Practice Questions

Q4. Pressure inside two soap bubbles are 1. 01 and 1. 02 atmosphere, respectively. The ratio of their volumes is : (1) 4 : 1 (2) 0. 8 : 1 (3) 8 : 1 (4) 2 : 1

Q4. A wheel is rotating freely with an angular speed ω on a shaft. The moment of inertia of the wheel is I and the moment of inertia of the shaft is negligible. Another wheel of moment of inertia 3I initially at rest is suddenly coupled to the same shaft. The resultant fractional loss in the kinetic energy of the system is: (1) 65 (2) 14 (3) 0 (4) 34

Q4. The acceleration due to gravity on the earth's surface at the poles is g and angular velocity of the earth about the axis passing through the pole is ω. An object is weighed at the equator and at a height h above the poles by using a spring balance. If the weights are found to be same, then h is: ( h ≪R, where R is the radius of the earth) (1) R2ω2 (2) R2ω2 2g g (3) R2ω2 (4) R2ω2 4g 8g

Q4. Mass per unit area of a circular disc of radius a depends on the distance r from its centre as σ(r) = A + Br . The moment of inertia of the disc about the axis, perpendicular to the plane and passing through its centre is: (1) 2πa4( A4 + aB5 ) (2) 2πa4( aA4 + B5 ) (3) πa4( A4 + aB5 ) (4) 2πa4( A4 + B5 )

Q4. A capillary tube made of glass of radius 0. 15 mm is dipped vertically in a beaker filled with methylene iodide (surface tension = 0. 05 N m−1 , density = 667 kg m−3 ) which rises to height h in the tube. It is observed that the two tangents drawn from observed that the two tangents drawn from liquid-glass interfaces (from opp. sides of the capillary) make an angle of 60º with one another. Then h is close to ( g = 10 m s−2 ) (1) 0. 049 m (2) 0. 087 m (3) 0. 137 m (4) 0. 172 m

Q4. Shown in the figure is a hollow ice-cream cone (it is open at top). If its mass is M, radius of its top is R and height, H , then its moment of inertia about its axis is (1) MR2 (2) M(R2+H2) 2 4 (3) MH2 (4) MR2 3 3

Q4. Three solid spheres each of mass m and diameter d are stuck together such that the lines connecting the centres form an equilateral triangle of side of length d . The ratio I0 of moment of inertia I0 of the system about an IA axis passing the centroid and about center of any of the spheres IA and perpendicular to the plane of the triangle is: (1) 13 (2) 15 23 13 (3) 23 (4) 13 13 15

Q4. A person pushes a box on a rough horizontal plateform surface. He applies a force of 200 N over a distance of 15 m. Thereafter, he gets progressively tired and his applied force reduces linearly with distance to 100 N . The total distance through which the box has been moved is 30 m. What is the work done by the person during the total movement of the box? (1) 3280 J (2) 2780 J (3) 5690 J (4) 5250 J

Q4. A rod of length l has non-uniform linear mass density given by ρ(x) = a + b( xl )2, where a and b are constants and 0 ≤x ≤l The value of x for the centre of mass of the rod is at: (1) 2 3 ( 2a+ba+b )L (2) 43 ( 3a+b2a+b )L (3) 3 4 ( 2a+3ba+b )L (4) 32 ( 3a+b2a+b )L

Q4. A uniform cylinder of mass M and radius R is to be pulled over a step of height a (a < R) by applying a force F at its centre ′O′ perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of F required is : (1) R−a 2 (2) R 2 Mg √( R−a ) −1 Mg √1 −( R ) (3) Mg Ra (4) Mg √1 −a2R2

Q4. As shown in figure. When a spherical cavity (centred at O ) of radius 1 is cut out of a uniform sphere of radius R (centred at C ), the centre of mass of remaining (shaded part of sphere is at G, i.e., on the surface of the JEE Main 2020 (08 Jan Shift 2) JEE Main Previous Year Paper cavity. R can be determined by the equation: (1) (R2 + R + 1)(2 −R) = 1 (2) (R2 −R −1)(2 −R) = 1 (3) (R2 −R + 1)(2 −R) = 1 (4) (R2 + R −1)(2 −R) = 1

Q5. On the x-axis and at a distance x from the origin, the gravitational field due to a mass distribution is given by Ax in the x-direction. The magnitude of the gravitational potential on the x-axis at a distance x, taking its (x2+a2)3/2 value to be zero at infinity is: (1) A (2) A (x2+a2)1/2 (x2+a2)3/2 (3) A(x2 + a2)1/2 (4) A(x2 + a2)3/2

Q5. The value of the acceleration due to gravity is g1 at a height h = R2 ( R = radius of the earth) from the surface of the earth. It is again equal to g1 at a depth d below the surface the earth. The ratio ( Rd ) equals: (1) 94 (2) 59 (3) 31 (4) 79

Q5. A uniform rod of length ' ℓ′ is pivoted at one of its ends on a vertical shaft of negligible radius. When the shaft rotates at angular speed ω the rod makes an angle θwith it (see figure). To find θ equate the rate of change of angular momentum (direction going into the paper) mℓ2 ω2 sin θ about the centre of mass (CM) to the torque 12 provided by the horizontal and vertical forces FH and Fv about the CM. The value of θ is then such that: (1) cos θ = 2g (2) cos θ = g 3𝓁ω2 2ℓω2 (3) cosθ = g (4) cos θ = 3g ℓω2 2ℓω2 JEE Main 2020 (03 Sep Shift 2) JEE Main Previous Year Paper

Q5. Four point masses, each of mass m , are fixed at the corners of a square of side I. The square is rotating with angular frequency ω, about an axis passing through one of the corners of the square and parallel to tis diagonal, as shown in the figure. The angular momentum of the square about the axis is (1) mℓ2ω (2) 4mℓ2ω (3) 3mℓ2ω (4) 2mℓ2ω JEE Main 2020 (06 Sep Shift 1) JEE Main Previous Year Paper

Q5. In an experiment to verify Stokes law, a small spherical ball of radius r and density ρ falls under gravity through a distance h in air before entering a tank of water. If the terminal velocity of the ball inside water is same as its velocity just before entering the water surface, then the value of h is proportional to: (ignore viscosity of air) (1) r4 (2) r (3) r3 (4) r2

Q5. Consider two uniform discs of the same thickness and different radii R1 = R and R2 = αR made of the same material. If the ratio of their moments of inertia I1 and I2 , respectively, about their axes is I1 : I2 = 1 : 16 then the value of α is : (1) 2√2 (2) √2 (3) 2 (4) 4

Q5. A box weighs 196N on a spring balance at the north pole. Its weight recorded on the same balance if it is shifted to the equator is close to (Take g = 10ms−2 at the north pole and the radius of the earth = 6400km ): (1) 195.66N (2) 194.32N (3) 194.66N (4) 195.32N

Q5. A satellite of mass M is launched vertically upwards with an initial speed u from the surface of the earth. After it reaches height R ( R = radius of the earth), it ejects a rocket of mass M10 so that subsequently the satellite moves in a circular orbit. The kinetic energy of the rocket is ( G is the gravitational constant; Me is the mass of the earth): JEE Main 2020 (07 Jan Shift 1) JEE Main Previous Year Paper (1) M 20 (u2 + 113200 GMeR ) (2) 5M(u2 −119200 GMeR ) (3) 3M 2 (4) M 2 8 20 3R (u + √5GMe6R ) (u −√2GMe )

Q5. When the temperature of a metal wire is increased from 0ºC to 10ºC, its length increases by 0. 02% .The percentage change in its mass density will be closed to: (1) 0. 06 (2) 2. 3 (3) 0. 008 (4) 0. 8

Q5. Two planets have masses M and 16 M and their radii are a and 2a , respectively. The separation between the centres of the planets is 10a . A body of mass m is fired from the surface of the larger planet towards the smaller planet along the line joining their centres. For the body to be able to reach at the surface of smaller planet, the minimum firing speed needed is : (1) 2√GMa (2) 4√GMa (3) √GM2ma (4) 23 √5 GMa

Q6. A air bubble of radius 1 cm in water has an upward acceleration of 9.8 cms–2 . The density of water is 1 gm cm–3 and water offers negligible drag force on the bubble. The mass of the bubble is (g = 980 cm/s2). JEE Main 2020 (04 Sep Shift 1) JEE Main Previous Year Paper (1) 4.51 gm (2) 3. 15 gm (3) 4.15 gm (4) 1.52 gm

Q6. A hollow spherical shell at outer radius R floats just submerged under the water surface. The inner radius of the shell is r . If the specific gravity of the shell material is 278 with respect to water, the value of r is: (1) 98 R (2) 49 R (3) 32 R (4) 13 R

Q6. Water flows m a horizontal tube (see figure). The pressure of water changes by 700 Nm−2 between A and B where the area of cross section are 40 cm2 and 20 cm2, respectively. Find the rate of flow of water through the tube. (density of water = 1000 kgm−3 ) (1) 3020cm3/s (2) 2720cm3/s (3) 2420cm3/s (4) 1810cm3/s

Q6. A particle of mass m with an initial velocity uˆi collides perfectly elastically with a mass 3 m at rest. It moves with a velocity vˆj after collision, then, v is given by (1) (2) v = u v = √23 u √3 (3) v = u (4) v = 1 u √2 √6