PHYSICS QUESTIONSIIT-ian’s PACE Education Ltd: DELHI/BHIWADI/MUMBAI/AKOLA/LUCKNOW/KOLKATA/NASHIK/GOA Kalu Sarai : 011-41022929, 41657741 | Mayur Vihar : 011-22751870, 43095612 1 ROTATIONAL DYNAMICS Question No. 1 to 4 (4 questions) The figure shows an isosceles triangular plate of mass M and base L. The angle at the apex is 90°. The apex lies at the origin and the base is parallel to X–axis 1. The moment of inertia of the plate about the z-axis is (A) 12 ML 2 (B) 24 ML 2 (C) 6 ML 2 (D) none of these 2. The moment of inertia of the plate about the x-axis is (A) 8 ML 2 (B) 32 ML 2 (C) 24 ML 2 (D) 6 ML 2 3. The moment of inertia of the plate about its base parallel to the x-axis is (A) 18 ML 2 (B) 36 ML 2 (C) 24 ML 2 (D) none of these 4. The moment of inertia of the plate about the y-axis is (A) 6 ML 2 (B) 8 ML 2 (C) 24 ML 2 (D) none of these 5. Find minimum height of obstacle so that the sphere can stay in equilibrium. (A) u + cos 1 R (B) u +sin 1 R (C) R (1– sinu) (D) R (1 – cosu) 6. A ring of mass m and radius R has three particles attached to the ring as shown in the figure. The centre of the ring has a speed v 0 . The kinetic energy of the system is: (Slipping is absent) (A) 6 2 0 mv (B) 12 2 0 mv (C) 4 2 0 mv (D) 8 2 0 mv 7. A yo-yo is resting on a rough horizontal table. Forces F 1 , F 2 and F 3 are applied separately as shown. The correct statement is (A) when F 3 is applied the centre of mass will move to the right. (B) when F 2 is applied the centre of mass will move to the left. (C) when F 1 is applied the centre of mass will move to the right. (D) when F 2 is applied the centre of mass will move to the right. 8. In the figure shown, the plank is being pulled to the right with a constant speed v. If the cylinder does not slip then: (A) the speed of the centre of mass of the cylinder is 2v. (B) the speed of the centre of mass of the cylinder is zero. (C) the angular velocity of the cylinder is v/R. (D) the angular velocity of the cylinder is zero. PHYSICS QUESTIONS IIT-ian’s PACE Education Ltd: DELHI/BHIWADI/MUMBAI/AKOLA/LUCKNOW/KOLKATA/NASHIK/GOA Kalu Sarai : 011-41022929, 41657741 | Mayur Vihar : 011-22751870, 43095612 2 9. Consider a sphere of mass ‘m’ radius ‘R’ doing pure rolling motion on a rough surface having velocity 0 v as shown in the Fig. It makes an elastic impact with the smooth wall and moves back and starts pure rolling after some time again. (A) Change in angular momentum about ‘O’ in the entire motion equals 2mv 0 R in magnitude. (B) Moment of impulse provided by the wall during impact about O equals 2mv 0 R in magnitude. (C) Final velocity of ball will be 0 v 7 3 (D) Final velocity of ball will be – 0 v 7 3 SUBJECTIVE 10. The uniform rod AB of mass m is released from rest when o 60 β = . Assuming that the friction force between end A and the surface is large enough to prevent sliding, determine (for time just after release) (a) The angular acceleration of the rod (b) The normal reaction and the friction force at A. (c) The minimum value of µ compatible with the described motion. 11. By pulling on the cord of a yo-yo just fast enough, a person manages to make the yo-yo spin counterclockwise, while remaining at a constant height above the floor. Denoting the weight of the yo-yo by W, the radius of the inner drum on which the cord is wound by r, and the radius of gyration of the yo-yo by k. determine: (a) The tension in the cord (b) The angular acceleration of the yo-yo 12. A rigid body is made of three identical thin rods each of length L fastened together in the form of letter H. The body is free to rotate about a horizontal axis that passes through one of the legs of the H. The body is allowed to fall from rest from a position in which the plane of H is horizontal. What is the angular speed of the body, when the plane of H is vertical. 13. A rod of length R and mass M is free to rotate about a horizontal axis passing through hinge P as in figure. First it is taken aside such that it becomes horizontal and then released. At the lowest point the rod hits the block B of mass m and stops. Find the ratio of masses such that the block B completes the circular track of radius R. Neglect any friction. 14. A uniform slender bar AB of mass m is suspended from two springs as shown. If spring 2 breaks, determine at that instant; (a) The angular acceleration of the bar. (b) The acceleration of point A. (c) The acceleration of point B PHYSICS QUESTIONS IIT-ian’s PACE Education Ltd: DELHI/BHIWADI/MUMBAI/AKOLA/LUCKNOW/KOLKATA/NASHIK/GOA Kalu Sarai : 011-41022929, 41657741 | Mayur Vihar : 011-22751870, 43095612 3 15. A double pulley is attached to a slider block by a pin at B. The 2cm radius inner pulley is rigidly attached to the 4cm radius outer pulley. Knowing that each of the two cords is pulled at a constant speed of 12 cm/sec. as shown, determine (a) The instantaneous centre of rotation of double pulley (b) The velocity of the slider block (c) The length of cord w r apped or unwrapped on each pulley per second 16. The end B of rod AB which makes angle θ with the floor is being pulled with a velocity v 0 as shown. Taking the length of the rod as l , calculate the following at the instant when θ = 37°. (a) The velocity of end A (b) Theangular velocityof rod (c) Velocity of CM of the rod 17. A rough plank of mass m is placed on a rough hollow cylinder of mass M. A constant force F making an angle θ with the horizontal (plank is also horizontal always) is applied to the plank. Assuming no slipping anywhere. Find acceleration of the cylinder and friction force at A and B. 18. Two steel ball of equal diameter are connected by a rigid bar of negligible weight as shown & are dropped in the horizontal position from height h above the heavy steel and brass base plates. If the coefficient of restitution between the ball & steel base is 0.6 & that between the other ball & the brass base is 0.4. Find the angular velocity of the bar immediately after rebound. Assume the two impacts are simultaneous. 19. A 500 g block P rests on a frictionless horizontal table at a distance of 400 mm from a fixed pin O. The block is attached to pin O by an elastic cord of constant k = 100N/m and of undeformed length 900 mm. If the block is set in motion perpendicularly, determine: (a) The speed in the beginning for which the distance from O to the block P will reach the maximum value of 1.2 m, (b) The speed when OP = 1.2 m (c) The radius of curvature of the path of the block when OP =1.2 m. 20. A bar of length L and mass m has a frictionless pivot through its mid point. There is an additional point mass 2m on the right end of the bar and an additional point mass m on the left end of the bar. The bar is held in horizontal position by a vertical cord attached at L/4 from the left end as shown in the Figure. The additional masses m & 2m remain fixed on the rod. The rod is initially horizontal. (a) Find the tension in the cord (b) Find the force that the pivot exerts on the bar. (c) If the cord is cut. what is the angular acceleration of the bar immediately after the cord is cut (d) When the bar has rotated through 90 o and is vertical, what is the linear velocity of the mass 2m. PHYSICS QUESTIONS IIT-ian’s PACE Education Ltd: DELHI/BHIWADI/MUMBAI/AKOLA/LUCKNOW/KOLKATA/NASHIK/GOA Kalu Sarai : 011-41022929, 41657741 | Mayur Vihar : 011-22751870, 43095612 4 21. A uniform rod of length ‘ l ’ is kept as shown in the figure. H is a horizontal smooth surface and W is a vertical smooth wall. The rod is released from this position. Find the angular acceleration of the rod just after the release. 22. A uniform thin rod with a mass M = 0.60 kg and a length of 0.30 m stands on the edge of a frictionless table, as shown in the figure. The rod is struck a horizontal impulsive blow. J = 6 N.s at a point 0.20 m above the table top, driving the rod directly off the table. Determine the orientation of the rod and the position of its C.M. 1 sec after the blow is struck. 23. Sphere A of mass m and radius r rolls without slipping with a velocity 1 v on a horizontal plane. It hits squarely an identical sphere B which is at rest. Denoting by x µ the co-efficient of kinetic friction between the sphere and the plane, neglecting friction between the spheres and assuming perfectly elastic impact (e = 1). determine; (a) The linear & angular velocity of each sphere immediately after the impact. (b) The velocity of each sphere after it has started rolling uniformly. (c) Discuss the special case when x 0 µ = . 24. A stationary free rod AB of mass m on a smooth horizontal plane ts struck at end A by a particle of the same mass moving on the same plane with velocity V 0 perpendicular to the rod. The coefficient of restitution is 0.5. Find the x and y coordinate of the end B as a function of time taking the origin to be fixed at the initial position of the centre of the rod. Take the initial position of the centre of rod as origin and positive y-axis along the rod and towards A and the x-axis along the initial velocity of the particle. 25. A uniform plate of mass ’m’ is suspended in each of the ways shown. For each case determine immediately after the connection at B has been released ; (a) The angular acceleration of the plate. (b) The acceleration of its center of mass. 26. Find the moment of inertia about x-axis of uniform thin plate of density ρ kg/m 2 as shown in the Figure. PHYSICS QUESTIONS IIT-ian’s PACE Education Ltd: DELHI/BHIWADI/MUMBAI/AKOLA/LUCKNOW/KOLKATA/NASHIK/GOA Kalu Sarai : 011-41022929, 41657741 | Mayur Vihar : 011-22751870, 43095612 5 27. A uniform disc rolls without sliding on a horizontal surface. Find the ratio of total kinetic energy of upper half part to the total kinetic energy of the disc. 28. A semicircular disc of radius ‘r’ is released from rest from the position shown. If no slipping occurs between the disc and the horizontal surface, determine the expression for the angular velocity e reached by the disc when its kinetic energy is maximum. 29. A ring rolls on a horizontal surface without sliding. The velocity of the centre is v. It encounters a step of height 0.3 R where R is the radius of the ring.Calculate the angular velocity of the ring just after the impact. Assume that the ring does not return back.(and there is sufficient friction to avoid slipping). Find tne minimum value of ’v’ so that the ring ascends the step. 30. A spherical ball of radius r and mass m collides with a fixed surface. Before impact, the centre of the ball has a velocity v 0 directed normal to the fixed surface and an angular velocity 0 e as shown. Assuming the normal velocity is reversed with same magnitude, calculate the net velocity of centre of mass and the angular velocity of the ball after impact The co-efficient of friction between the surfaces is µ. 31. A solid body starts rotating about a stationary axis with an angular acceleration | = at, where a = 2.0 × 10 -2 rad/s 3 . How soon after the beginning of rotation will the total acceleration vector of an arbitrary point of the body form an angle o = 60° with its velocity vector.. 32. A solid body rotates with deceleration about a stationary axis with an angular deceleration | · e , where e is its angular velocity. Find the mean angular velocity of the body averaged over the whole time of rotation if at the initial moment of time its angular velocity was equal to 0 e . 33. A solid body rotates about a stationary axis so that its angular velocity depends on the rotation angle ¢ as 0 e= e ÷ ¢ a where 0 e and a are positive constants. At the moment t=0 the angle ¢ = 0. Find the time dependence of (a) the rotation angle; (b) the angular velocity 34. A rotating disc (figure) moves in the positive direction of the x-axis. Find the equation y(x)describing the position of the instantaneous axis of rotation, if at the initial moment the axis C of the disc was located at the point O after which it moved. (a) with a constant velocity v. while the disc started rotating counter clockwise with a constant angular acceleration | (the initial angular velocity is equal to zero); (b) with a constant acceleration w (and the zero initial velocity), while the disc rotates counterclockwise with a constant angular velocity e. 35. A cylinder rolls without slipping over a horizontal plane. The radius of the cylinder is equal to r. Find the curvature radii of trajectories traced out by the points A and B Fig. PHYSICS QUESTIONS IIT-ian’s PACE Education Ltd: DELHI/BHIWADI/MUMBAI/AKOLA/LUCKNOW/KOLKATA/NASHIK/GOA Kalu Sarai : 011-41022929, 41657741 | Mayur Vihar : 011-22751870, 43095612 6 36. A disc A of mass m sliding over a smooth horizontal surface with velocity v experiences a perfectly elastic collision with a smooth stationary wall at a point O. The angie between the motion direction of the disc and the normal of the wall is equal to o . Find: (a) the points relative to which the angular momentum M of the disc remains constant in this process. (b) the magnitude of the increment of the vector of the disc’s angular momentum relative to the point O’ which is located in the plane of the disc’s motion at the distance l from the point O. 37. A small ball of mass m suspended from the ceiling at a point O by a thread of length l moves along a horizontal circle with a constant angular velocity e. Relative to which points does the angular momentum M of the ball remain constant? Find the magnitude of the increment of the vector of the ball’s angular momentum relative to the point O picked up during half a revolution. 38. A thin uniform rod AB of mass m = 1.0 kg moves transitionally with acceleration w = 2.0 m/s 2 due to two antiparallel forces F 1 and F 2 (Fig.). The distance between the points at which these forces are applied is equal to a = 20 cm. Besides, it is known that F 2 = 5.0. N. Find the length of the rod. 39. A force F = Ai + Bj is applied to a point whose radius vector relative to the origin of coordinates O is equal to r = ai + bj, where a. b &, A, B are constants, and i, j are the unit vectors of the x and y axes. Find the moment N and the arm l of the force relative to the point O. 40. Three forces are applied to a square plate as shown in Figure. Find the modulus, direction, and the point of application of the resultant force, if this point is taken on the side BC. 41. (i) A uniform ring of radius R is spinned to the angular velocity e and then carefully placed on a horizontal surface such that the plane of the ring is horizontal. How long will the ring be rotating on the surface it the friction coefficient is equal to k? The pressure exerted by the ring on the surface can be regarded as uniform. (ii) A uniform disc of radius R is spinned to the angular velocity e and then carefully placed on a horizontal surface such that the plane of the disc is horizontal. How long will the disc be rotating on the surface it the friction coefficient is equal to k? The pressure exerted by the disc on the surface can be regarded as uniform. PHYSICS QUESTIONS IIT-ian’s PACE Education Ltd: DELHI/BHIWADI/MUMBAI/AKOLA/LUCKNOW/KOLKATA/NASHIK/GOA Kalu Sarai : 011-41022929, 41657741 | Mayur Vihar : 011-22751870, 43095612 7 42. A spool with a thread would on it is placed on an inclined smooth plane set at an angle o = 30°to the horizontal. The free end of the thread is attached to the wall as shown in Fig. The mass of the spool is m = 200 g, its moment of inertia relative to its own axis l = 0.45 g.m 2 , the radius of the wound thread layer r = 3.0 cm. Find the acceleration of the spool axis. 43. A uniform solid cylinder of mass m rests on two horizontal planks. A thread is wound on the cylinder. The hanging end of the thread is pulled vertically down with a constant force F. Find the maximum magnitude of the force F which still does not bring about any sliding of the cylinder, if the coefficient of friction between the cylinder and the planks is equal to k. What is the acceleration w max of the axis of the cylinder rolling down the inclined plane. 44. In the arrangement shown in Fig a weight A possesses mass m.a. a pulley B possesses mass M. Also known are the moment of inertia I of the pulley relative to its axis and the radii of the pulley R and 2R. The mass of the threads is negligible. Find the acceleration of the weight A after the system is set free. 45. A uniform solid cylinder A of mass m 1 can freely rotate about horizontal axis fixed to a mount B of mass m 2 . A constant horizontal force F is applied to the end K of a light thread tightly wound on the cylinder. The friction between the mount and the supporting horizontal plane is assumed to be absent. Find: (a) the acceleration of the point K; (b) the kinetic energy of this system t seconds after the beginning of motion. 46. Determine the kinetic energy of a tractor crawler belt of mass m if the tractor moves with velocity v. PHYSICS QUESTIONS IIT-ian’s PACE Education Ltd: DELHI/BHIWADI/MUMBAI/AKOLA/LUCKNOW/KOLKATA/NASHIK/GOA Kalu Sarai : 011-41022929, 41657741 | Mayur Vihar : 011-22751870, 43095612 8 ANSWERS 1. C 2. A 3. C 4. C 5. D 6. A 7. C 8. B,C 9. A,B,D 10. (a) 3g (clockwise) 4L (b) 13mg 3 3 N , f mg 16 16 = ↑ = → (c) 3 3 13 11. T = w (b) 2 rg / k (anticlockwise) o = 12. 9g w 4 = 5, w 5 rad / s = 13. M 15 m = 14. (a) 3g (cw) L (b) o 3 ˆ ˆ i j g 1.323g 49.1 2 | | + = Z | | \ . (c) o 3 ˆ ˆ i 2j g 2.18g 66.6 2 | | ÷ = Z÷ | | \ . 15. (a) 1 cm to right B (b) 4cm/ sec + (c) out pulley unwraped 16 cm/sec. inner pulley unwrapped 8 cm/sec 16. (a) 0 4v 3 (b) 0 5v 3 (c) 0 x v v 2 = (d) 0 y 2v v 3 = 17. A B a (Fcos ) / (M/ 2m), f 0, f Ma = u = = 18. 0.28 rad/sec 19. (a) 4.5 m/s (b) 1.5 m/s (c) 3.75 cm 20. (a) T = 2mg (b) N = 6 mg (c) 3g 5L (d) 6g L 3gL 5L 2 10 × = 21. 3gcos 2 u o = 22. (10, – 4.75 m) w.r.t. initial position of lower end of the rod, 200/3 rad with upward vertical. 23. (a) A 1 e = e , clockwise B B 1 0v v e = = ÷, A v 0 = (b) 1 A 2v v 7 = ÷ , 1 B 5v v 7 = ÷ (c) same as part (a) 24. 0 0 centre B 2 9V 3V , V , X V t (Lsin ) / 2 5L 10 e= = = ÷ u , B Y (Lcos t) / 2 = ÷ e 25. (i) (a) 1.2g (cw) c (b) ˆ ˆ 0.3(i 2j)g ÷ + (ii) (a) 24g (cw) 17c (b) 12g 17 + (iii) (a) 2.4g (cw) c (b) 0.5g + PHYSICS QUESTIONS IIT-ian’s PACE Education Ltd: DELHI/BHIWADI/MUMBAI/AKOLA/LUCKNOW/KOLKATA/NASHIK/GOA Kalu Sarai : 011-41022929, 41657741 | Mayur Vihar : 011-22751870, 43095612 9 26. 7 2 a 21 µ 27. 9 16 18 t+ 28. 9 4 rad / sec (9 16)r e= t ÷ 29. min 1.7v 2 V 0.3gR 2R 1.7 e= = 30. 2 2 2 0 0 v v 4 v = + µ ; 1 tan 2 ÷ u = µ with normal, 0 0 5 v r µ e= e ÷ 31. 3 t (4 / a) tan 7s = o = 32. 0 3 e < e>= 33. (a) at 0 (1 e ) / a ÷ | ÷ e (b) at 0 e ÷ e= e 34. (a) 2 v y (Hyperbola) x = | (b) y 2wx / (Parabola) = e 35. A B R 4r, R 2 2r = = 36. (a) Relative to all points of the straight line drawn at right angles to the wall through the point O; (b) | m| 2m cos A = o 37. Relative to the centre of the circle 2 2 g mg | m| 2 1 | | A = ÷ | e e \ . 38. 2 2aF / mw 1.0m = = 39. N=(aB–bA)k, where k is the unit vector of the z-axis 2 2 | aB bA| / A B = ÷ + 40. F res = 2F. This force is parallel to th diagonal AC and is app;ied at the midpoint of the side BC. 41. (i) R kg e (ii) 0 2 R 4kg e 42. 2 2 w gsin / (1 1/ mr ) 1.6m/ s = o + = 43. max F 3kmg / (2 3k) = ÷ ; max W 2kg / (2 3k) = ÷ 44. 2 w 3g(M 3m) / (M 9m I / R ) = + + + 45. (a) 1 2 1 1 2 F(3m 2m ) w m (m m ) + = + ; (b) 2 2 1 2 1 1 2 F t (3m 2m ) T 2m (m m ) + = + 46. 2 T mv =