Practice Questions

Q2. Consider a force F = −xˆi + yˆj . The work done by this force in moving a particle from point A(1,0) to B(0,1) along the line segment is : (all quantities are in SI units) (1) 2 (2) 21 (3) 1 (4) 32

Q2. A spring mass system (mass m, spring constant k and natural length l ) rests in equilibrium on a horizontal disc. The free end of the spring is fixed at the centre of the disc. If the disc together with spring mass system rotates about it's axis with an angular velocity ω, (k >> mω2) the relative change in the length of the spring is best given by the option: (1) mω2 (2) 2mω2 k √23 ( k ) (3) mω2 (4) mω2 k 3k

Q2. When a car is at rest, its driver sees rain drops falling on it vertically. When driving the car with speed v, he sees that rain drops coming at an angle 60∘ from the horizontal. On further increasing the speed of the car to (1 + β)v, this angle changes to 45∘ . The value of β is close to : (1) 0. 50 (2) 0. 41 (3) 0. 37 (4) 0. 73 B of mass m2

Q2. A particle of mass m is fixed to one end of a light spring having force constant k and unstretched length l. The other end is fixed. The system is given an angular speed ω about the fixed end of the spring such that it rotates in a circle in gravity free space. Then the stretch in the spring is: (1) mlω2 (2) mlω2 k−ωm k−mω2 (3) mlω2 (4) mlω2 k+mω2 k+mω

Q2. An insect is at the bottom of a hemispherical ditch of radius 1m . It crawls up the ditch but starts slipping after it is at height h from the bottom. If the coefficient of friction between the ground and the insect is 0. 75, then his : (g = 10 m s−2) (1) 0. 20 m (2) 0. 45 m (3) 0. 60 m (4) 0. 80 m

Q2. A balloon is moving up in air vertically above a point A on the ground. When it is a height h1, a girl standing at a distance d (point B) from A (see figure) sees it at an angle 45° with respect to the vertical. When the balloon climbs up a further height h2 , it is seen at an angle 60° with respect to the vertical if the girl moves further by a distance 2. 464 d (point C). Then the height h2 is (given tan 30°= 0. 5774): (1) 1. 464 d (2) 0. 732 d (3) 0 .464 d (4) d

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. 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

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. 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. 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. 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. 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 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. 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. 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. 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. 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. 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 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. 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)

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. Consider a uniform rod of mass M = 4m and length l pivoted about its centre. A mass m moving with velocity v making angle θ = π4 to the rod’s long axis collides with one end of the rod and sticks to it. The angular speed of the rod-mass system just after the collision is: (1) 3 v (2) 3 v 7√2 l 7 l (3) 3√2 v (4) 4 v 7 l 7 l

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. 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