Center of mass and momentum Questions and Answers

A cubical vessel with open top of side length 1m is filled with water Now it start moving with constant acceleration 19 6 m s2 in horizontal direction Then the amount of water split out is 1 250 kg 3 750 kg 2 500 kg 4 900 kg

25 In a rectangle ABCD BC 2 AB The moment o inertia is minimum along axis through F a BC HE A E B H b BD FC D C G

ment of The pulley of vertical oscillation of its centre of mass The spring has spring constant k and the string does not slip over inertia I about its axis and mass m Find the time period the pulley Th elle

A spring loaded toy sits at rest on horizontal frictionloss surfaco Whon the spring roloases the toy breaks into three equal mass pieces A B and C which slide along the surface Place A inoves off in tho nogalivo x diroction whilo pioco B movos off in tho nogativo y diroction Which of the throo pieces is moving the fastest NTSE Stage 1 2016 A A C C 8 B D They move with identical speeds

A ball collides with another identical ball which was initially at rest The speed of the second ball after collision is V a b 3v W C DE V m e 1 2 0 U 0 d V m res

4 FORCE AND NEWTO LAW OF MOTION A cart of mass M moves at a speed u on a frictionless surface At regular intervals of length L M blocks of mass ma drops vertically into the cart How much time is taken to cover a distance of 2 4 9 2 L SL R SL 19L C NTSE Stage 11 2018 17L D

figure 10 107 S 10 B 20 NTSE Stage 2019 Which one of the following statements is correct for the interval from 0 to 20 s A The momentum of the object decreases by 75 kg m s B The momentum of the object increases by 75 kg mis C The momenturn of the object increases by 125 kg m s D The change in momentum cannot be found as initial speed is unknown

40 Two equal masses are situated at a separation ro One of them is imparted a velocity vo perpendicular to the line joining them both are free to move Treating motion only under mutual gravitatio force Find the ratio of maximum and minimum separation between them

The block of mass M moving on the frictionless horizontal surface collides with a spring of spring constant K and compresses it by length L The maximum momentum of the block after collision is 1 VMKL 3 zero 2 4 KL 2M ML K M m

A ball of mass m strikes a rigid wall with speed v and gets reflected without any loss of speed as shown in the figure 30 30 a What is the magnitude of the impulse imparted to the ball by the wall b What is the direction of the force on the wall due to the ball

3 does not change 79 The graph between the angular momentum J and angular velocity for a body will be 3 2 4 CI a work Indicate of the 1 4 8 3 2 84 A bod rotatin then 1 1 3 5

1 2 A particle of mass 1 kg is thrown vertically upward with speed 100 ms After 5 s it explodes into two parts One part of mass 400 g emerges with speed 25 ms in downward direction what is the ma velocity of other part just after explosion a 100 ms upward 1 b 600 ms upwarded d 300 ms upward 1 c 100 ms 1 downward

2 A person holding a rifle mass of person and rifle toge is 100 kg stands on a smooth surface and fires 10 horizontally in 5 s Each bullet has a mass of 10 muzzle velocity of 800 ms The final velocity ac by the person and the average force exerted on the 8 V are a 0 08 ms 16 N 1 c 1 6 ms 16 N b 0 8 ms 8 N d 1 6 ms 8 N

3 A cart of mass M is at rest on a frictionless horizontal surface and a pendulum bob of mass m hangs from the roof of the cart figure 9 E6 The string breaks the bob falls on the floor makes several collisions on the floor and finally lands up in a small slot made in the floor The horizontal distance between the string and the slot is L Find the displacement of the cart during this process

Particles of mass m 2m 3m nm gram are placed on the same line at distances l 21 31 ne cm from a fixed point The distance of centre of mass of the particles from the fixed point is n n 1 2n 1 6 n n 1 3 2n 1 e 3 n n 1 e 2 l n 1 2l 4 n n 1

7 hasses are attached by a lightweight rod This object orbits a planet the length of the rod is smaller than the radius of the orbit but not negligible The rod rotates about its axis in such a way that it remains vertical with respect to the planet Assume in each case perturbation maintains the rate of rotation so that in the co rotating frame the rod is still stationary but at an angle to the vertical 1 There is no force in the rod the equilibrium is neutral with respect to a small perturbation in the angle of the rod 2 The rod is in tension the equilibrium is stable with respect to a small perturbation in the angle of the rod 3 The rod is in compression the equilibrium is stable with respect to a small perturbation in the angle of the rod 4 The rod is in tension the equilibrium is unstable with respect to a small turbation in the angle of the rod

3 In the figure given below the position time graph of a particle of mass 0 1 kg is shown The impulse at t 2 sec is A 0 2 kgms 1 B 0 2 kgms 1 C 0 1 kgms D 0 4 kgms X m 6 642 2 4 t seconds

JEE Main 2013 4 16 Distance of the centre of mass of a solid uniform cone from its vertex is zo If the radius of its base is R and JEE Main 2015 its height is h then zo is equal to 5h 3h h 1 2 3h 8 8R 4R 4 4 17 A particle of mass m moving in the x direction with speed 2u is hit by another particle of mass 2m moving in th u direction with snood y If the collisions perfectly inolastic the 3

A plate of uniform thickness is shown in figure The coordinates of its centre of mass are 1 3 2a 5a 5a 6 6 4a 4a 5 5 9 y4 Sh 2a My FP 2 X 2a 2a 3 3 4 a a

9 An ornament weighing 36 g in air weighs only 34 g in water Assuming that some copper is mixed with gold to prepare the ornament find the amount of copper in it Specific gravity of gold is 19 3 and that of copper is 8 9

4 kg 3 A fan of moment of inertia 2 4 kgm2 is to run upto a working speed of 0 5 revolution per second Indicate the correct value of the angular momentum of the fan 1 4 8 kg x 3 2 41 kg x metre sec 2 metre 2 9 6 kg x 4 TC 2 metre sec metre kg x ITIONAL MOTere ATIONAL MOTION 02 ROTATIONAL

A parallel beam of monochromatic light of wavelength 663 nm is incident on a totally reflecting plane mirror The angle of incidence is 60 and the number of photons striking the mirror per second is 10 x 10 Calculate the force exerted by the light beam on the mirror 19

A small block of mass m is placed on a plank of equal mass m All the surfaces are smooth The block is pushed and given an initial speed of u as shown in the diagram The block slides on the horizontal surface and then breaks off the plank at a negligible height in the vertical direction with respect to the plank U A the velocity of the plank when the block breaks the contact is 2 Y U 2 B the velocity of the block when it breaks the contact with the plank is Im 1 m horizontal surface 2 C the ratio of kinetic energies of the block and the plank when the block breaks the contact is 3 1 D the path of the block while it is in the air is a parabola with respect to the ground and a vertical path with respect to the plank

A thin and uniform rod of mass M and length L is held vertical on a floor with large friction The rod is released from rest so that it falls by rotating about its contact point with the floor without slipping Which of the following statement s is are correct when the rod makes an angle 60 with vertical g is the acceleration due to gravity A The radial acceleration of the rod s center of mass will be B The angular speed of the rod will be 3g V2L 2g C The angular acceleration of the rod will be L 3g 4 Mg D The normal reaction force from the floor on the rod will be 16 mg s I J5o 2019

Four identical particles each of mass 1 kg are arranged at the corners of a square of side length 2 2 m 13 1 If one of the particles is removed the shift in the centre of mass is 8 1 m 4 m 2 zm m 4 2m

37 A man of mass m moves with a constant speed on a plank of mass M and length 1 kept initially at rest on a frictionless horizontal surface from one end to the other in time t The speed of the plank relative to ground while man is moving is 2 M t c M t M m b m tm M d None of these

Six identical balls are lined in a straight groove made on a horizontal frictionless surface Two similar balls each moving with a velocity v collide elastically with the row of 6 balls from left What will happen 1 One ball from the right rolls out with a speed 2v and the remaining balls will remain at rest 2 Two balls from the right rolls out with speed v each and the remaining balls will remain stationary 3 All the six balls in the row will roll out with speed v 6 each and the two colliding balls will come to rest 4 The colliding balls will come to rest and no hall rolls out from right

A photoelectric material having work function is illuminated with light of wavelength gel No The fastest photoelectron has a de Broglie wavelength 2d A change in wavelength of the incident light by A results in change Ad in d then the ratio A2 A2 is proportional to A 2d 2 2017 B 22 2 C 23 2 D 23 2 2 x hc

MCQ Single Correct 1 O Consider regular polygons with number of sides n 3 4 5 as shown in the figure The center of mass of all the polygons is at height h from the ground They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted The maximum increase in height of the locus of the center of mass for each polygon is A Then A depends on n and h as A A h sin C A h sin RI n 2 n B A h 1 COS D A h tan T TC 2n 2017

If the resultant of all the external forces acting on a system of particles is zero then from an inertial frame one can surely say that 2009 A linear momentum of the system does not change in time B kinetic energy of the system does not change in time C angular momentum of the system does not change in time D potential energy of the system does not change in time

50 In the given arrangement what can be th maximum mass of block C so that blocks A and B move together um 1 1 4 m 3 1 11 bremem A 16 B TTTT 3m Smooth C 3 m 2 1 2 m 4 1 11

A thin disc of non uniform mass distribution is shown in x y plane Surface mass density of disc varies according to relation Gosine where do is a positive constant and e is the angle made by position vector of any point on disc with positive x axis The coordinates of centre of mass of disc are A 0 2R T B 0 R 6 C 0 3nR 2 D R 2 3R 4 R AY be you a retardado a t v 4 9 20x30 C X

A large heavy sphere and a small light sphere are dropped Comprehension I together onto a flat surface from a height h The radius of spheres is much smaller than height h The large sphere collides with the surface with velocity vo and immediately thereafter with the small sphere The spheres are dropped so that all motion is vertical before the second collision and the small sphere hits the larger sphere at an angle a from its uppermost point as shown in the diagram All collisions are perfectly elastic and there is no surface friction between the spheres h A C 30 2 a B 2a D zero a Prob 1 The angle made by velocity vector of small sphere with the vertical just after the second collision in the frame of large sphere l

1 In the rectangular lamina shown in the fig AB BC 2 The moment of inertia of the lamina is minimum along the axis passing through A E D FF B 1 AB A fly whaal is 2 BC TO G H 3 EG 4 FH structed that almost whol

1 A circular plate of diameter d is kept in contact with a square plate of edge d as shown in figure The density of the material and the thickness are same everywhere The centre of mass of the composite system will be d d 1 inside the circular plate 2 inside the square plate 3 at the point of contact 4 outside the system 9 mass of the 1 horizon 2 centre 3 line jo 4 point The co shown

A ball of mass m is released from A inside a smooth wedge of mass m as shown in Fig 8 269 What is the speed of the wedge when the ball reaches point B A B 4 45 R Smooth

31 As per this diagram a point charge q is placed at the origin O Work done in taking another point charge Q from the point A co ordinates 0 a to another point B co ordinates a 0 along the straight path AB is AIPMT 2005 1 zero 3 A qQ 1 a 2 B 2 4 X 9Q 4 E 2 qQ 1 2a 1 Za 2a

System of Particles and Rotational Motion 67 Two masses m and 2m are at a distance and if they revolve due to their mutual attraction then the ratio of their angular velocities and the linear velocities respectively are 1 1 1 1 1 3 1 1 2 1 2 2 1 1 1 4 1 1 1 2 7

In the arrangement shown in figure wedge of mas M moves towards left with an acceleration a A surfaces are smooth The acceleration of mass relative to wedge is 1 2 2 3 2Ma E 2 M ma E M ma E M 2H m fo sinto 60 A pendulum of mass 1kg and length 1m

The centre of mass of rigid body always lie inside the body Is this statement true or false

A block A of mass 2m is placed on another block B of mass 4m which in turn is placed on a fixed table The two blocks have a same length 4d and they are placed as shown in figure The coefficient of friction both static and kinetic between the block B and table is u There is no friction between the two blocks A small object of mass m moving horizontally along a line passing through the centre of mass CM of the block B and perpendicular to its face with a speed v collides elastically with the block B at a height d above the table Han Sin Svit 1991 4 4M a What is the minimum value of v call it vo required to make the block A to topple 106 m 11 di A 2m B 4m 2d 4d b If v 2 vo find the distance from the point P in the figure at which the mass m falls on the table after collision Ignore the role of friction during the

A cone and sphere are placed together as shown in fig Density of material of cone is 1 18 times that of sphere The position of centre of mass on the line of symmetry from base of cone is a 4 6 r b 4 3 r c 4 27 r d 3 4 r

A bomb of mass 3 0 kg at rest explodes in a into two pieces of masses 2 0 kg and 1 0 kg The smaller mass goes at a speed of 80 m s The total energy imparted to the two fragments is 2 2 14 KJ 1 1 07 kJ 3 2 4 kJ 4 4 8 kJ

Two equal vectors have a resultant to eithe side what is the angle between them A 45 C 90 B 60 D 120

A tennis ball is dropped on a horizontal smooth surfa It bounces back to its original position after hitting the surface The force on the ball during the collision is proportional to the length of compression of the ball Which one of the following sketches describes the variation of its kinetic energy K with time t most appropriately The figures are only illustrative and not to the scale m r a K b d K

Dall of mass m is released from the top of an inclined plane of inclination 8 as shown It strikes a rigid 3 from top elastically Impulse imparted to ball by the rigid surface is 4 at a distances 1 m 32 gh Smooth 2 m 3gh h 1 2 m 3 20 m Two steel bal on a smooth 1 8 m s 3 11 m s 30 Two ivory bal as shown in 29 1 Third ba

A body moving with speed v in space explodes into two piece of masses in the ratio 1 3 If the smaller piece comes to rest the speed of the other piece is 4v 3v

A body of mass M hits normally a rigid wall with velocity V and bounces back with the same velocity The impulse experienced by the body is a MV b 1 5MV c 2MV d zero 2011 A conveyer belt is moving at a constant speed of 2 ms

Two particles having masses 1 kg and 7 k respectively attract each other Initially they are a rest and infinite separation The velocity c approach of the particles are at a separation of tr is G universal gravitational constant 1 2 Gmvs 2 4 G m s 3 m s S 04 mys

Two uniform hemispheres A and B are concentrically fixed to each other and the composite body thus formed is placed on a horizontal floor as shown Masses of the hemisphere A and B are mA and me and their radii are TA and r respectively Find conditions for stable unstable and neutral equilibria of the composite body in the situation shown