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  British Physics Olympiad 2018-19BPhO Round 1 Section 1 16 th November 2018This question paper must not be taken out of the exam roomInstructions Time :  1 hour 20 minutes  for this section. Questions : Students may attempt any parts of   Section 1 , but are not expected to complete all parts. Working : Working, calculations, explanations and diagrams, properly laid out, must be shown for fullcredit. The final answer alone is not sufficient. Writing must be clear. Marks : A  maximum of 50 marks  can be awarded for  Section 1 . There is a total of   87  marks allocatedto the problems of Question 1 which makes up the whole of   Section 1 . Instructions : You are allowed any standard exam board data/formula sheet. Calculators : Any standard calculator may be used, but calculators cannot be programmable and mustnot have symbolic algebra capability. Solutions : Answers and calculations are to be written on loose paper or in examination booklets.Graph paper and formula sheets should also be made available. Students should ensure that their  name and their  school/college  are clearly written on each and every answer sheet. Number each questionclearly. Setting the paper : There are two options for sitting BPhO Round 1:a.  Section 1  and  Section 2  may be sat in one session of 2 hours 40 minutes plus 5 minutes reading time(for  Section 2 ).  Section 1  should be collected in after 1 hour 20 minutes and then  Section 2  given out.b.  Section 1  and  Section 2  may be sat in two sessions on separate occasions, with 1 hour 20 minutesplus 5 minutes reading time allocated for  Section 2 . If the paper is taken in two sessions on separateoccasions,  Section 1  must be collected in after the first session and  Section 2  handed out at thebeginning of the second session.  Important Constants Constant Symbol Value Speed of light in free space  c  3 . 00 × 10 8 ms − 1 Elementary charge  e  1 . 60 × 10 − 19 C Planck constant  h  6 . 63 × 10 − 34 Js Mass of electron  m e  9 . 11 × 10 − 31 kg Mass of proton  m p  1 . 67 × 10 − 27 kg atomic mass unit  u 1 . 661 × 10 27 kg (1u is equivalent to 931.5 MeV)Gravitational constant  G  6 . 67 × 10 − 11 m 3 kg − 1 s − 2 Acceleration of free fall at Earth’s surface  g  9 . 81ms − 2 Permittivity of free space  ε 0  8 . 85 × 10 − 12 Fm − 1 Permeability of free space  µ 0  4 π × 10 − 7 Hm − 1 Avogadro constant  N  A  6 . 02 × 10 23 mol − 1 Mass of Sun  M  S  1 . 99 × 10 30 kg Radius of Earth  R E  6 . 37 × 10 6 m T  (K)  =  T  ( ◦ C)  + 273 Volume of a sphere  =  43 πr 3 For small angles,  sin θ ≈ tan θ ≈ θ For  x  1 ,  (1 + x ) n ≈ 1 + nx  Question 1 a) The Milky Way galaxy has a period of rotation of   240 × 10 6 years . The Sun is 26 light years fromthe centre of the galaxy. How fast is the Sun moving with respect to the centre of the galaxy, givenin units of   ms − 1 ?A light year is the distance that light travels in one year of 365.25 days. [3] b) Asmoothsphereofradius 6 . 0cm issuspendedfromathreadoflength 9 . 0cm attachedtoasmoothwall as shown in shown in  Figure 1 . If the mass of the sphere is  0 . 5kg , calculate the tension,  T  ,in the thread. Figure 1 [3] c) The displacement of an object is determined by the following function: s  = 2 t 3 − 9 t 2 + 12 t + 4 where  s  is the displacement in metres, and  t  the time elapsed in seconds. Determine(i) the times when the object comes to rest,(ii) the time when the acceleration is zero,(iii) the object’s velocity when its acceleration is zero,(iv) the object’s accelerations when its velocity is zero. [4] d) The distance in which a train can be stopped is given by: s  =  av  + bv 2 where  s  is the stopping distance,  v  the initial velocity, and  a  and  b  are constants. When moving at 40kmhr − 1 , the train can be stopped in  100m , and at  80kmhr − 1 it can be stopped in  280m .Find the greatest speed such that the train can be stopped in  500m . [4] 1  e) Two planes set out at the same time from an aerodrome. The first flies north at  360kmh − 1 , thesecond south-east at  300kmh − 1 . After 40 minutes they both turn and fly towards each other.Calculate(i) the bearing, and(ii) the distanceof the meeting point from the aerodrome. [7] f) A neutron moving through heavy water strikes an isolated and stationary deuteron (the nucleus of an isotope of hydrogen) head-on in an elastic collision.(i) Assuming the mass of the neutron is equal to half that of the deuteron, find the ratio of thefinal speed of the deuteron to the initial speed of the neutron.(ii) What percentage of the initial kinetic energy is transferred to the deuteron?(iii) How many such collisions would be needed to slow the neutron down from  10MeV  to 0 . 01eV ? [6] g) A uniform chain of mass per unit length,  µ , is suspended from one end above a table, with thelower end just touching the surface. The chain is released, falls and comes to rest on the tablewithout bouncing.(i) Determine an expression, in terms of   µ  and the gravitational field strength  g , for the reactionforce exerted by the table on the chain as a function of time,  t .Hint: you might consider  F   in the form  F   =  ∆ m ∆ t  v .(ii) In terms of the total weight  W   of the chain, what is the maximum reaction force exerted bythe table, and at what time during the fall does this occur? [6] h) A small particle of mass  m  can slide without friction round the inside of a cylindrical hole of radius  r , in a rectangular shaped object of mass  M  . The rectangular object is held between rigidwalls by small wheels so that it can slide up and down without friction, as shown in  Figure 2 . If the small particle  m  is initially at rest at the bottom of the cylindrical hole, and is then given animpulse to give it a speed  v , what is the minimum speed  v  needed to just lift the rectangular mass M   off the ground?  M m Figure 2 [5] 2
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