Practice Questions

Q5. Let the moment of inertia of a hollow cylinder of length 30 cm (inner radius 10 cm and outer radius 20 cm ), about its axis be I . The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also I, is: (1) 16 cm (2) 14 cm (3) 12 cm (4) 18 cm

Q5. A force acts on a 2 kg object so that its position is given as a function of time as x = 3t2 + 5. What is the work done by this force in first 5 seconds? (1) 875 J (2) 850 J (3) 950 J (4) 900 J

Q5. 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 ms−1, from the ground. The bullet gets JEE Main 2019 (10 Jan Shift 1) JEE Main Previous Year Paper embedded in the wood. Then the maximum height to which the combined system reaches above the top of the building before falling below is: (g = 10 ms−2) (1) 40 m (2) 20 m (3) 10 m (4) 30 m

Q5. A person of mass 𝑀 is sitting on a swing of length 𝐿 and swinging with and an angular amplitude θ0 . If the person stands up when the swing passes through its lowest point, the work done by him, assuming that his centre of mass moves by a distance 𝑙 𝑙< < 𝐿, is close to: (1) 𝑀𝑔𝑙 (1 - θ02 ) (2) θ02 𝑀𝑔𝑙(1 + ) 2 (3) 𝑀𝑔𝑙 (4) 𝑀𝑔𝑙(1 + θ02 )

Q6. An alpha- particle of mass m suffers 1- dimensional elastic collision with a nucleus at rest of unknown mass. It is scattered directly backwards losing 64% of its initial kinetic energy. The mass of the nucleus is (1) 1.5m (2) 4m (3) 3.5m (4) 5m

Q6. An equilateral triangle ABC is cut from a thin solid sheet of wood. (See figure) D, E and F are the mid-points of its sides as shown and G is the centre of the triangle. The moment of inertia of the triangle about an axis passing through G and perpendicular to the plane of the triangle is I0 . If the smaller triangle DEF is removed JEE Main 2019 (11 Jan Shift 1) JEE Main Previous Year Paper from ABC, the moment of inertia of the remaining figure about the same axis is I . Then (1) I = 1516 I0 (2) I = 43 I0 (3) I = 169 I0 (4) I = I04

Q6. A thin circular plate of mass 𝑀 and radius 𝑅 has its density varying as ρ ( 𝑟) =ρ0r with 𝜌0 as constant and 𝑟 is the distance from its centre. The moment of Inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is 𝐼= 𝑎𝑀𝑅2. The value of the coefficient 𝑎 is: (1) 3 (2) 1 (3) 8 (4) 3 5 2 5 2

Q6. A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of θ, where θ is the angle by which it has rotated, is given as kθ2 . If its moment of inertia is I then the angular acceleration of the disc is: (1) 2k θ (2) k θ I 2I (3) k θ (4) k θ 4I I

Q6. A body of mass 𝑚1 moving with an unknown velocity of 𝑣1 ^i, undergoes a collinear collision with a body of mass 𝑚2 moving with a velocity 𝑣2 ^i . After the collision, 𝑚1 and 𝑚2 move with velocities of 𝑣3 ^i and 𝑣4 ^i, respectively. If 𝑚2 = 0.5 𝑚1 and 𝑣3 = 0.5 𝑣1, then 𝑣1 is: 𝑣2 𝑣2 (1) 𝑣4 - 4 (2) 𝑣4 - 2 (3) 𝑣4 + 𝑣2 (4) 𝑣4 - 𝑣2

Q6. A thin smooth rod of length L and mass M is rotating freely with angular speed ω0 about an axis perpendicular to the rod and passing through center. Two beads of mass m and negligible size are at the center of the rod initially. The beads of mass m and negligible size are at the center of the rod initially. The beads are free to slide along the rod. The angular speed of the system, when the beads reach the opposite ends of the rod, will be: (1) M ω0 (2) M ω0 M+3m M+2m (3) M ω0 (4) M ω0 M+6m M+m

Q6. A metal coin of mass 5 g and radius 1 cm is fixed to a thin stick AB of negligible mass as shown in the figure. The system is initially at rest. The constant torque, that will make the system rotate about AB at 25 rotations per second in 5 s , is close to: (1) 1.6 × 10-5 N m (2) 2.0 × 10-5 N m (3) 7.9 × 10-6 N m (4) 4.0 × 10-6 N m

Q6. To mop-clean a floor, a cleaning machine presses a circular mop of radius R vertically down with a total force F and rotates it with a constant angular speed about its axis. If the force F is distributed uniformly over the mop and if coefficient of friction between the mop and the floor is μ, the torque, applied by the machine on the mop is: (1) 2 μFR/3 (2) μFR/3 (3) μFR/6 (4) μFR/2

Q6. A circular disc D1 of mass M and radius R has two identical discs D2 and D3 of the same mass M and radius R attached rigidly at its opposite ends (see figure). The moment of inertia of the system about the axis OO', passing through the centre of D1 , as shown in the figure, will be JEE Main 2019 (11 Jan Shift 2) JEE Main Previous Year Paper (1) MR2 (2) 3MR2 (3) 5 4 MR2 (4) 32 MR2

Q6. A smooth wire of length 2πr is bent into a circle and kept in a vertical plane. A bead can slide smoothly on the wire. When the circle is rotating with angular speed ω about the vertical diameter AB, as shown in figure, the JEE Main 2019 (12 Apr Shift 2) JEE Main Previous Year Paper bead is at rest with respect to the circular ring at position P as shown. Then the value of ω2 is equal to: (1) 2g/r (2) √3g 2r (3) 2g/(r√3) (4) (g√3)/r

Q6. The value of acceleration due to gravity at Earth's surface is 9.8 m s−2 . The altitude above its surface at which the acceleration due to gravity decreases to 4.9 m s−2 , is close to: (Radius of earth = 6.4 × 106 m ) (1) 1.6 × 106 m (2) 2.6 × 106 m (3) 6.4 × 106 m (4) 9.0 × 106 m

Q6. Two identical spherical balls of mass M and radius R each are stuck on two ends of a rod of length 2R and mass M (see figure). The moment of inertia of the system about the axis passing perpendicularly through the centre of the rod is JEE Main 2019 (10 Jan Shift 2) JEE Main Previous Year Paper (1) 209 MR2 . (2) 152 MR2 . 15 15 (3) 137 MR2 . (4) 17 MR2 . 15 5

Q6. A satellite of mass M is in a circular orbit of radius R about the center of the earth. A meteorite of the same mass, falling towards the earth, collides with the satellite completely inelastic. The speeds of the satellite and the meteorite are the same, just before the collision. The subsequent motion of the combined body will be: (1) In an elliptical orbit (2) Such that it escapes to infinity (3) In a circular orbit of a different radius (4) In the same circular orbit of radius R

Q6. Two masses 𝑚 and are connected at the two ends of a massless rigid rod of length 𝑙. The rod is suspended by 2 a thin wire of torsional constant 𝑘 at the centre of mass of the rod-mass system (see figure). Because of torsional constant 𝑘, the restoring torque is 𝜏= 𝑘𝜃 for angular displacement 𝜃. If the rod is rotated by 𝜃0 and released, JEE Main 2019 (09 Jan Shift 1) JEE Main Previous Year Paper the tension in it when it passes through its mean position will be: (1) 𝑘𝜃02 (2) 3𝑘𝜃02 𝑙 (3) 2𝑘𝜃02 (4) 𝑘𝜃02 𝑙 𝑙

Q6. A rod of length 50 cm is pivoted at one end. It is raised such that it makes an angle of 30o from the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in rad s−1 ) will be (g = 10 ms−2) JEE Main 2019 (09 Jan Shift 2) JEE Main Previous Year Paper (1) √20 (2) √30 3 2 (3) √302 (4) √30

Q7. A satellite is revolving in a circular orbit at a height h from the carth surface, such that h << R where R is the radius of the earth. Assuming that the effect of earth"s atmosphere can be neglected the minimum increase in the speed required so that the satellite could escape from the gravitational field of earth is (1) √2gR (2) √gR (3) √gR2 (4) √gR(√2 −1)

Q7. 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: 2 1 (1) tan⁡𝜃= (2) tan⁡𝜃= √3 3 (3) tan⁡𝜃= 1 (4) tan⁡𝜃= 1 2 2√3

Q7. A long cylindrical vessel is half filled with a liquid. When the vessel is rotated about its own vertical axis, the liquid rises up near the wall. If the radius of vessel is 5 cm and its rotational speed is 2 rotations per second, then the difference in the heights between the center and the sides, in cm, will be: (1) 0.4 (2) 2.0 (3) 1.2 (4) 0.1

Q7. Four identical particles of mass M are located at the corners of a square of side ‘𝑎’ . What should be their speed if each of them revolves under the influence of other’s gravitational field in a circular orbit circumscribing the JEE Main 2019 (08 Apr Shift 1) JEE Main Previous Year Paper square? (1) GM (2) GM (3) GM (4) GM 1.35√ a 1.21√ a 1.41√ a 1.16√ a

Q7. The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal plane: (i) a ring of radius R, (ii) a solid cylinder of radius R2 and (iii) a solid sphere of radius R4 . If, in each case, the speed of the center of mass at the bottom of the incline is same, the ratio of the maximum heights they climb is: JEE Main 2019 (09 Apr Shift 1) JEE Main Previous Year Paper (1) 2 : 3 : 4 (2) 20 : 15 : 14 (3) 4 : 3 : 2 (4) 14 : 15 : 20

Q7. A homogeneous solid cylindrical roller of radius R and mass M is pulled on a cricket pitch by a horizontal force. Assuming rolling without slipping, angular acceleration of the cylinder is: (1) F (2) 3F 3 mR 2 m R (3) 2F (4) F 3 m R 2 m R