Assume e 1 1 45 2 30 3 53 A

Three particles of different masses and velocities collide with each other and stick together to form a single particle Find the speed of the newly formed particle O u 2m D U 60 60 m 2u 3m u

A smooth disc of radius R is placed on a smooth surface Another particle of same mass strikes it as shown elastically After collision angle between linear velocities of particle and disc is 530 37 74 90 2R

A ball of mass m is fired from a cannon of mass nm without the ball with a relative speed of vo The ball is projected horizontally smooth A The total energy released is B The speed of ball is nvo n 1 C The speed of cannon is 1 mn va 2 n 1 Vo n 1 D The total energy released is 1 mv 2n 1

A person slowly pulls a water bucket of mass 2m from a well with a uniform rope of mass m and length I starting with free end in his hand Then work done by the person is mgl O 5 2 mgl O 2mgl O 3 2 mgl

man of mass M stands at one end of a plank of length I of mass 2M that rests on a smooth horizontal surface If the man starts walking towards the other end of the ank with a constant acceleration a relative to the plank the velocity of the plank at the instant the man reaches its other end is 2aL 9 14a L 9

A point mass of 2 kg collides elastically with a stationary point mass of 5 kg The 2 kg mass reverses its direction and moves with a speed of 4 m s after their collision Kinetic energy of center of mass will be in J

Two equal masses of 4 kg each have velocities v 21 21 ms and V 41 41 ms initially 1 If a constant force F 241 N is applied on the system for 5 seconds then the final velocity of centre of mass after the action of force is 1 18i j 2 18 5j 3 31 5j 4 31 18

Two point masses M are kept fixed on the x axis at a distance a from th origin another point mass m is moving in a circular path of radius R in y plane under the influence of gravitational force of attraction then speed m will be Assume no forces are acting on m other than the gravitational forces by tw M A C 2GMR R a 3 2 2 3GMR 3 R a 2 2 M B D M GMR 3 R a 5GMR R 2 a 3 MIN 2

Three blocks A B and C are placed as shown in figure Block A has mass m and moving with velocity v Block B of mass m and block C of mass 4 m are initially at rest Neglect friction Assuming all collisions are elastic the final velocity of block A is 1 3v 5 2 zero 3 v to the left 3v 4 A m to the left to the right B m C 4m

0 A mass of 3 kg is placed on the massless piston of area 10 cm of a cylindrical vessel containing a gas Find the fractional change in surface area of sphere if its bulk modulus is 2 x 104 N m and placed in the given vessel 1 0 5 2 1 5 3 1 4 Zero

option in this case will result in 1 marks Selecting any incorrect option s second option in this case with or without selection of any correct option s will result in The figure shows a L shape object of uniform mass per unit area x coordinate of center of mass is 5a 6 y coordinate of center of mass is 5a 6 x coordinate of center of mass is 2a 3 a

A uniform solid sphere of mass M and radius R is lying on a rough M horizontal surface A particle of mass M strikes the sphere at an angle with the horizontal as shown in the figure During the collision A momentum of the particle is conserved along the direction sin ei cos 01 1 2 B momentum of the sphere is conserved in the direction C momentum of the particle is conserved along the direction sin 01 cos 01 D none of the above M AY 0 0

A bomb at rest explodes into two pieces of mass 2 kg and 4 kg respectively If the relative velocity of pieces after explosion is 60 m s then kinetic energy of lighter particle is Only One Correct Answer A B C P 1600 J 800 J 600 J 1200

b The magnitude and direction of velocities of two identical frictionless balls before they strike each other is shown in Fig 9 b Assume e 0 9 determine the magnitude and direction of velocities of each ball after impact 30 UA 3 m s B 60 Line of impact 4 m s I

31 Blocks of masses m 2m 4m and 8m are arranged in a line of a frictionless floor Another block of mass m moving with speed along the same line see figure collides with mass m in perfectly inelastic manner All the subsequent collisions are also perfectly inelastic By the time the last block of mass 8m starts moving the total energy loss is p of the original energy Value of p is close to A 87 B 37 m m 2m C 77 4m 8m D 94

38 A tennis ball is released from a height h and after falling freely on a wooden floor it rebounds and reaches height h 2 The velocity versus height of the ball during its motion may be represented graphically by graphs are drawn schematically and on not to scale A h 2 h B h 2 h C h 2 D 1 8 28