JEE Main Previous Year Papers Questions With Solutions Physics CENTRE OF MASS, MOMENTUM & IMPULSE

Multiple Choice with ONE correct answer:

1.Two particles A and B, initially at rest, move to ­  wards each other under mutual force of attraction. At the instant when the speed of A is v and the speed of B is 2v, the speed of the centre of mass of the system is    [1982-3 marks]

a) 3v             b) v

c) 1.5v          d) zero

Ans: d

2.An isolated particle of mass m is moving in  horizontal plane (x-y) along the x-axis at a certain height above the ground. It suddenly explodes into two fragments of masses m/4 and 3m/4. An instant later, the smaller fragment is at y = + 15 cm. The larger fragment at this instant is at [1997-1 mark]

a) y = – 5cm                  b) y = + 20cm

c) y = + 5cm              d) y = – 20cm

Ans:a

3.Two particles of masses mt and m2, in projectile motions have velocities vt < v2 respectively at time t=0. They collide at time t0. Their velocities become v, and v2 at time 2t0 while moving in air. The  value of  Ans:c

4.Two blocks of masses 10 kg and 4 kg are con ­nected by a spring of negligible mass and placed on a frictionless horizontal surface. An impulse gives a velocity of 14 m/s to the heavier block in the direction of the lighter block. The velocity of the centre of mass is :           

a) 30 m/s                   b) 20 m/s

c) 10 m/s                    d) 5 m/s

Ans:c

5.Two small particles of equal  masses start moving in opposite directions from a point A in a horzontal circular orbit. Their tangential velocities are v and 2v, respectively, as shown in the figure. Between collisions, the particles move with constant speeds. After making how many elastic collisions, other than that at A, these two particles will again reach the point A? a) 4                      b) 3

c) 2                       d) 1

Ans:c

6.Look at the drawing given in the figure which has been drawn with ink of uniform line-thickness. The mass of ink used to draw each of the two inner circles, and each of the two line segments is m.  The mass of the ink used to draw the outer circle is 6m. The coordinates of the centres of the different parts are; outer circle (0,0), left inner circle (-a, a), right inner circle (a, a), vertical line (0, 0) and horizontal line (0, -a). The y-corrdinate of the centre of mass of the ink in this drawing is  Ans:a

Multiple Choice with ONE or MORE THAN ONE correct answers

7. A ball hits the floor and rebounds after an inelastic collision. In this case                 [1986-2 marks]

(a)the momentum of the ball just after the collision is the same as that just before the collision
(b)the mechanical energy of the ball remains the same in the collision
(c)the total momentum of the ball and the earth is conserved
(d)the total energy of the ball and the earth is conserved.

Ans:c,d

8.A shell is fired from a cannon with a velocity v (m/sec) at an angle 0 with the horizontal direction. At the highest point in its path it explodes into two pieces of equal mass. One of the pieces ‘ retraces its path to the cannon and the speed (in m/sec) of the other piece immediately after the explosion is                 [1986-2 marks] Ans:a

9.A uniform bar of length 6a and mass 8m lies on a smooth horizontal table. Two point masses m and 2m moving in the same horizontal with speed 2v and v respectively, strike the bar [as shown in the fig.] and stick to the bar after collision.  Denoting angular velocity (about the centre of mass), total energy and centre of mass velocity by co,E and vc respectively, we have after collision.    [1991-2 marks]  Ans:a,c,d

10.Two blocks A and B, each of mass m, are connected by a massless spring of natural length L and spring constant K. The blocks are initially resting on a smooth horizontal floor with the spring at its natural length, as shown in figure. A third identical block C, also of mass m, moves on the floor with a speed v along the line joining A and B, and collides elastically with A. Then. [1993-2 marks]  Ans:b,d
11.Two balls, having linear momenta pt = pi and p2 = -pi, undergo a collision in free space. There is no external force acting on the balls. Let P1 and P2 be their final momenta. The following option(s) is are NOT ALLOWED for any non-zero value of p,a1,a2,b1,b2,c1 and c2 Ans:a,d

12.A point mass of 1 kg collides elastically with a stationary point ipass of 5 kg. After their collision, the 1 kg mass reverses its direction and moves with a speed of 2 ms4. Which of the following statements(s) (are) correct for the system of these two masses ?

(a)Total momentum of the system is 3 kg ms-1
(b)Momentum of 5kg mass after collision is 4kg ms-1
(c)Kinetic energy of the centre of mass is 0.75 J
(d)Total kinetic energy of the system is 4 J

Ans:a,c

H Assertion & Reasoning type Instructions :
Each question contains statement-1 (assertion) and statement-2 (reason). Of these statements, mark correct choice if

13.Statement-1 ;If there is no external torque on a body about its centre of mass, then the velocity of the centre of mass remains constant.
Statement-2 :The linear momentum of an isolated system remains constant.         [2007-3 marks]

(a).Statement-1 and 2 are true and statement-2 is a correct explanation for statement-1
(b).Statements-1 and 2 are true and statement-2 is not a correct explanation for statement-1
(c).Statement-1 is true, statement-2 is Base
(d).Statement-1 is false, statement-2 is true.

Ans:d

14.Statement -1 In an elastic collision between two bodies, the relative speed of the bodies after colision is equal to the relative speed before the collision
because
Statement -2 In an elastic collision, the linear momentum of the system is conserved * [2009-3 marks]

(a).Statement -1 is True, Statement -2 is True; Statement-2 is a correct explanation for Statement-1
(b).Statement -1 is True, Statement -2 is True; Statement-2 is not a correct explanation for Statement-1
(c).Statement -1 is True, Statement -2 is False
(d).Statement -1 is False, Statement -2 is True

Ans:b

Subjective / Numerical integer type

15.A body of mass m moving with a velocity v in the x-direction collides with another body of mass M moving in the y-direction with a velocity V. They coalese into one body during collision. Find.

(a)The direction and magnitude of the momentum of the composite body,
(b)The fraction of the initial kinetic energy transformed into heat during the collision          

Ans:  16.When a ball is thrown up, the magnitude of its momentum decreases and then increases. Does this violate the conservation of momentum principle?  

Ans :No

17.A 20g bullet pierces through a plate of mass Mplkg and then comes to rest inside a second plate of mass M2 = 2.98 kg as shown in the figure. It is found that the two plates initially at rest, now move with equal velocities. Find the percentage loss in the initial velocity of the bullet when it is between Mj and M2. Neglect any loss of material of the plates due to the action of bullet. Ans:25%

18.A circular plate of uniform thickness has a diameter of 56cm. A circular portion of diameter 42cm is removed from one edge of the plate as shown in the figure. Find the position of the centre of mass of the remaining portion Ans:9 cm to left of centre of bigger circle
19.A body of mass 1kg, initially at rest, explodes and breaks into three fragments of masses in the ratio 1:1:3. The two pieces of equal mass fly off per ­pendicular to each other with a speed of 30 m/sec each. What is the velocity of the heavier fragment?  [1981-3 marks]

Ans:  10âˆš2m/s

20.Three particles A,B and C of equal mass move with equal speed v along the medians of an equilateral traingle as shown in fig. They collide at the cen ­troid of the traignle. After the collision, A comes to rest, B retraces its path with the speed v. What is the velocity of C ?        (1982-2Marks) Ans:  v’= v = velocity of C

21.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 re ­leased from rest at the top point A, as shown in figure. The cylinder slips on the semicircular fric ­tionless track. How far has the block moved when the cylinder reaches the bottom (point B) of the track? How fast is the block moving when the cyl ­inder reaches the bottom of the track?        [1983-7 marks] Ans 22.Two bodies A and B of masses m and 2m respectively are placed on a smooth floor. They are connected by a spring  A third body C of mass m moves with velocity v0 along the line joining A and B and collides elastically with A as shown in fig. At a certain instant ‘ of time t0 after collision, it is found that the instantaneous velocities of A and B are the same. Further at this instant the compression of the spring is found to be jc0 . Determine
i) the common velocity of A and B at time t0 and
ii) the spring constant. Ans: 23.A ball of mass lOOg is projected vertically upwards from the ground with a velocity of 49m/sec. At the same time another identical ball is dropped from a height of 98m to fall freely along the same path as that followed by the first ball. After some time the two balls collide and stick  together and finally fall to the ground. Find the time of flight of the masses. [1985-8 marks]

Ans:6.53sec

24.A simple pendulum is suspended from a peg on a vertical wall. The pendulum is pulled away from the wall to a horizontal position (sec fig.)  and released. The ball hits the wall  the co-efficient of restituion being  2â?„ âˆš5   What is the minimum  number of collisions after which theamplitude of oscillations becomes  less than 60 degree?                                         [1987-7 marks] Ans:4

25.A bullet of mass M is fired with a velocity 50 m/s at an angle 0with the horizontal. At the highest point of its trajectory, it collides head-on with a bob of mass 3M suspended by a massless string of length (10/3)metre and gets embedded in the bob, After the collision, the string moves through an angle of 120 °. Find
i)the angle 0;
ii)the vertical and horizontal coordinates of the initial position of the bob with respect to the point of firing of the bullet. Take g = 9.8m/s2                                                  [1988-6 marks]

Ans:  (i)Î¸=47 °     (ii):   46cm, 122.6cm

26.A thin uniform bar lies on a  frictionless horizontal surface and is free to move in  any way on the surface.Its  mass is 0.16kg and length  âˆš3   metre. Two particles  each of mass 0.08 kg, are moving on the same surface and towards the bar in a direction perpen ­dicular to the bar, one with a velocity of lOm/s, and other with 6m/s as shown in fig. The first par ­ticle strikes the bar at point A and the other at point B. Points A and B are at a distance of 0.5m from the centre of the bar. The particles strike the bar at the same instant of time and stick to the bar on collision. Calculate the loss of the kinetic energy of the system in the above collision process.  [1989-8 marks] Ans:2072j

27.An object’of mass 5kg is projected with a velocity  of 20m/s at an angle of 60 ° to the horizontal. At the highest point of its path the projectile explodes and breaks up into two fragments of masses 1kg and 4kg. The fragments separate horizontally after the explosion. The explosion releases internal energy such that the kinetic energy of the system at the highest point is doubled. Calculate the separation between the two fragments when they reach the ground.                           [1990-8 marks]

Ans:44.25m

28.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 same length 4d and they are placed as shown in fig. The coefficient of friction (both static and kinetic) between the block B and table is |Li. 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 (COM), 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.    [1991-8 marks] a)What is the minimum value of v(call it v0) required to make the block A topple ?
b)If v = 2v0, 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 collision)

Ans:   Ans: 30.A uniform thin rod of mass M and length L is standing vertically, along the y-axis, on a smooth horizontal surface, with its lower end at the origin (0,0). A slight disturbance at t=0 causes the lower end to slip on the smooth surface along the positive x-axis and the rod starts falling

i)What is the path, followed by the centre of mass of the rod, during its fall ?
ii)Find the equation of the trajectory of a point on the rod, located at a distance r, from die lower end. What is the shape of the path of this point?    [1993-5+1 marks]

Ans: (i): Stright line (ii): Ellipse

31.A small sphere of radius R is held against the inner surface of a larger sphere of radius 6R. The masses of large and small spheres are 4M and M respectively. This arrangement is placed on a horizontal table. There is no friction between any surfaces of contact. The small sphere is now released. Find the coordinates of the centre of the larger sphere when the smaller sphere reaches the other extreme position.    [1996-3 marks] Ans:(L+2R,0)

32.A cart is moving along +x direction with a velocityof 4m/s. A person in the cart throws a stone with a velocity of 6m/s relative to himself. In the frame of reference of the cart the stone is thrown in y-z plane making an angle of 30 ° with the vertical z-axis. At the highest point of its trajectory, the stone hits an object of equal mass hung vertically from the branch of a tree by means of a string of length L. A completely inelastic collision occurs, in which the stone gets embedded in the object. Determine                                                                                 [1997-5 marks]

(a).the speed of the combined mass immediately after the collision with respect to an observer on the ground.

(b).the length L of the string such that the tension in the string becomes zero when the string becomes horizontal during the subsequent motion of the combined mass.

Ans:(I)2.5M/S

(II)L=0.318m

33.A wedge of mass m and triangular cross section (AB=BC=CA=2R) is moving with a constant velocity vi Ë†  towards a sphere of radius R fixed on a smooth horizontal table, as shown in the figure. The wedge makes an elastic collision with the fixed sphere and returns along the same path,any rotation* Neglect all friction and suppose that the wedge remains in contact with the sphere for a very short time Î”t during which the sphere exerts a constant force Fâ€¾  on the wedge. [1998-8 marks] (a)Find the force p and also the normal force N exerted by the table on the wedge during the time Î”t
(b)Let h denote the perpendicular distance between the centre of mass of the wedge and the line of action of F ¯. Find the magnitude of the torque due to the normal force n about the centre of the wedge dining the intervalÎ”t

34Two blocks of mass 2kg  and M are at rest on an inclined plane and are separated by a distance of 6.0m as shown. The coefficient of friction between each of the blocks and the inclined plane is 0.25. The 2kg block is given a velocity of lO.Om/s up the inclined plane. It collides with M, comes back and has a velocity of l.Om/s when it reaches its initial position. The other block M after the collision moves 0.5m up and comes to rest. Calculate the coefficient of restitution between the blocks and the mass of the block M. [Take sinÎ¸ â‰ˆ tanÎ¸= 0.05 and g = 10m/s2] [1999-10 marks] Ans: 0.84,  15.02kgs

35.A rod AB of mass M and length L is lying on a horizontal frictionless surface. A particle of mass m travelling along the surface hits the end ‘A’ of the rod with a velocity v0 in a direction perpen ­dicular to AB. The collision is elastic. After the ‘collision the particle comes to rest.      [2000-10 marks]

(a)Find the ratio m/M .
(b)A point P on the rod is at rest immediately after the collision. Find the distance AP.
(c)Find the linear speed of the point P after a time Ï€L/(3v0) after the collision.

Ans:m/M=1/4  , (b) AP=2L/3 , (c)Vp=Vo/2âˆš2

36.A car P is moving with auniform speed of 5(31/2) m/s towards a carriage of mass 9 Kg at rest kept on the rails at a point B as shown in fig. The height AC is 120 m. Cannon balls of 1 Kg are fired from the car with an initial velocity 100 m/s at an angle 30 ° with the horizontal. The first canon ball hits the stationary carriage after a time ^ and sticks to it. Determine t0. At t0, the second cannon ball is fired. Assume that the resistive force between the rails and the carriage is constant and ignore the vertical motion of the carriage throughout. If the second ball also hits and sticks to the carriage. What will be the horizontal velocity of the carriage just after the second impact?         [2001-10 marks] Ans:to=12Seconds,15.75m/s

37.Two heavy metallic plates are joined together at 90 ° to each other. A laminar sheet of mass 30 Kg is hinged at the line AB joining the two heavy metallic plates. The hinges are frictionless. The moment of inertia of the laminar sheet about an axis parallel to AB and passing through its centre of mass is 1.2 Kg-m2. Two rubber obstacles P and Q are fixed, one on each metallic plate at a distance 0.5 m from the line AB. This distance is chosen so that the reaction due to the hinges on the laminar sheet is zero during the impact. Initially the laminar sheet hits one of the obstacles with an angular velocity 1 rad/s and turns back. If the impulse on the sheet due to each obstacle is 6 N-s
[2001-10 marks] (a).Find the location of the centre of mass of the laminar sheet from AB.
(b).At what angular velocity does the laminar sheet come back after the first impact?
(c).After how many impacts, does the laminar sheet come to rest ?

Ans: (a):0.1m , (b): 1rad/sec , (c): Sheet swings Infinitly

38.A particle’ of mass m, moving in a circular path of radius R with a qpnstant speed v2is located at point (2R,0) at time t=0 and a man starts moving with a velocity V ¹ along the +ve y-axis from origin at time t=0. Calculate the linear momentum of the particle with respect to the man as a function of time.        [2003-2 marks] Ans: 39.Two masses m} and m2 are connected by a light spring of natural length l0 which is compressed completely and tied by a string. This system while moving with a velocity v0 along +ve x-axis passes through the origin at t=0. At this position the string snaps. Position of mass mt at time t is given by the equation Xj (t) = v0t – A (l – cos cot) calculate[2003-4 marks]

Ans: (a).position of the particle as a function of time
(b)l0 in terms of A.

40.There is a rectangular plate of mass M kg of dimensions (a x b). The plate is held in  horizontal position by striking n small balls, each of mass m, per unit area per unit time. These are striking in the shaded half region of the plate. The balls are colliding elastically with velocity v. What is v?    It is given
n = 100, M = 3 kg, m = 0.01 kg; b = 2 m; a = 1m; g = 10 m/s2.         [2006-6 marks] Ans:10m/s
41. Three objects A, B and C are kept in a straight line on a frictionless horizontal surface. These have masses m, 2m and m, respectively. The object A moves towards B with a speed 9 m/s and makes ‘ an elastic collision with it. Thereafter, B makes completely inelastic collision with C. All motions occur on the same straight line. Find the final speed (in m/s) of the object C. Ans:4

True / False Type

42.Two particles of mass 1kg and 3kg move towards each other under their mutual force of attraction. No other force acts on them. When the relative velocity of approach of the two particles is 2m/s, their centre of mass has a velocity of 0.5m/s. When the relative velocity of approach becomes 3m/s, the velocity of the centre of mass is 0.75m/[1989-2 marks]

Ans:False

43.A particle of mass 4m which is at rest explodes  into three fragments. Two of the fragments each of mass m are found to move with a speed v each in mutually perpendicular directions. The total energy released in the process of explosion is          [1987-2 marks]

Ans:3/2mv ²

Fill in the blanks typeV

44.The magnitude of the force (in newton) acting on a body varies with time t (in microsecond) as shown in the fig. AB, BC and CD are straight line segments. The magnitude of the total impulse of the force on the body from t = 4Î¼s to t = 16Î¼s …..     [1994-2 marks] Ans:0.005