Name:. Set:. Don: Physics: Pre-U Revision Toytime 2015-16 Rotational and Circular Motion
1 19 (ii) Place ticks in the table below to identify the effect on waves of light as they refract from diamond into air at A. wave property of the light effect increase unchanged decrease speed wavelength frequency [3] (c) A very thin phosphor-bronze disc is used to saw through rough uncut diamonds. The disc rotates about a horizontal axis at 4000 revolutions each minute. (i) Calculate the angular speed w of the disc. w =... rad s 1 [2] (ii) The rim of the disc is initially impregnated with diamond dust, which is replenished as the diamond is cut. Without this dust, the disc would fail to cut through the diamond. What does this tell us about the relative hardness of diamond and phosphor-bronze?...[1] (d) Laws of rotational motion can be deduced by comparison with Newton s laws of linear motion. Complete the table below by stating the equivalent formulae, in words, for rotational motion. linear motion rotational motion work = force displacement momentum = mass velocity UCLES 2010 9792/03/M/J/10 [2] [Turn over
20 (e) Fig. 7.2a and Fig. 7.2b show a phosphor-bronze cutting disc of mass M and thickness t with radius R. The uniform density of the disc is r. R R Fig. 7.2a (front view) t Fig. 7.2b (side view) (i) integration to derive an expression for the moment of inertia I of the disc. You may draw on Fig. 7.2a to help illustrate your working. [4] (ii) The disc has mass 35.4 g and a moment of inertia of 1.13 10 4 kg m 2. Calculate the radius R of the disc. R =... m [2] UCLES 2010 9792/03/M/J/10
(iii) 21 Determine the rotational kinetic energy E of the disc. E =... J [3] UCLES 2010 9792/03/M/J/10 [Turn over
2 4 Section A Answer all questions in this section. You are advised to spend about 1 hour 30 minutes on this section. 1 (a) A body is travelling in a circular orbit of radius r with constant speed v as shown in Fig. 1.1. v r Fig. 1.1 a vector diagram to show that the acceleration a of the body is given by towards the centre of the circle. a = v2 r UCLES 2011 9792/03/M/J/11 [4]
(b) 5 The drum of a spin drier has a rate of rotation of 4.0 revolutions per second. An object in the drum has a mass of 0.20 kg and rotates in a vertical circle of radius 0.16 m. (i) Calculate the magnitude of the acceleration of the object. acceleration =... m s 2 [2] (ii) Calculate the magnitude of the resultant force on the object. resultant force =... N [1] (iii) each of the three positions shown in Fig. 1.2 draw arrows to represent the weight W of the object and the force D that the drum exerts on the object. Indicate how these two forces always add to produce the resultant force of constant magnitude calculated in (ii). Fig. 1.2 [6] [Total: 13] UCLES 2011 9792/03/M/J/11 [Turn over
3 22 10 Fig. 10.1 shows part of an array of wind turbines on farmland. Fig. 10.1 (a) Each turbine converts the kinetic energy of the wind into electrical energy. 1.16 10 7 kg of air, travelling at a speed of 20 m s 1, pass through the turbine each minute. Calculate the maximum power of the turbine in MW assuming all of the kinetic energy of the wind is transferred to the turbine. maximum power =... MW [2] (b) Blades of length l sweep out area A with each rotation. When air of density ρ and wind speed v passes the blades then the maximum power P transferred to the turbine is also given by the expression: P = 1 2 Aρv3 UCLES 2011 9792/03/M/J/11
23 Fig. 10.2 is a graph of lnp against lnv for different wind speeds. The gradient of the line is 3. 18.0 16.0 In (P / W) 14.0 12.0 10.0 8.0 0.0 1.0 2.0 In (v / m s 1 ) 3.0 Fig. 10.2 (i) State the expression for the intercept on the lnp axis in terms of A and ρ.... [1] (ii) State the value of this intercept.... [1] (iii) the value of the intercept stated in (ii) to determine the blade length l. The density of air ρ is 1.23 kg m 3. blade length =... m [2] UCLES 2011 9792/03/M/J/11 [Turn over
(c) (d) 24 As the wind passes each blade a torque is exerted on it causing it to rotate about its horizontal axis. The blade has a moment of inertia. State, in words, the equation that relates torque to moment of inertia....... [2] Fig. 10.3 shows a typical waterwheel used in a mill to generate rotational kinetic energy. Water exerts a torque on the wheel as it pours from above into the buckets built into the rim of the wheel. Fig. 10.3 Assume the wheel is a large ring of dense oak wood, of mass M. Fig. 10.4 is a diagram of the wheel. waterwheel O R Fig. 10.4 UCLES 2011 9792/03/M/J/11
(i) Show that the moment of inertia of the waterwheel can be taken to be 25 I = MR 2 as long as the waterwheel is treated as a thin ring of average radius R. [2] (ii) The waterwheel rotates at 4 revolutions per minute. Calculate its angular speed in rad s 1. angular speed =... rad s 1 [1] (iii) The maximum power of the waterwheel is 6.5 kw. The wheel comes to rest in 30 minutes once the water stops flowing. As it slows down its average output power is 3.25 kw. 1. Show that the loss in rotational kinetic energy of the wheel as it comes to rest is approximately 6 MJ. 2. State an assumption made in the calculation in 1. [2] UCLES 2011... [1] 9792/03/M/J/11 [Turn over
(iv) 26 Using information given in (ii) and (iii), determine the moment of inertia of the wheel. Give the unit with your answer. moment of inertia =... unit... [4] (v) The radius of the wheel is 5.5 m. Determine the mass M of the waterwheel. mass of waterwheel =... kg [2] [Total: 20] UCLES 2011 9792/03/M/J/11
4 28 (f) A rod is rolled down an inclined plane as shown in Fig. 9.5. 2.5 m end view of rod 25 Fig. 9.5 (not to scale) The plane of length 2.5 m is inclined at an angle of 25 to the horizontal. (i) The rod has a radius R and a moment of inertia I about its central axis. Briefly explain what is meant by the phrase moment of inertia about its central axis. You may add to the diagram below to help illustrate your answer. rod central axis of rotation... [2] UCLES 2012 9792/03/M/J/12
(ii) 29 The rod has a mass M of 0.20 kg and its moment of inertia I is 0.10 10 4 kg m 2. It starts from rest and rolls down the inclined plane without slipping. Determine the angular speed ω of the rod at the bottom of the inclined plane where its linear speed v is 3.72 m s 1. angular speed =... rad s 1 [4] [Total: 20] UCLES 2012 9792/03/M/J/12 [Turn over
5 17 Section B Answer any three questions in this section. You are advised to spend about 1 hour 30 minutes on this section. 8 (a) An object moving in a circular path of radius r experiences an acceleration a even when travelling at constant speed v. (i) Explain how it is possible for the object to accelerate and yet at the same time have constant speed.... [3] (ii) State an expression for this acceleration.... [1] (b) Fig. 8.1 shows the forces acting on a child who is riding backwards and forwards on a swing that follows a circular arc of radius r. r R circular arc Fig. 8.1 (not to scale) The child s weight is mg. R is the force of the seat on the child. As the instantaneous speed v of the child changes, R also varies. As the child swings through the lowest point on the circular arc, θ = 0, her instantaneous speed is 4.7 ms 1. The child weighs 200 N and the radius r is 2.8 m. Calculate the value of R at this instant. m g UCLES 2013 9792/03/M/J/13 R =... N [2] [Turn over
18 (c) Fig. 8.2 shows a roundabout. handle roundabout central hub soft ground Fig. 8.2 The roundabout consists of a solid disc of mass M supported on a central hub. (i) integration to derive an expression for the moment of inertia I of a thin uniform disc of radius R about its centre. You may annotate the diagram of the disc in Fig. 8.3 to define the terms you use. thin uniform disc R Fig. 8.3 UCLES 2013 9792/03/M/J/13 [4]
(ii) 19 The moment of inertia of the roundabout in Fig. 8.2 is 44.8 kg m 2. A torque of 10.1 N m is applied. Show that the time taken to accelerate the disc from rest to 1.40 rad s 1 is approximately 6 s. (iii) time =... s [2] Two teenagers of equal mass sit directly opposite each other on the roundabout. The moment of inertia of the roundabout and the teenagers is now 118 kg m 2. Calculate how much longer than the time determined in c(ii) it will now take to accelerate from rest to 1.40 rad s 1. Assume the same torque is applied as in c(ii). (iv) time increase =... s [2] The roundabout continues to rotate at 1.40 rad s 1. The teenagers then lean outwards. 1. Explain why the period of rotation of the roundabout increases.... [3] UCLES 2013 9792/03/M/J/13 [Turn over
20 2. The period of rotation increases by 0.66 s. Calculate the new moment of inertia. new moment of inertia =... kg m 2 [3] [Total: 20] UCLES 2013 9792/03/M/J/13
6 16 Section B Answer any three questions. You are advised to spend 1½ hours on this section. Examiner's 8 A bus uses a spinning flywheel which is brought up to full rotational speed by a motor when the bus stops at a station. The flywheel is a solid cylinder of mass 900 kg and diameter 2.2 m. Its maximum angular velocity is 2800 revolutions per minute. (a) Calculate the angular velocity in rad s -1. angular velocity = rad s -1 [2] (b) Calculate the maximum speed of particles on the rim of the spinning flywheel. maximum speed = [2] (c) Fig. 8.1 shows a cylinder of radius R and length L and of density ρ. The cylinder contains a small cylindrical shell of radius r and thickness δr. Show that the mass of the small cylindrical shell is given by δm = 2πrδrLρ. [2] UCLES 2007 9792/03/SP/10
17 (d) By integration show that the moment of inertia I of the whole cylinder is given by I = 2 1 MR 2 Examiner's where M is the mass of the cylinder. [4] (e) The cylinder has mass 900 kg and diameter 2.2 m. Calculate the moment of inertia of the cylinder. I = [2] (f) Calculate the maximum rotational kinetic energy of the flywheel. maximum rotational kinetic energy = [2] (g) The flywheel is brought up to maximum speed at a station by a motor connected to an overhead power line. The electric power is then disconnected and the energy stored in the flywheel is used to drive the bus. The average power required to operate the bus is 2.0 x 10 4 W. how many minutes can the bus operate between stops? time between stops = minutes [3] UCLES 2007 9792/03/SP/10 [Turn over
18 (h) To increase the time between stops, the flywheel needs to store more kinetic energy. It is suggested that the flywheel dimensions and angular velocity remain the same, but a material with 20% larger density be used. Examiner's (i) What would be the new time between stops if the same average power were used to operate the bus? time between stops = minutes [1] (ii) Give two reasons why the new flywheel might reduce the performance of the bus. 1. 2. [2] [Total: 20] UCLES 2007 9792/03/SP/10