Practice Questions

Q3. A particle moves such that its position vector →r(t) = cos ωtˆi + sin ωtˆj where ω is a constant and t is time. Then which of the following statements is true for the velocity →v(t) and acceleration →a(t) of the particle: (1) →vis perpendicular to →rand →a is directed away (2) →vand →a both are perpendicular to →r from the origin (3) →vand →a both are parallel to →r (4) →vis perpendicular to →rand →a is directed towards the origin

Q3. As shown in the figure, a bob of mass m is tied to a massless string whose other end portion is wound on a fly wheel (disc) of radius r and mass m. When released from rest the bob starts falling vertically. When it has covered a distance of h, the angular speed of the wheel will be: (1) 1 gh (2) 3 r √4 3 r√ 2 gh (3) 1 gh (4) 3 r √2 3 r√ 4 gh

Q3. A bead of mass m stays at point P(a, b) on a wire bent in the shape of a parabola y = 4Cx2 and rotating with angular speed ω (see figure). The value of ω is (neglect friction) (1) 2√2gC (2) √2gC (3) √2gCab (4) √2gC

Q3. A mass of 10 kg is suspended by a rope of length 4 m , from the ceiling. A force F is applied horizontally at the mid-point of the rope such that the top half of the rope makes an angle of 45° with the vertical. Then F equals: (Take g = 10 m s−2 and the rope to be massless) (1) 100N (2) 90N (3) 70N (4) 75N

Q3. Particle A of mass m1 moving with velocity (√3ˆi +ˆj) ms−1 collides with another particle which is at rest initially. Let →v1 and →v2 be the velocities of particles A and B after collision respectively. If + and →v2 is : m1 = 2m2 and after collision →v1 −(ˆi √3ˆj) ms−1 , the angle between →v1 (1) 15° (2) 60° (3) −45° (4) 105°

Q3. Starting from the origin at time t = 0, with initial velocity 5 ⌢j ms−1 , a particle moves in the x - y plane with a constant acceleration of + 4⌢j . At time t, its coordinates are (20 m, y0 m). The values of t and y0 (10⌢i )ms−2 are, respectively: (1) 2 s and 18 m (2) 4 s and 52 m (3) 2 s and 24 m (4) 5 s and 25 m

Q3. The coordinates of the centre of mass of a uniform flag-shaped lamina (thin flat plate) of mass 4 kg. (The coordinates of the same are shown in the figure) are: (1) (1.25 m, 1.50 m) (2) (0.75 m, 1.75 m) (3) (0.75 m, 0.75 m) (4) (1 m, 1.75 m)

Q3. A helicopter rises from rest on the ground vertically upwards with a constant acceleration g. A food packet is dropped from the helicopter when it is at a height h. The time taken by the packet to reach the ground is close to [ g is the acceleration due to gravity]: (1) 2 h (2) h t = 1. t = 3 g ) g ) √( 8√( (3) h t = 3. (4) t = √2h3g g ) 4√(

Q3. A satellite is moving in a low nearly circular orbit around the earth. Its radius is roughly equal to that of the earth's radius Re. By firing rockets attached to it, its speed is instantaneously increased in the direction of its motion so that it become times larger. Due to this the farthest distance from the centre of the earth that the √32 satellite reaches is R. Value of R is : (1) 4Re (2) 2. 5Re (3) 3Re (4) 2Re

Q3. A particle starts from the origin at t = 0 with an initial velocity of 3.0ˆim/s and moves in the x −y plane with a constant acceleration + The x− coordinate of the particle at the instant when its y− (6.0ˆi 4.0ˆj)m/s2. coordinate is 32m is D meters. The value of D is: (1) 32 (2) 50 (3) 60 (4) 40

Q3. Two particles of equal mass m have respective initial velocities uˆi and u( ˆi+ˆj2 ) . They collide completely inelastically. The energy lost in the process is: (1) 1 mu2 (2) 1 mu2 3 8 (3) 3 mu2 (4) 4 √23 mu2

Q3. The height ‘ h’ at which the weight of a body will be the same as that at the same depth ‘h’ from the surface of the earth is (Radius of the earth is R and effect of the rotation of the earth is neglected) (1) √5 2 R −R (2) R2 (3) √5R−R (4) √3R−R 2 2

Q3. Hydrogen ion and singly ionized helium atom are accelerated, from rest, through the same potential difference. The ratio of final speeds of hydrogen and helium ions is close to: (1) 1 : 2 (2) 10 : 7 (3) 2 : 1 (4) 5 : 7

Q3. A small ball of mass m is thrown upward with velocity u from the ground. The ball experiences a resistive force mkv2 where v is it speed. The maximum height attained by the ball is : ku2 (1) + 2k 1 tan−1 ku2g (2) k1 ln(1 2g ) ku2 (3) + k 1 tan−1 ku22g (4) 2k1 ln(1 g )

Q3. A spaceship in space sweeps stationary interplanetary dust. As a result, its mass increases at a rate dM(t) v(t) is its instantaneous velocity. The instantaneous acceleration of the satellite is: dt = bv2(t), where (1) −bv3(t) (2) M(t)−bv3 (3) −2bv3M(t) (4) − 2M(t)bv3

Q3. If the potential energy between two molecules is given by U = A + B , then at equilibrium, separation r6 r12 between molecules, and the potential energy are: (1) ( 2AB ) 1/6, −A22B (2) ( AB ) 1/6, 0 1 A2 , B ) 6 (3) ( 2BA ) 1/6, 4BA2 (4) ( 2A 2B

Q4. The linear mass density of a thin rod AB of length L varies from A to B as λ(x) = λ0(1 + Lx ), where x is the distance from A . If M is the mass of the rod then its moment of inertia about an axis passing through A and perpendicular to the rod : (1) 12 5 ML2 (2) 187 ML2 (3) 2 5 ML2 (4) 37 ML2

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

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 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. The radius of gyration of a uniform rod of length l, about an axis passing through a point 4l away from the centre of the rod, and perpendicular to it, is: (1) 14 l (2) 18 l (3) (4) l l √748 √38

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. 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. 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. 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