Practice Questions

Q5. A particle of mass m is moving with speed 2v and collides with a mass 2m moving with speed v in the same direction. After the collision, the first mass is stopped completely while the second one splits into two particles each of mass m, which move at an angle 45o with respect to the original direction. The speed of each of the moving particle will be (1) √2v (2) v √2 (3) 2√2v (4) v (2√2)

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 solid sphere of mass M and radius R is divided into two unequal parts. The first part has a mass of and is 8 converted into uniform disc of radius 2R . The second part is converted into a uniform solid sphere. Let I1 be the moment of inertia of the disc about its axis and I2 be the moment of inertia of the new sphere about its axis. The ratio I1 / I2 is given by: (1) 140 (2) 185 (3) 65 (4) 285 JEE Main 2019 (10 Apr Shift 2) JEE Main Previous Year Paper

Q5. Three blocks A, B and C are lying on a smooth horizontal surface, as shown in the figure. A and B have equal masses, 𝑚 while C has mass 𝑀. Block A is 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 . 5 of the initial kinetic 6th energy is lost in the whole process. What is the value of 𝑀/ 𝑚? (1) 3 (2) 4 (3) 5 (4) 2 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. 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 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 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

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. A straight rod of length L extends from x = a to x = L + a. The gravitational force it exerts on a point mass 'm' at x = 0, if the mass per unit length of the rod is A + Bx2, is given by: JEE Main 2019 (12 Jan Shift 1) JEE Main Previous Year Paper (1) Gm[A( a+L1 −1a ) + BL] (2) Gm[A( a+L1 −1a ) −BL] (3) Gm[A( a1 − a+L1 ) −BL] (4) Gm[A( a1 − a+L1 ) + BL]

Q7. A circular disc of radius 𝑏 has a hole of radius a at its centre(see figure). If the mass per unit area of the disc 𝜎0 varies as then, the radius of gyration of the disc about its axis passing through the center is 𝑟 (1) 𝑎+ 𝑏 (2) + 𝑏2 + 𝑎𝑏 3 √𝑎2 3 + 𝑏2 + 𝑎𝑏 (3) 𝑎+ 𝑏 (4) √𝑎2 2 2

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

Q8. Two stars of masses 3 × 1031 kg each, and at distance 2 × 1011 m rotate in a plane about their common centre of mass O. A meteorite passes through O moving perpendicular to the stars,s rotation plane. In order to escape from the gravitational field of this double star, the minimum speed that meteorite should have at O is ( Take Gravitational constant G = 6.67 × 10−11 N m2 kg−2 ) (1) 2.4 × 104 m s−1 (2) 3.8 × 104 m s−1 (3) 2.8 × 105 m s−1 (4) 1.4 × 105 m s−1

Q8. A test particle is moving in a circular orbit in the gravitational field produced by a mass density ρ(r) = K . r2 Identify the current relation between the radius R of the particle’s orbit and its period T : JEE Main 2019 (09 Apr Shift 2) JEE Main Previous Year Paper (1) TR is a constant (2) T 2/R3 is a constant (3) T/R is a constant (4) T/R2 is a constant

Q8. A satellite is moving with a constant speed v in circular orbit around the earth. An object of mass 'm' is ejected from the satellite such that it just escapes from the gravitational pull of the earth. At the time of ejection, the kinetic energy of the object is: (1) 23 m v2 (2) m v2 (3) 12 m v2 (4) 2 m v2

Q9. Ice at −20∘C is added to 50 g of water at 40∘C, When the temperature of the mixture reaches 0∘C, it is found that 20 g of ice is still unmelted. The amount of ice added to the water was close to (Specific heat of water = 4.2 J/g/∘C Specific heat of Ice = 2.1 J/g/∘C Heat of fusion of water at 0∘C = 334 J/g) (1) 50 g (2) 100 g (3) 60 g (4) 40 g

Q10.A 25 × 10−3 m3 volume cylinder is filled with 1 mol of O2 gas at room temperature (300 K) . The molecular diameter of O2 , and its root mean square speed, are found to be 0.3 nm and 200 m/s , respectively. What is the average collision rate (per second) for an O2 molecule? (1) ~1011 (2) ~1012 (3) ~1010 (4) ~1013

Q11.The specific heats, Cp and Cv of a gas of diatomic molecules, A, are given (in units of J mol−1 K−1 ) by 29 and 22, respectively. Another gas of diatomic molecules, B, has the corresponding values 30 and 21. If they are treated as ideal gases, then: (1) A has one vibrational mode and B has two (2) A has a vibrational mode but B has none. (3) Both A and B have a vibrational mode each. (4) A is rigid but B has a vibrational mode.

Q11.Two light identical springs of spring constant k are attached horizontally at the two ends of a uniform horizontal rod AB of length l and mass m. The rod is pivoted at its center 'O' and can rotate freely in horizontal plane. The other ends of the two springs are fixed to rigid supports as shown in figure. The rod is gently pushed through a small angle and released. The frequency of resulting oscillation is: (1) 1 (2) 1 2π √3km 2π √km (3) 1 (4) 1 2π √6km 2π √2km

Q12.A submarine A travelling at 18km hr-1 is being chased along the line of its velocity by another submarine B travelling at 27 km hr-1 . B sends a sonar signal of 500 Hz to detect A and receives a reflected sound of frequency v. The value of v is closed to (Speed of sound in water 1500 m s-1 ) (1) 504 Hz (2) 499 Hz (3) 502 Hz (4) 507 Hz

Q12.A rod of mass M and length 2L is suspended at its middle by a wire. It exhibits torsional oscillations. If two masses, each of mass m, are attached at a distance L/2 from its centre on both sides, it reduces the oscillation frequency by 20%. The value of ratio m/M is close to (1) 0.17 (2) 0.77 (3) 0.57 (4) 0.37

Q13.A uniformly charged ring of radius 3a and total charge q is placed in x‐y plane centred at origin. A point charge q is moving towards the ring along the z− axis and has speed v at z = 4a . The minimum value of v such that it crosses the origin is: JEE Main 2019 (10 Apr Shift 1) JEE Main Previous Year Paper (1) √2m ( 151 4πϵ0aq2 ) 1/2 (2) √2m ( 154 4πϵ0aq2 ) 1/2 (3) √2m ( 51 4πϵ0aq2 ) 1/2 (4) √2m ( 152 4πϵ0aq2 ) 1/2

Q14.A simple pendulum of length 1 m is oscillating with an angular frequency 10rad/s. The support of the pendulum starts oscillating up and down with a small angular frequency of 1rad/s and an amplitude of 10−2 m . The relative change in the angular frequency of the pendulum is best given by : (1) 10−3rad/s (2) 1rad/s (3) 10−1rad/s (4) 10−5rad/s

Q14.Two electric dipoles, A, B with respective dipole moments dA→ = −4qaˆi and dB→ = −2qaˆi are placed on the x - axis with a separation R, as shown in the figure The distance from A at which both of them produce the same potential is: (1) R (2) √2R √2 −1 √2−1 (3) √2 R (4) R √2+1 √2+1 JEE Main 2019 (10 Jan Shift 1) JEE Main Previous Year Paper

Q14.Charge is distributed within a sphere of radius R with a volume charge density ρ(r) = A e−2ra , where A and a r2 are constants. If Q is the total charge of this charge distribution, the radius R is: (1) a 1 (2) 1 a Q Q 2 log( 1− 2πaA ) log( 1− 2πaA ) − 2πaA − 2πaA (3) a log(1 Q ) (4) a2 log(1 Q ) q2(−25 μC) are placed on the x -axis at x = 1 m and x = 4 m