Center of mass and momentum Questions and Answers

Three particles A B and C start moving simultaneously with constant velocities from three places The starting places are collinear and that of B is somewhere in between those of A and C In the absence of C particles A and B would have collided t time after they started and in the absence of B particles A and C would have collided to time after they started What would have happened if A were not present a If t t particles B and C may or may not collide b If t ti particles B and C collide in the interval t t c If t ti particles B and C may collide at an instant t t d If t t particles B and C may collide in the interval t tel

10 A ball of mass m 1kg is hung vertically by a thread of length 1 1 50m Upper end of the thread is attached to the ceiling of a trolley of mass M 4kg Initially the trolley is stationary and it is free to move along horizontal rails without friction A shell of mass m moving horizontally with velocity Vo 6m s collides with the ball and gets stuck with it As a reusit the thread starts to deflect towards right 1kg 08 1 The maximum deflection of the thread with the vertical is COS 3 1 50 m 3 M Your Answer

1 5 A vertically oriented uniform rod of mass M and length I can rotate about its upper end A horizontally flying bullet of mass m strikes the lower end of the rod and gets stuck in it as a result the rod swings through an angle a M 10 1 Assuming that m M find at what distance a from the upper end of the rod the bullet must strike for the momentum of the system bullet rod to remain constant during the impac B IA 418 Your Answer

A long slender rod of mass 2kg and length 4m is placed on a smooth horizontal table Two particles of masses 2kg and 1kg strike the rod simultaneously and stick to the rod after collision as shown in the figure Angular velocity of the rod after collision is 11 17 rad s 4 1 kg 10 m s Your Ans Correct Ans

The Marking scheme to evaluate the Answer to each question Full Marks 4 If the answer is correct Partial Marks 3 If all the four options are correct but only 3 are chosen Partial Marks 2 If THREE or More options are correct but only 2 correct answers are chosen Partial Marks 1 If TWO or More options are correct but only 1 correct answer is chosen Zero Marks 0 If the question is unanswered Negative Marks 1 In all other cases 1 A circular ring of radius R and mass 9 m is placed on a perfectly smooth horizontal surface An ant of mass m starts moving along the ring Select the correct conclusion s A trajectory of centre of ring is a circle of radius R 10 9R 10 B trajectory of ant is also a circle but of radius C Ratio of linear speed of centre of ring and that of ant is 9 1 D Ratio of angular speeds about a common fixed point is 1 1

inextensible strings which are just taut connect AB and BC and ZABC is 135 An impulse J is imparted to the particle C in the direction BC Mass of each particle is m Choose the correct options ec Sr Super60 All A 135 B Space for rough work i Chaitanya IIT Academy A Speed of A just after the impulse imparted is 17 06 2021 Sr Super60 All Jee Adv 2019 P2 GTA 10 Q P 2J 7m Page 3 10J B Speed of B just after the impulse imparted is 7m

Ex 8 The masses of 5 balls at rest in a straight line form a GP where common ratio is 2 and their coefficient of restitution are each 2 3 if the first ball he started towards the second with velocity u show that the velocity communicated to the fifth ball is 5 9 A GRY

A bottle of dry ammonia and a bottle of dry hydrogen chloride connected through a long tube are opened simultaneously at both ends the white ammonium chloride ring first formed will be 1 at the centre of the tube 2 near the hydrogen chlrodie bottle 3 near the ammonia bottle 4 throughout the length of the tube The

1 2 6 x 10 ms 2 6 2 10 ms 3 10 x 10 ms 4 10 ms 22 An object is projected with a speed 10 ms at an angle of 30 with the horizontal It breaks into n equal fragments during its motion One fragment strikes the ground at a distance of 3m from the point of projection The center of mass of the remaining fragments strikes the ground at a distance of 7 3m from the point of projection If all fragments strike the ground at the same time Find the value of n 1 2 2 3 3 4 4 525

A cricket ball of mass 100 g has initial velocity 37 4 m s and final velocity v 37 4 m s after being hit by the bat The magnitude of impulse on the bat by the ball will be Ns

a pv pg h y b pg h y pv c pg h y pv d pg h y pv shell of mass m in flight explodes into three equal fragments One of the fragment reaches th ound earlier than the other fragments The acceleration of the centre of mass of the shell system ju ter first one reaches ground is b 2g 3 c 3g 4 d g 3

30 A solid cylinder of mass 20kg has length 1 mand radius 0 2 m Then its moment of inertia in kg m about its geometrical axis is 1 0 8 kg m vods snimisl 3 0 2 kg m 2 0 4 kg m ingmom 4 20 2 kg m Php 2

The moment of inertia of a thin uniform rod of length L and mass M about an axis passing mrough a point at a distance of 1 3 from one of its ends and perpendicular to the rod is ML 12 B ML 9 7ML 48 D ML 48

A cylinder of mass m and radius R is rolling without slipping on a horizontal surface with angular velocity 0 The velocity of centre of mass of cylinder is R The cylinder comes across a step of height R 4 Assume required friction is present at edge of the step Answer the following questions based on above information 12 O B wo D 600 5 000 R R R 4 Then the angular velocity of cylinder just after the collision is Assume cylinder remains in contac and no slipping occurs on the edge of the step A 500 6 C 2000 3

3 A uniform rod of length 1 0 metre is bent at its midpoint to make 90 angle The distance of the centre of mass from the centre of the rod is a 36 1 cm c 17 7 cm b 25 2 cm d zero AMU Med

4A uniform equilateral triangular lamina of side 1 has a mass m Its moment of inertia about the axis through the centroid and perpendicular to its plane is 72 a ml c ml b ml d None of these

A freight car of mass M contains a mass of sand m At t 0 a constant horizontal force F is applied in the direction of rolling and at the same time a port in the bottom is opened to let the sand flow out at constant rate dm dt Find the speed of the freight car when all the sand is gone Assume the freight car is at rest at t 0

50 b The tangent to A c A line d None of the above perpendicular to the plane of rotation B A B Initially After Take a bicycle rim and extend its axle on both sides Tie two strings at both ends A and B as shown in the figure Hold both the strings together in one hand such that the rim is vertical If you leave one string the rim will tilt Now keeping the rim in vertical position with both the strings in one hand put the wheel in fast rotation around the axle with the other hand Then leave one string say B from your hand and observe what happens What will happen if we leave string B a The rim will stop rotating b The rim will rotate in a vertical plane and the plane of rotation will precesses about string A c The rim will rotate in a horizontal plane d None of the above

16 Two identical blocks A and B each of mass m resting on the smooth horizontal floor are connected by a light spring of natural length L and spring constant K A third block C of mass m moving with a speed v along the line joining A and B collides with A The maximum compression in the spring is C m 1 3 M 2K mv K A m Sol 1 C comes to rest Official Ans by NTA 1 Vem of A B ret 2 2 1 kx 2 4 mv 2K m 2K B m Sol 19

13 Figure 9 27 shows a uni form square plate of edge length 6d 6 0 m from which a square piece of edge length 2d has been removed What are a the x coor dinate and b the y coordinate of the center of mass of the remaining 34 piece 14 An atomic nucleus at rest at the origin of an xy coordi nate system transforms into 3d to 0 d 2d 2d cater 3 3d 3d Figure 9 27 Problem 1

40 Three bricks each of length L and mass M are arranged as shown in figure The distance of the centre of mass of a system from the wall is a 20 3 L 12 C L Wall L L 2 b 117 12 13 d L L 4 Bricks

ntal surface the x axis cm s and f its center face nt A C 2g sin 0 3r 36 With reference to figure of a cube of edge a and mass m Which of the following is the incorrect statement O is the centre of the cube 1 1 Z 2 I I Z ma A 2 D 2 z H c I I X y d None of theor X E d 2g cos 0 3r a Of a The moment of inertia of cube about z is 2 ma Z B C b The moment of inertia of cube about z is z G F y 38 A thin freely maxim maxim 1 0 a 2 C 68 Po 38 39 Three conta line I distar a A

A ball of mass m 1 kg is released on an incline plane of length and inclination 8 It hits a rigid A board elastically which is fixed at a distance 4 from the bottom Impulse in SI unit imparted by board on the ball is Given fg sine 4 m s 114 0

Two blocks of masses 1 kg and 3 kg are moving with velocities 2 m s and 1m s respectively as shown If the spring constant is 75 N m the maximum compression of the spring is 2 m s 1 m s 1 kg mmm 3 kg Smooth

An inextensible string connects two masses m and 4m and the system rests on a smooth horizontal floor An impulse J is imparted to B as shown in figure The tension in the string in the subsequent motion just after this instant is 1 m B J 4m O J ml O J 5ml O 4J 5ml O J 20ml

A block of mass m is connected to another block of mass Mby a massless spring of spring constant k The blocks are kept on a smooth horizontal plane and the blocks are at rest and th spring is unstretched when a constant force F starts acting o the block of mass M to pull it Find the maximum extension the spring M F

Sphere A of mass m moving with a constant velocity u hits another stationary sphere B of the same mass If e is the coefficient of restitution then ratio of velocities of the two spheres VA VB after collision will be 1 2 3 1 e 1 e e 1 1 e 4 1 e 1 e e 1 1 e

10 A circuit has section AB as shown in figure If the charge on each capacitor is 10 C then potential difference V VB is A A 1 5 V 3 10 V 1 F 10 V 2 F 2 5 V 4 10 X B

Q29 Two balls of equal masses have a head on collision with speed 6 ms each If the coefficient of restitution is 1 3 the speed of each ball after impact will be a 18 ms b 2 ms c 6 ms 1 d data insufficient

Seven homogeneous bricks each of length L mass M are arranged as shown Projection L X then x co ordinate of C M is 10 X X X X i X 0 22 1 L 2 X 32 L Cu 3 42 12 4

A body is dropped from rest from height of h 5 0 m After rebounding twice from a horizontal floor to wha height will it rise if coefficient of restitution is 0 8 3 5 m 2 05 m 1 4 m 2 9 m

Que 12 A circular disc of radius R is removed from a bigger circular disc of radius 2R such that the circumference of the discs coincide The centre of mass of the new disc is aR from the centre of the bigger disc The value of a is 1 3 1 2 O 1 6

placed in touch The distance of the centre of mass of the system from the centre of sphere A is 3 3a 4 None of these 1 a 2 2a Three identical spheres lie at rest along a line on a smooth horizontal surface The separation between any two adjacent spheres is L The first sphere is moving with a velocity u towards the second sphere at time t 0 The coefficient of restitution for a collision between any two blocks is 1 3 The correct statements is 1 The third sphere will start moving at t 5L 4L 2 The third sphere will start moving at t 2u U 3 The centre of mass of the system will have a final speed 2u 4 The centre of mass of the system will have a final speed u

Que 5 Two particles A and B initially at rest move towards each other under a mutual force of attraction The speed of centre of mass at the instant when the speed of A is v and the speed of B is 2v is V 3v 2 O Zero O 2v

41 A ball is dropped on the floor from a height of 10 m It rebounds to a height of 2 5 m If the ball is in contact with the floor for 0 01 sec the average acceleration during contact is a 2100 m sec downwards b 2100m sec upwards c 1400 m sec d 700m sec

A rod of length 3 m has its mass per unit length directly proportional to distance from one of its ends then its centre of gravity from that end will be at A B C 1 5 m 2m 2 5 m 30m Correct Answer

A spherical hollow cavity is made in a lead sphere of radius R such that its surface touches the outside surface of lead sphere and passes through the centre What is the shift in the centre of mass of lead sphere due to the hollowing 1 3 R R 2 0 0 2 R 14 4 R

Find the position of centre of mass of the quarter solid sphere from C in which where r is radial distance unit volume is given as p r Po 1 F R from centre and Rois radius of solid quarter sphere mass per

3 Assertion It is better to wash the clothes in cold soap solution Reason The surface tension of cold solution is more than the surface tension of hot solution

A particle of mass m is initially at rest at the origin It is subjected to a force and starts moving along the X axis Its kinetic energy K changes with time as dK dt yt where y is a positive constant of appropriate dimensions Which of the following statements is are true A The force applied on the particle is constant B The speed of the particle is proportional to time C The distance of the particle from the origin increases linearly with time D The force is conservative

4 The workdone on the system must be zero A rigid rod of mass 3m and length I is under translational motion on a smooth horizontal surface If the rod collides inelastically with a particle of mass m place at point O as shown in the figure then the angular velocity of the rod just after collision will be 9v0 1 4 T B A 12vo 0 m 3vo 1 3v0

and length is under A rigid rod of mass translational motion on a smooth horizontal surface If the rod collides inelastically with a fixed hinged O as shown in the figure then the angular velocity of the rod just after collision will be B 1 3 13vo 51 12 71 1 41 A 0 2 11vo 71 4 None of these

14 In a resonance tube the first resonance with a tuning fork occurs at 16 cm and second at 49 cm If the velocity of sound is 330 m s they frequency of tuning fork is 2 300 3 330 1 500 4 165

A hemispherical surface of mass M is placed on smooth horizontal floor A man of mass m starts moving on the surface from position A with constant speed v relative to hemisphere and man doesn t slip on it While man moves from position A to C relative to surface B my M Question Type Multiple Correct Type 1 speed of hemispherical will first increase and then decrease 2 net work done by contact force between man and the hemispherical on system containing man and hemisphere during the walking of man is zero 3 2mR hemisphere will get displaced by M m towards left after man reaches C speed of hemisphere will be maximum

Two identical metal balls connected at the ends of a light spring of force constant k form a dumbbell like structure The dumbbell rests on a frictionless horizontal floor and third identical ball is placed at distance I from the right ball of the dumbbell All the three balls are in a line A fourth identical ball moving with velocity u collides with left ball of the dumbbell If all collisions are elastic and rightmost ball aquries a velocity u The minimum value of I is Tu m Find x Vxk ooooooooo

A thin uniform flat plate is in shape of L as shown The mass of lamina is 3 kg The position of centre of mass from point O y 0 2 O 0 0 mm 7 m m E 5 3 E 2 m 6 m 6 1 2 1 1 2 1 Plate X 2 0

32 The system shown in the figure can move on a smooth surface The spring is initially compressed by 6 cm and then released The amplitude of 3 kg block will be 6 kg a 2 cm c 6 cm K 800N m 3 kg 6000000 b 4 cm d None of these

A body of mass 1 kg stri another body at rest and continues to move in the same direction with one forth of its initial velicity The mass of the other body is 1 0 6 kg 2 24 kg 3 3 kg 4 4 kg

Three identical balls A B and C each of mass m 3kg are connected by strings AB and BC as shown in the figure The whole system is placed on a smooth horizontal surface 18 B Im Im Now the ball B is given an initial velocity VE m s perpendicular to the string and along the horizontal surface Find the tension in newton in the string just before the balls A and C collide

4 13 A particle 1 moving with velocity v 10 m s experienced a head on collision with a stationary particle 2 of the same mass As a result of collision the kinetic energy of the system decreased by 50 Find the magnitude and direction of the velocity of the particle 1 after collision Ans 5m s