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HKAL Exercise : Part 1 Mechanics Chapter 3 Circular Motion 3.1 Definition 1. The turntable of a gramophone rotates at a steady angular speed ω. A record is dropped from rest on the turntable. Initially, the record slips but it undergoes uniform angular acceleration and eventually moves with the same angular speed as the turntable. The turntable turns through an angle θ while the record is slipping on its surface. Find the angular acceleration of the record and the angle which it turns through before it attains the steady speed of the turntable. Angular acceleration Angle 2 A. ω /θ 2θ 2 B. ω /θ θ/2 C. D. ω2/(2θ) ω2/(2θ) 2θ θ/2 B Centripetal Force 2. 4. H A A car of mass m travels into a region where the track is an arc of a vertical circle of radius r. At the bottom of this arc, the car travels at speed v. At this position the vertical force exerted upwards by the track on the car is A. mv2/r. B. mg. C. mv2/r – mg. D. mg + mv2/r. 3. A small object of mass 0.3 kg is released from rest at the rim of a heavy, smooth semi-spherical bowl of radius 25 cm. Find the force acting on the object by the bowl when it passes the bottom of the bowl. A. 6 N B. 9 N C. 12 N D. 15 N HKAL Extra Exercise_Chapter 3 Circular Motion 80 m B 40 m Terminal platform The diagram shows part of the route of a roller-coaster in an amusement park. The cart descends from H, completes a circular loop A and moves to B. If the cart of passengers is to complete the central circular track safely, what is the minimum velocity of the cart at the bottom of the circular track A ? (Assume that there is no friction between the cart and the track.) A. 24.5 m/s B. 31.6 m/s C. 40 m/s D. 80 m/s 1/6 5. Which of the following pairs of forces is/are example(s) of action and reaction ? 6. (1) The forces of repulsion experienced by two parallel wires carrying currents in the opposite direction. (2) The centripetal force keeping a satellite in orbit round the earth and the weight of the satellite. (3) The air resistance acting on an object falling through the air with terminal velocity and the weight of the object. A. (1) only kept rotating at a constant angular speed about a vertical axis through its centre. The distance of B from the axis of rotation is twice that of A from the axis. Which of the following statements is/are correct ? (1) The frictional force acting on B is double that acting on A. (2) The kinetic energy of B is double that of A. (3) If the angular speed of the turntable B. C. D. 7. (3) only (1) and (2) only (1), (2) and (3) Two small identical coins A and B are placed on a horizontal turntable which is gradually increases, B will slip before A. A. (2) only B. (3) only C. (1) and (2) only D. (1) and (3) only A student whirls a small bucket of water in a vertical circle of radius 0.7 m. For no spilling, what is the minimum speed of the bucket at the highest point of its path ? A. 2.65 m s-1 B. 3.74 m s-1 C. D. 4.41 m s-1 5.29 m s-1 Whirling a ball with a string 8. glass tube L rubber bung W A student performing a centripetal force experiment whirls a rubber bung attached to one end of a string which passes through a glass tube with smooth openings, and has a weight W hanging at its other end. The weight of the rubber bung is much smaller than W. The rubber bung is set into a horizontal uniform circular motion with angular speed ω while the length of the string beyond the upper opening of the glass tube is L and this portion of the string makes an angle θ with the vertical as shown. Which of the following statements is/are correct ? HKAL Extra Exercise_Chapter 3 Circular Motion 2/6 (1) If L is kept constant, θ will increase with ω. (2) When W increases, θ will increase. (3) If θ is kept constant, L will increase with ω. A. B. C. D. (1) only (3) only (2) and (3) only (1), (2) and (3) 3.2 Uniform Motion in a Horizontal Circle 9. A 25O A. B. (1) only (3) only C. D. (1) and (3) only (2) and (3) only 11. Q B A smooth conical container rotates about the axis AB as shown. A marble remains at rest relative to the container at a radial distance r from the axis. If the angular velocity of the marble is ω, then ω2 is equal to A. B. C. D. g tan 25∘/r. g / (r tan 25∘). g cos 25∘/r. g / (r cos 25∘). 10. A particle P is moving in a horizontal circle in a clockwise direction as shown (top view). The following diagrams show the direction of the acceleration a of the particle. Which of them is/are possible ? (1) (2) (3) a P a P a v P A small ball-bearing is projected with velocity v from the lowest position P of a vertical circular track which is not smooth. The ball-bearing starts to leave the track at Q. Which of the following diagrams best represents all the forces acting on the ball-bearing at Q ? A. B. Q Q P P C. D. Q Q P P P HKAL Extra Exercise_Chapter 3 Circular Motion 3/6 Conical pendulum 12. A spring of unstretched length 14.0 cm, Level road without banking 13. has one end fixed to the ceiling. A mass is suspended at the other end, and the extension so produced is 3.0 cm. When the mass is set to rotate in a horizontal circular path the length of the spring is 19.0 cm. The angle between the spring and the vertical is A. 9∘. B. 37∘. C. 53∘. D. P The diagram represents the rear view of a motor car moving, away from an observer on a level road at a constant speed around a bend of which the centre of curvature is at P. Which of the arrows best represents the direction of the resultant of the forces exerted by the road on the car? 81∘. Banked Road 14. A. B. C. D. 15. N centre of mass 5m r A mg 30 o O The figure shows a car moving round a corner with a radius of 5 m on a banked road of inclination 30∘. At what angular speed would there be no friction acting on the car along OA ? A. 0.9 rad s-1 B. 1.1 rad s-1 C. 1.3 rad s-1 D. 1.5 rad s-1 HKAL Extra Exercise_Chapter 3 Circular Motion A car of mass m is moving with speed v on a banked road along a circular path of horizontal radius r. The angle of inclination of the road is θ . If the centripetal force arises entirely from a component of the normal reaction N from the road, which of the following relations is correct ? A. N cos mg B. v2 gr sin C. v2 gr tan D. N mv 2 sin r 4/6 Aircraft 16. An aircraft flies along a horizontal circle of radius 20 km with a constant angular speed of 0.012 rad s-1. Calculate the angle between its wings and the horizontal. A. 15.0∘ B. 15.5∘ C. 16.0∘ D. 16.5∘ Bicycle on a level road 17. A motorcyclist going round a corner on a level road leans over at an angle to the horizontal. The reason for this is A. to lower his centre of mass. B. C. D. to provide the centripetal force. to reduce the radius of curvature of his path. to allow his weight to exert a torque about the contact point on the ground. 3.4 Non-Uniform Motion in a Vertical Plane Looping the loop 18. 19. A P h A particle P is placed at a height h on a smooth loop-the-loop track, as shown. The radius of the loop is r. P is now released from rest. If P is to complete the loop, the minimum value of h should be A. 1.5 r. B. 2 r. C. 2.5 r. D. 3 r. HKAL Extra Exercise_Chapter 3 Circular Motion h A ball bearing is released from rest at a height h on a smooth track and completes the circular loop of the track. If R is the reaction acting on the ball bearing at the highest point A of the loop, which of the following graphs correctly shows the variation of R with h ? A. B. R R 0 h 0 C. D. R R 0 h 0 h h 5/6 Whirling freely (with a rod) 20. A stone of weight w tied to a piece of string is swung in a vertical circle. At the lowest point of its path, the tension in the string is T and the centripetal force is F. Which of the following statements is true ? A. F = w + T. B. F = w – T. C. The net force acting upwards on the stone is F + T + w. D. The net force acting upwards on the stone is F – T – w. 21. O 0.7 m P A small object P of mass 0.4 kg is attached to one end of a light, rigid rod of length 0.7 m, which is free to rotate about the other end O as shown. The object is swung to rotate in a vertical circle so that it attains a speed of 4 m s-1 at its lowest position. What is the force exerted on one end of the rod at this instant ? A. a compressive force of 5.1 N B. a compressive force of 13.1 N C. D. a tensional force of 5.1 N a tensional force of 13.1 N KEY : 1 – 10 : BDBBA, DBBBC, 11 – 20 : DCCBA, CDCAB, 21 : D. HKAL Extra Exercise_Chapter 3 Circular Motion 6/6