Center of mass and momentum Questions and Answers

Four particles of masses m m m and m are placed at the vertices A B C and D as respectively of a square shown The COM of the system will lie at diagonal AC if Bm a m m3 OBAM 3 m l b m my c m m d m m m4 m C

41 A man of mass M stands at one end of a plank of length L which lies at rest on a frictionless horizontal surface The man walks to the other end of the plank If the mass of the plank is M 3 the distance that the man moves relative to the ground is 1 3 L 4 2 4 L 5 Which of the following is incorrect ST 3 L 4 4 none of these

A smooth narrow semicircular tube in vertical plane contains a small ball of mass m at its top as shown System is initially at rest and now ball is slightly pushed to its right such that system starts moving If the acceleration of the tube just before the ball hits the 4g floor is find 1 Take mass of tube 2 m and neglect the n break off of the tube from floor R smooth

In the arrangement shown in figure pulley A and B are massless and the thread is inextensible Mass of pulley C is equal to m If friction in all the pulleys is negligible then A tension in thread is equal to 1 2 mg B acceleration of pulley C is equal to g 2 downward C acceleration of pulley A is equal to g 2 upward D acceleration of pulley A is equal to 2g upward A

A small wooden box of mass m 1 50 kg and initial speed v 5 00 m s collides head on with a large metal block of mass m2 12 0 kg which is initially at rest on a frictionless horizontal surface Immediately after the collision the 1 50 kg box bounces back with a speed of 1 00 m s while the large block moves to the right and goes up a frictionless 20 inclined plane The acceleration of gravity is g 9 80 m s Calculate the speed of the 12 0 kg metal block immediately after the collision a 6 pts b 5 pts Calculate how far up the incline the metal block my will go until it stops momentarily distance d c 5 pts If the collision lasts 0 020 s calculate the average force on the wooden box and the average force on the metal block during the collision and state their directions left or right d 4 pts Calculate the amount of kinetic energy lost in the collision For full credit show all your work Ma d

A 500 g ball is released from a height of 4 m Each time it makes contact with the ground it loses 25 of its energy Find the KE it possess after 3rd hit B 11 25 J A 15 J C 8 44 J D none of these

20 From a circular disc of radius R a square is cut out with a radius as its diagonal The distance of the centre of mass of the remainder from the centre of the disc is R a 2 R 2 2 1 b d RERN 2

22 A bullet is fired from a gun The force on the bullet is given by F 600 2 x 105 t Here F is in newton and t in second The force on the bullet becomes zero as soon as it leaves the barrel The impulse imparted to the bullet is a 8 N s c 1 8 N s b 0 9 N s d 2 4 N s

49 Seven particles are placed at the seven corners of a cube of side a except at origin The mass of each particle is m The centre of mass of the system is wa a 3 4a c Comm N ZA a j 3 4a 4 A G a 3 4a 7 k B k O D F E X a b 5a d j a 3 5a 5a k 7 7

44 An infinite number of bricks are placed one over the other as shown in the figure Each succeeding brick having half the length and breadth of its preceding brick and the mass of each succeeding bricks being 4 th of the preceding one Take O as the origin The x coordinate of centre of mass of the system of bricks is a c a 7 3a 7 b M b 2 a b M 4 a 2 3a 7 d 2a 0

23 A uniform rod of length I is kept vertically on a rough horizontal surface at x O It is rotated slightly and released When the rod finally falls on the horizontal surface the lower end will remain at X 0 a x 1 2 b x 2 x axis x 1 2 d x 0

42 A smooth sphere is moving on a horizontal surface with velocity vector 2 i 2 jimmediately before it hits a vertical wall The wall is parallel to j vector and the coefficient of restitution between the sphere and the 1 wall is e The velocity vector of the sphere after it 2 hits the wall is a c 1 21 d 21 1

47 A block of mass M with a semicircular track of radius R rests on a horizontal frictionless surface A uniform cylinder of radius r and mass m is released from rest from the top point A The cylinder slips on the semicircular frictionless track The distance travelled by the block when the cylinder reaches the point B is M R r adw a 7 c M m nol M m R of leup M A m b B m R r M m d None of these C R M

34 A ball of mass m collides on ground at an angle a with the vertical If the collision lasts for time t the average force exerted by the ground on the ball is e coefficient of restitution between the ball and the ground a c e mu cos a t 1 e mu cos a b d 2 1 e mu cos a t e mu

51 A body of mass m is dropped from a height of h Simultaneously another body of mass 2m is thrown up vertically with such a velocity v that they collide at the height h 2 If the collision is perfectly inelastic the velocity of combined mass at the time of collision with the ground will be a b gh 5gh V4 c gh V4 d 13gh 9

45 A bullet of mass m moving vertically upwards with a velocity u hits the hanging block of mass m and gets embedded in it The height through which block rises after the collision assume sufficient space above block a u 2g c u 8g b u g d u 4g m E

42 Displacement time graph of a particle moving in a straight line is as shown in figure 10 Mass of the particle is 2 kg The total impulse O imparted to the particle in a time interval from t 0 to t 6s in N s will be a 30 b 15 c 30 d 15 s m 2 2 6 t s

33 A bullet of mass m moving with velocity u passes through a wooden block of mass M nm The block is resting on a smooth horizontal floor After passing through the block the velocity of bullet is v Its velocity relative to block is 1 n v u a c n nu v n 1 b d u M nm nv u n 1 n 1 u v 2n 1 mod

41 A ball collides at B with velocity 10 m s at 30 with vertical There is a flag at A and a wall at C Collision of ball with ground is perfectly inelastic e 0 and that with wall is elastic A 30 a 4 b 5 c 6 d ball will not collide with the flag B e 1 Given AB BC 10 m The ball will collide with the flag after time t S

A binary star consists of two stars A mass 2 2 Ms and B mass 11 Ms where Ms is the mass of the sun They are separated by distance d and are rotating about their centre of mass which is stationary Find the ratio of the total angular momentum of the binary star to the angular momentum of star B about the centre of mass

de with elastic ht h 4 song och LEI 9 A force exerts an impulse I on a particle changing its speed from u to 2u The applied force and the initial velocity are oppositely directed along the same line The work done by the force is c lu 3 a lu 2 6 14 lu d 2 u

is S 7 A ball collides with an inclined plane of inclination 0 after falling through a distance h If it moves horizontally just after the impact the coefficient of restitution is a tan e b cot 0 c tane d cote

height of its n the mpact illed 8 Three blocks are initially placed as shown in figure block A has mass m and initial velocity v to the right Block B with mass m and block C with mass 4 m are both initially at rest Neglect friction All collisions are elastic The final velocity of block A is ya 0 6 V A m a 0 6 v to the left c v to the left B m C 4m b 0 4 v to the left d 0 4 v to the right a c 6 14 Disp stra Imp inte a c

5 There are hundred identical blocks equally spaced on a frictionless track as shown in the figure Initially all the blocks are separate Each collision is perfectly inelastic The final velocity will be a 99 1 2 V you 100 b 3 c zero 100 d v

2 Two particles having mass ratio n 1 are interconnected by a light inextensible string that passes over a smooth pulley If the system is released then the acceleration of the centre of mass of the system is a n 1 g 1 2 A n 1 9 b n n

3 A ball of mass 50 g is dropped from a heigh h 10 m It rebounds losing 75 per cent of it kinetic energy If it remains in contact with th ground for At 0 01 s the impulse of the impa force is a 1 3 N s c 1300 N s 6 1 06 N s d 105 N s

4 In the arrangement shown bob A on the left is pulled aside It is then released and allowed to collide with other bob B which is at rest A perfectly inelastic collision occurs and the system rises to a height h 4 The ratio of the masses of the bobs is a 1 b 2 A www B c 3 cavig d 4

1 The centre of mass of the shaded portion of the disc is The mass is uniformly distributed in the shaded portion Lay 6 to the left of A c to the right of A 6 A B 842 R 2 b d R 12 R 12 to the left of A to the right of A

A continuous flow water heater geyser has an electrical power rating 2 kW and efficienty of conversion of electrical power into heat 80 If water is flowing through the device at the rate of 100 cc s and the inlet temperature is 10 C the outlet temperature will be 1 12 2 C 3 20 C 2 13 8 C 4 16 5 C

Two spherical bodies of masses m and 6m and radii R and 2R respectively are released in free space with initial Separation between their centrem equal to 10R If they attract each other due to gravitational force only then the distance covered by smaller sphere just before collisions will be 6R 7 5R 25R 9R

A carrom coin of mass 5 gram moving with speed 2 m s strikes elastically with the board wall at 45 If it remains in contact with the wall for 0 02 second then the average force acting on the coin is approximately 1 2 N 3 N 2 2 2N 1 N

21 A quarter sector of a uniform disc of radius R is shown in figure The coordinates of center of mass is 1 2 3 2R 2R 3 3 2R 4R 3 3 9 9 4R 4R 3 3 9 4R 2R 3 3 ya R R X

t 28 3R M O If collision between sphere and particle is elastic ther find speed of sphere just after collision 1 14 12 3 u 2 4 2 m s 3u 2 sphere of radius R mass M at rest

Find the moment of inertia of this cross section of a uniform guide rail of uniform thickness and uniform surface density o 0 5 kg cm2 about its centroid 10 cm I 10 cm 2 cm 2 cm 2 cm

A light rigid rod of length 4m is connected rigidly with two identical particles each of mass m 2kg the free end of the rod is smoothly pivoted at O The rod is released from rest from its horizontal position at t 0 Find the angular acceleration of the rod at t 0 in rads m m I 2m I 2 m

A 30 kg child standing on 50 kg stationary plank which is placed on a smooth horizontal surface jumps off the plank with horizontal component of velocity 16 m s relative to the plank Find recoil velocity of the plank in meter per seconds

Q A uniform rectangular thin sheet ABCD of mass M has length a and breadth b as shown in the figure If the shaded portion HBGO is cut off the coordinates of the center of mass of themaining portion will be 0 b LIBEK JA a 2a 2b 3 3 c 3a 3b b PRA 12 12 D 0 0 H 0 F CSAHOLICS a b B a

1 A particle of mass 4 kg starts its motion from rest with a constant acceleration 1 m s In the system there is another particle of same mass which is moving with a constant velocity of 2 m s Which among the following is not a possible value of the speed of COM after 2 s 1 2 m s 3 1 m s 2 4 m s 4 0 5 m s

Three bodies a ring R a solid cylinder C and a solid sphere S having same mass and same radius roll down the inclined plane without slipping They start from rest if VR Vc and vs are velocities of respective bodies on reaching the K CET 2016 bottom of the plane then VR VC Vs VR VC Vs A VR Vc Vs C VR VC Vs B D

1 Three particles of masses 0 50 kg 10 kg and 1 5 kg are placed at the three corners of a right angled triangle of sides 30 cm 40 cm and 5 0 cm as shown in figure 9 W1 Locate the centre of mass of the system 1 5 kg dauer bevoni 3 0 cm 5 0 cm 0 50 kg 4 0 cm Figure 9 W1 1 0 kg

Three blocks A B and C are kept in a straight line on a frictionless horizontal surface These have masses m 2m and 3m respectively The object A moves towards B with speed V All collisions are elastic then match the list and choose the CORRECT code P List I Final speed of A is Q Final speed of B is R Final Speed of C is S A P 2 Q 4 R 3 S1 C P 2 Q3 R 4 S1 A B C mv 2m 3m Final speed of centre of mass is 1 v 6 2 List II wo 4 4 3 3 8v 15 2v 15 B P 3 Q4 R 2 S 1 D P 3 Q 2 R 3 S4

41 In the circuit shown the potential differences V between A and B and V between C and D are 14 RRR wwwwwwww R C www 1 6iR 3iR 2 3 R 3iR 3 3iR iR ir 2R R wwwwwww 3R D www 1 2 24 V B

2 A sphere of 4 cm radius is suspended within a hollow sphere of 6 cm radius The inner sphere is charged to a potential 3 e s u When the outer sphere is earthed the charge on the inner sphere is 1 2 e s u 4 4 36 e s u 1 54 e s u 3 30 e s u

A uniform rod of mass m and length L is at rest on a smooth horizontal surface A ball of mass m moving with velocity vo hits the rod perpendicularly at its on end and sticks to it The angular velocity of rod after collision is A V 6 un Correct Answer 12 to 5 L

94 A boat of mass 100 kg is of mass 40 kg and 60 kg are at a distance 10m on the boat Each move distance 2m towards each other over the boat Then find displacement of Centre of Mass of the boat 1 2m towards 60 kg man 2 0 2 m away from 60 kg man 3 0 2 m towards 40 kg man words 60 kg man

Eour holes of radius R are cut from a thin square plate of side 4R and mass M Determine the moment of inertia of the remaining portion about Z axis Square M m MR Solution M Mass of the square plate before the holes were cut M TM 16R2 16 Moment of inertia of the remaining portion Mass of each hole m TR 2 9 41 10 F 11

3 50 N 4 60 N 142 A bullet of mass 200 g moving with a velocity v strikes a block of mass 1800 g at rest and remains in it The block moves a distance of 10 m on horizontal surface of friction coefficient 0 5 The value of v is 0 1 50 m s 2 100 m s 0 2 kg 1 8 kg 3 150 m s 4 200 m s

Three point particles of masses 1 0 kg 1 5 kg and 2 5 kg are placed at three corners of a right angle triangle of sides 4 0 cm 3 0 cm and 5 0 cm as shown in the figure The center of mass of the system is at a point 2 5 kg 4 cm 5 cm 1 0 kg 1 5 kg 3 cm 1 1 5 cm right and 1 2 cm above 1 kg mass 2 0 9 cm right and 2 0 cm above 1 kg mass 3 0 6 cm right and 2 0 cm above 1 kg mass 4 2 0 cm right and 0 9 cm above 1 kg mass

3 The electric field in a region is radially outward and at a point is given by E 250 r V m where ris the distance of the point from origin Calculate the charge contained in a sphere of radius 20 cm centred at the origin E Rov 1 2 22 x 10 6 C N 250x0 2 4 x 2 2 22 x 10 8 C 3 2 22 10 1 C 0 22 109

A uniform solid square plate ABCD of mass m and side a is moving in x y horizontal smooth plane The velocity of centre of mass is vo 21 4 m s The end A of square plate is suddenly fixed by a pin find the new velocity of centre of mass of square A vo 2 4 y Vcm Vo 2 47 B 3vo B 30 3vo C 30 i 507 4 4 2 D 3voi 3voj