PHYSICS-17-09- 11th (PQRS)

XI (PQRS) PHYSICS REVIEW TEST-4 Q.1 Three carts move on a frictionless track with inertias and velocities as shown. The carts collide and stick together after successive collisions. (a) Find loss of mechanical energy when B & C stick together. (b) Find magnitude of impulse experienced by A when it sticks to combined mass (B & C). [3+3] [Sol.(a) After first collision velocity of combined mass is u1 (m2 + m3)u1 = 2 × (–2) + 1 × 1  u1 = –3/3 = –1 [Ans. (a) 3 J, (b) ] KEi = 1 × 2 × (2)2 + 2 1 (1) × 12 = 4.5 J 2 KEf = 1 × 3 × (1)2 = 1.5 J 2 Loss = 3 J (b) Impulse experienced by A = P 1 – 2 × 5 12 – 2 = – 5 12 Magnitude = 5 N·s] Q.2 Two cats are running along two perpendicular streets of a town. The mass of one (Gullu) is 2 kg and its speed is 4 m/s. The mass of the other (Pullu) is 6 kg and its speed is 2 m/s. (a) What is the speed of Gullu relative to Pullu? (b) Find speed of their centre of mass (c) Find KE of system in reference frame of their combined centre of mass [Sol. → vG = → vP = → 4ˆj 2ˆi ˆ ˆ ˆ ˆ [Ans. (a) , (b) [2+2+3] 13 , (c) 15 ] 2 (a) vG / P → = 4 j 2i =  2i  4 j  vG / P = → 2 4ˆj  6  2ˆi 3ˆi ˆ (b) vCM = 8 → =  j 2 13 vCM = → → = 2 →  3 ˆi  3ˆj (c) vG / CM = vG – vCM = 2 → vP / CM → = vP – → vCM = 1 ˆi  ˆj 2 1  9  1  1  45 15 KEsys/CM = (2)  9   6 1 = 2 4 2 4 4  4 = 15 ]     Q.3 An object of mass 6.0 kg is free to move along the x-axis on a frictionless horizonal track. It starts form rest at x = 0, t = 0. It moves 3.00 m under the action of a horizontal force F = (3 + 4x)N where x is in m (a) What velocity does it acquire? (b) What is its acceleration at that point? (c) What power is being delivered to it at that point? [2+2+3] [Ans. (a) 3 m/s, (b) 2.5 m/s2, (c) 45 W] [Sol.(a) W. D. on object = KE 3  (3  4x) dx = 0 1 mv2 – 0 2 27 = 1 × 6 v2  v = 3 m/s ] 2 (b) F = 3 + (4 × 3) = 15 N a = 15/6 = 2.5 m/s2 → (c) P = → = 15 × 3 = 45 W ] F·v Q.4 Find the distance of the centre of mass of the uniform plate ABCD from edge AD. [7] [Sol. Considering three triangular plates ABP, PCB & PCD [Ans. 2a 3 ] 9 1  a sin 60  a CQ =    = sin 60 3  9 CP = CQ + QP = a sin 60 + 9 a sin 60 = 3 4a  9 3 = 2a 3 ] 2 9 Q.5 An inclined plane of angle  is fixed onto a horizontal turntable, with its line of greatest slope in same plane as a diameter of turntable. Asmall block is placed on the inclined plane a distance r from the axis of rotation of the turntable and the coefficient of friction between the block and the inclined plane is . The turntable along with incline plane spins about its axis with constant minimumangular velocity . (a) Draw a free body diagram for the block from reference of ground, showing the forces that act on it. (b) Find an expression for the minimum angular velocity, c, to prevent the block from sliding down the plane, in terms of g, r,  and the angle of the plane . (c) Now a block of same mass but having coefficient of friction (with inclined plane) 2  is kept instead of the original block. Find ratio of friction force acting between block and incline now to the friction force acting in part (b). [2+3+2] [Sol.(a) (b) N sin + N cos – mg = 0 N sin – N cos = m2r (sin   cos) [Ans. (a) , (b) , (c) 1] mg (sin   cos) = m2r   = (c) Since requirement offriction is same as in part (b). Force of frictionwill remain unchanged. Hence ratio is 1] Q.6 A projectile of 2 kg is launched from ground at an angle of 30° from horizontal. At the highest point of its trajectory the radius of curvature is 16 km. (a) Find its speed at highest point. (b) Calculate its range on level ground (c) Calculate impulse due to gravity during total time of flight. [2+2+2] (u cos 30)2 [Sol.(a) g [Ans. (a) 400 m/s, (b) = 16 × 103  u cos30° = 4 × 102 = 400 m/s m, (c) N] (2u sin 30)  2  800  1  800  3 (b) R =   g  (u cos 30)  = 10 2 2 = m 1 (c) mg dt = mu sin 30° – (–mu sin 30°) = 2 × 2 × × 2 = N ] Q.7 A uniform sphere ofmassmheld at a height 2R between awedge ofsame mass mand a rigid wall, is released. Assume that all the surfaces are frictionless. (a) Considering wedge and sphere as a system can momentum of the system be conserved in horizontal direction. Give reason for your answer. (b) Find displacement of wedge when sphere touches the ground. (c) Find speed ofboth the bodies just before the sphere hits the ground. [2+2+3] [Ans.(a) No, (b) Rcot , (c) 2gR sin ] [Sol.(a) No, momentum cannot be conserved in horizontal direction as normal contact from wall will be acting. (b) y/x = tan   x = y cot   x = R cot (c) W.D. by gravity = mgR mgR = 1 mv2 + 1 mv2 2 1 2 2 v2/v1 = tan  2gR = v2 + v2 tan2 1 1 v1 = cos v2 = 2gR sin  ] Q.8 A block of mass m placed on a smooth horizontal surface is attached to a spring and is held at rest by a force P as shown. Suddenly the force P changes its direction opposite to the previous one. How many times is the maximum extension l2 of the spring longer compared to its initial compression l1? [Sol. P = kl1 [7] [Ans. 3]  1 kl 2  0  1 kl 2  0 P(l + l ) =  2  2  –  1    2  kl 2 + kl l = k l 2  l 2  1 2l 2 1 2 2 2 + 2l1l2 – 2 1 2 = 0 3 l 2 + 2l1l2 – 2 = 0 l2 = 3l1 ] Q.9 In the figure shown the spring is compressed by 'x0' and released. Two blocks 'A' and 'B' of masses 'm' and '2m' respectively are attached at the ends of the spring. Blocks are kept on a smooth horizontal surface and released. (a) Find displacement of block A by the time compression of the spring is reduced to x0/2. (b) Find the ratio of KE of blocks A & B by the time compression of the spring reduced to x0/2. (c) Find work done by force of spring on block B when spring reaches natural length. [Sol.(a) xA + (l0 – x0) + xB = l0 – x0/2 xA + xB = x0/2 SCM = 0, since Fext = 0 & uCM = 0 m(–xA) + 2mxB = 0 xA [3+2+2] kx2 [Ans. (a) x /3, (b) 2, (c) 0 ] 6 = 2 B xB = x0/6 xA = x0/3 (b) –mvA + 2mvB = 0 Conserving momentum Writing work energy eq. for the system kA (1/ 2)mv2 2 B = (1/ 2)  2mv2 = 1 (c) 1 kx2 = 1 m v2 + 1 m v2 2 0 2 A A 2 B B vA = 2vB 1 kx2 = 1 m (2v )2 + 1 m v2 2 0 2 A B 2 B B 1 kx2 = 1 m × 4 v2 + 1 × 2m × v2 2 0 2 B 2 B 1 2 kx2 × 2mv = 0 ] 2 6 Q.10 A homogeneous flexible chain rests on a wedge whose side edges make angles  and  with the horizontal (refer figure). The central part of the chain lies on the upper rib C of the wedge. With what acceleration should the wedge be pulled to the left along the horizontal plane in order to prevent the displacement of the chain with respect to the wedge? [Consider all surfaces to be smooth] [7] g[sin   sin ] left part N1 cos  T sin   mg / 2  0 [Ans. a = [cos  cos] ] [Sol. N sin   T cos  ma / 2  1 right part T cos  N2 sin   ma / 2 T sin   N cos  mg / 2  0 Solving we get, a = g[sin   sin ] [cos  cos] ] Q.11 The speed of an object undergoing uniform circular motion is 4 m/s. The magnitude of the change in the velocity during 0.5 s is also 4 m/s. (a) Find minimum possible angular velocity. (b) Find centripetal acceleration for the angular velocity found in part (a). (c) Find radius of the circle for the angular velocity of part (a). [2+3+2] 2 [Ans. (a) 3 8 rad s–1, (b) 3 m/s2, (c) 6 m]  → → [Sol. | v2  v1 | = | v | | v | = 2 v sin(t/2)   0.5  4 = 2 × 4 sin    2   0.5 2 2 (a)  = 3  = 6 rad s–1 2 8 (b) ac = v = 3 × 4 = 3 m/s2 v (c) r =  = 4 (2 / 3) 6 =  m ]