Collision & Variable Mass PYQs Set 3 (PDF)

Set 3 of 4 · Questions 61-90 of 119

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Mere Teacher · Previous Year Questions JEE Main Physics · Collision & Variable Mass · Set 3 (Q61-90)
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61. $A uniform rectangular thin sheet ABCD of mass M has length a and breadth b, as shown in the figure. If the shaded portion HBGO is cut off, the coordinates of the centre of mass of the remaining portion will be:$ (1) $(\frac{3 a}{4}, \frac{3 b}{4})$ (2) $(\frac{5 a}{3}, \frac{5 b}{3})$ (3) $(\frac{2 a}{3}, \frac{2 b}{3})$ (4) $(\frac{5 a}{12}, \frac{5 b}{12})$ 62. $An L-shaped object, made of thin rods of uniform mass density, is suspended with a string as shown in figure. If AB = BC, and the angle made by AB with downward vertical is θ, then:$ (1) $tan θ = \frac{1}{2 3}$ (2) $tan θ = \frac{1}{2}$ (3) $tan θ = \frac{2}{3}$ (4) $tan θ = \frac{1}{3}$ 63. $A particle of mass m with an initial velocity ui collides perfectly elastically with a mass 3m at rest. It moves with a velocity v ı ˋ after collision, then, v is given by$ (1) $v = \frac{1}{6} u$ (2) $v = \frac{u}{3}$ (3) $v = \frac{2}{3} u$ (4) $v = \frac{u}{2}$ 64. $A large block of wood of mass M=5.99 kg is hanging from two long massless cords. A bullet of mass m = 10 g is fired into the block and gets embedded in it. The (block + bullet) then swing upwards, their Centre of mass rising a vertical distance h = 9.8 cm before the (block + bullet) pendulum comes momentarily to restate the end of its arc. The speed of the bullet just before collision is:(Take: g = 9.8 ms^{- 2})$ (1) $841.4 m/s$ (2) $811.4 m/s$ (3) $831.4 m/s$ (4) $821.4 m/s$ 65. $A particle of mass m is projected from the ground with an initial speed u_{0} at an angle α with the horizontal. At the highest point of its trajectory, it makes a completely inelastic collision with another identical particle, which was thrown vertically upward from the ground with the same initial speed u_{0} . The angle that the composite system makes with the horizontal immediately after the collision is :$ (1) $\frac{π}{4}$ (2) $\frac{π}{4} + a$ (3) $\frac{π}{4} - a$ (4) $\frac{π}{4} + \frac{a}{2}$ 66. $On a frictionless surface, a block of mass M moving at speed v collides elastically with another block of same mass M which is initially at rest. After collision, the first block moves at an angle θ to its initial direction and has a speed v /3 . The second block's speed after the collision is$ (1) $\frac{2 2}{3} v$ (2) $\frac{3}{4} v$ (3) $\frac{3}{2} v$ (4) $\frac{3}{2} v$ 67. $A body of mass M moving at speed V_{0} collides elastically with a mass 'm' at rest. After the collision, the two masses move at angles θ_{1} and θ_{2} with respect to the initial direction of motion of the body of mass M. The largest possible value of the ratio M/m, for which the angles θ_{1} and θ_{2} will be equal, is:$ (1) $4$ (2) $1$ (3) $3$ (4) $2$ 68. $A mass m moving horizontally (along the x ‐axis) with velocity v collides and sticks to a mass of 3 m moving vertically upward (along the y ‐axis) with velocity 2 v . The final velocity of the combination is (Mains 2011)$ (1) $\frac{3}{2} v i + \frac{1}{4} v j$ (2) $\frac{1}{4} v i + \frac{3}{2} v j$ (3) $\frac{1}{3} v i + \frac{2}{3} v j$ (4) $\frac{2}{3} v i + \frac{1}{3} v j$ 69. $A piece of wood of mass 0.03 kg is dropped from the top of a 100 m height building. At the same time, a bullet of mass 0.02 kg is fired vertically upward, with a velocity 100 m s^{- 1}, from the ground. The bullet gets embedded in the wood. Then the maximum height of which the combined system reaches above the top of the building before falling below is: (g = 10 m s^{- 2})$ (1) $20 m$ (2) $30 m$ (3) $40 m$ (4) $10 m$ 70. $Four particles A, B, C and D with masses m_{A} = m, m_{B} = 2 m, m_{C} = 3 m and m_{D} = 4 m are at the corners of a square. They have accelerations of equal magnitude with directions as shown. The acceleration of the centre of mass of the particles is:$ (1) $\frac{a}{5} (i - j)$ (2) $a (i - j)$ (3) $Zero$ (4) $a10i-j$ 71. $As per the given figure, a small ball P slides down the quadrant of a circle and hits the other ball Q of equal mass which is initially at rest. Neglecting the effect of friction and assume the collision to be elastic, the velocity of ball Q after collision will be: (g = 10 m/ s^{2})$ (1) $0$ (2) $0.25 m/s$ (3) $2 m/s$ (4) $4 m/s$ 72. $There block, A, B and C are lying on a smooth horizontal surface, as shown in figure. A and B have equal masses, m while C has mass M . A is a given an initial speed v towards B due to which it collides with B perfectly inelastically. The combined mass collides with C, also perfectly inelastically \frac{5}{6} th of the kinetic energy is lost in whole process. What is value of M/m?$ (1) $5$ (2) $2$ (3) $4$ (4) $3$ 73. $Two bodies A and B of same mass undergo completely inelastic one-dimensional collision. The body A moves with velocity v_{1} while body B is at rest before collision. The velocity of the system after collision is v_{2} . The ratio v_{1} : v_{2} is [NEET 2024]$ (1) $1 : 2$ (2) $2 : 1$ (3) $4 : 1$ (4) $1 : 4$ 74. $Blocks of masses m, 2 m, 4 m and 8 m are arranged in a line on a frictionless floor. Another block of mass m, moving with speed v along the same line (see figure) collides with mass m in perfectly inelastic manner. All the subsequent collisions are also perfectly inelastic. By the time the last block of mass 8 m starts moving the total energy loss is p% of the original energy. Value of 'P' is close to$ (1) $37$ (2) $77$ (3) $87$ (4) $94$ 75. $A particle of mass m, moving with a velocity v makes an elastic collision in one dimension with a stationary particle of mass m . During the collision, they remain in contact with each other for an extremely small time T . Their force of contact, with time is shown in the figure. Then, F_{0}$ (1) $\frac{2 mv}{T}$ (2) $\frac{4 mv}{3 T}$ (3) $\frac{mv}{T}$ (4) $\frac{3 mv}{4 T}$ 76. $Two billiard balls of mass 0.05 kg each moving in opposite directions with 10 m s^{- 1} collide and rebound with the same speed. If the time duration of contact is t = 0.005 s, then what is the force exerted on the ball due to each other?$ (1) $100 N$ (2) $200 N$ (3) $300 N$ (4) $400 N$ 77. $A particle of mass m moving in the x direction with speed 2v is hit by another particle of mass 2m moving in the y direction with speed v. If the collision is perfectly inelastic, the percentage loss in the energy during the collision is close to :$ (1) $44%$ (2) $50%$ (3) $56%$ (4) $62%$ 78. $As shown in figure. When a spherical cavity (centered at O) of radius 1 is cut out of a uniform sphere of radius R (centered at C), the centre of mass of remaining (shaded) part of sphere is at G, i.e on the surface of the cavity. R can be determined by the equation:$ (1) $(R^{2} + R + 1) (2 - R) = 1$ (2) $(R^{2} - R - 1) (2 - R) = 1$ (3) $(R^{2} - R + 1) (2 - R) = 1$ (4) $(R^{2} + R - 1) (2 - R) = 1$ 79. $A particle of mass m is projected with a speed u from the ground at an angle θ = π /3 w.r.t. horizontal (x -axis). When it has reached its maximum height, it collides completely in-elastically with another particle of the same mass and velocity u i^. The horizontal distance covered by the combined mass before reaching the ground is:$ (1) $\frac{3 3}{8} \frac{u ^{2}}{g}$ (2) $22 \frac{u ^{2}}{g}$ (3) $\frac{5}{8} \frac{u ^{2}}{g}$ (4) $\frac{3 2}{4} \frac{u ^{2}}{g}$ 80. $The disc of mass M with uniform surface masse density σ is shown in the figure. The centre ofmass of the quarter disc (the shaded area) is atthe position \frac{x}{3} \frac{a}{π}, \frac{x}{3} \frac{a}{π} where x is ________ .(Round off to the Nearest Integer) [a is an area as shown in the figure]$ 81. $Two bodies of the same mass are moving with the same speed, but in different directions in a plane. They have a completely inelastic collision and move together thereafter with a final speed which is half of their initial speed. The angle between the initial velocities of the two bodies (in degree) is$ 82. $A bullet of 10g, moving with velocity υ, collides head-on with the stationary bob of a pendulum and recoils with velocity 100m/s. The length of the pendulum is 0.5m and mass of the bob is 1kg. The minimum value of υ = m / s so that the pendulum describes a circle. (Assume the string to be inextensible and g = 10 m / s^{2})$ 83. $A rod of mass M and length L is lying on a horizontal frictionless surface. A particle of mass 'm' travelling along the surface hits at one end of the rod with a velocity 'u' in a direction perpendicular to the rod. The collision is completely elastic. After collision, particle comes to rest. The ratio of masses (\frac{m}{M}) is \frac{1}{x} . The value of 'x' will be ___$ 84. $Consider uniform cubical box of side a on a rough floor that is to be moved by applying minimum possible force F at a point b above its centre of mass (see figure). If the coefficient of friction is μ = 0.4, the maximum possible value of 100 \times \frac{b}{a} for box not to topple before moving is$ 85. $A solid sphere of mass 1 kg rolls without slipping on a plane surface. Its kinetic energy is 7 x 1 0^{- 3} J. The speed of the center of mass of the sphere is______ cm s^{- 1} .$ 86. $The projectile motion of a particle of mass 5g is shown in the figure. The initial velocity of the particle is 52 ms^{- 1} and the air resistance is assumed to be negligible. The magnitude of the change in momentumbetween the points A and B is x \times 10^{- 2} kgms^{- 1} . The value of x, to the nearest integer, is ___.$ 87. $A block starts moving up an inclined plane of inclination 30^{o} with an initial velocity of v_{0} . It comes back to its initial position with velocity \frac{v _{0}}{2} . The value of the coefficient of kinetic friction between the block and the inclined plane is close to \frac{1}{1000} . The nearest integer to 1 is_______.$ 88. $The identical spheres each of mass 2 M are placed at the corners of a right-angled triangle with mutually perpendicular sides equal to 4 m each. Taking point of intersection of these two sides as origin, the magnitude of position vector of the centre of mass of the system is \frac{4 2}{x}, where the value of x is_____$ 89. $Four identical discs each of mass^{‘} M' and diameter 'a' are arranged in a small plane as shown in figure. If the moment of inertia of the system about OO' is \frac{x}{4} M a^{2} . Then, the value of x will be _____ .$ 90. $A ball with a speed of 9 m / s collides with another identical ball at rest. After the collision, the direction of each ball makes an angle of 30^{\circ} with the original direction. The ratio of velocities of the balls after collision is x : y, where x is$