Mark Scheme (Results) January 2011
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1 Mark Scheme (Results) January 011 GCE GCE Mechanics M3 (6679) Paper 1 Edexcel Limited. Registered in England and Wales No Registered Office: One90 High Holborn, London WC1V 7BH
2 Edexcel is one of the leading examining and awarding bodies in the UK and throughout the world. We provide a wide range of qualifications including academic, vocational, occupational and specific programmes for employers. Through a network of UK and overseas offices, Edexcel s centres receive the support they need to help them deliver their education and training programmes to learners. For further information, please call our GCE line on , our GCSE team on , or visit our website at If you have any subject specific questions about the content of this Mark Scheme that require the help of a subject specialist, you may find our Ask The Expert service helpful. Ask The Expert can be accessed online at the following link: January 011 Publications Code UA06583 All the material in this publication is copyright Edexcel Ltd 011
3 General Instructions for Marking 1. The total number of marks for the paper is 75.. The Edexcel Mathematics mark schemes use the following types of marks: M marks: method marks are awarded for knowing a method and attempting to apply it, unless otherwise indicated. A marks: Accuracy marks can only be awarded if the relevant method (M) marks have been earned. B marks are unconditional accuracy marks (independent of M marks) Marks should not be subdivided. 3. Abbreviations These are some of the traditional marking abbreviations that will appear in the mark schemes. bod benefit of doubt ft follow through the symbol will be used for correct ft cao correct answer only cso - correct solution only. There must be no errors in this part of the question to obtain this mark isw ignore subsequent working awrt answers which round to SC: special case oe or equivalent (and appropriate) dep dependent indep independent dp decimal places sf significant figures The answer is printed on the paper The second mark is dependent on gaining the first mark
4 January 011 Mechanics M Mark Scheme Question 1. dv v 7 x dx = 1 v = x x + c 7 ( ) x= 0 v= 6 c= 18 v= 0 x 7x 18 = 0 ( x+ )( x 9) = 0 x = 9 [6] GCE Mechanics M3 (6679) January 011 1
5 . (a) Mass ratio 4m km (4 + k)m B1 3 Dist from O 8 r 1 r 0 B1 Moments about O: 3 1 r 4m= r km 8 k = 3 (4) (b) O 30 o G 7mg λ mg OG tan 30 = r λ OG = r 7 + λ ( ) 1 λ 1 = r 3 7 r ( + λ ) 7+ λ = 3λ 7 λ = 3 1 ( ) (o.e.) or.84 B1 (4) [8] GCE Mechanics M3 (6679) January 011
6 3. (a) Vol = π y dx= π e x dx x = π e 1 = 1 4 e e π (4) (b) π yxdx 1 C of M = vol x 1 x 1 x e x dx= xe e dx x x 1 1 = xe e = e 1e e e = e e π e e 4 4 C of M = = π ( e e ) = 1.66 (3 sf) (6) [10] GCE Mechanics M3 (6679) January 011 3
7 4. (a) πt x = 5sin 3 π πt x = 5 cos 3 3 π πt x = 5 sin 3 3 π x= x S.H.M. 9 ( ) (3) (b) period = ππ = 6 B1 3 amplitude = 5 B1 () (c) π πt... = 5 cos 3 3 5π max. v = 3 or vmax = aω () (d) πt At A x= = 5sin 3 π sin t = 3 sin 1 t A = 0.4 π 3 1 At B x= 3 sin t B = 0.6 π time A B = sin 0.6 sin 0.4 π π = = 0. s accept awrt 0. (4) [11] GCE Mechanics M3 (6679) January 011 4
8 5. A T a T b B mg (a) l B1 r = R T cos 45 = T cos 45 + mg ( ) a b Ta Tb = mg (1) R ( ) Tacos 45 + Tbcos 45 = mrω 1 1 l Ta + Tb = m ω Ta + Tb = mlω () Ta Tb = mg (1) Ta = m( lω + g ) 1 Ta = m( lω + g ) T = mlω T b a = 1 ( ) mlω g (8) (b) 1 Tb > 0 m( lω g ) > 0 g ω > * l () [10] GCE Mechanics M3 (6679) January 011 5
9 6. (a) A C 3 l 4 B T a P θ T b 3mg 5 B1 length AP= length BP= l 4 1 kmg ( 4 l) 1 Ta = Tb = = kmg ( or T =...) l 4 R ( ) Tacosθ + Tbcosθ = 3mg ( or Tcosθ = 3mg) kmg + kmg = 3mg or kmg = 3mg kmg 3mg 10 = k = 10 * (7) (b) 1 5 l 13 8 initial extn = l l = l 5 5 x 10mg 8l 18mgl λ E.P.E. lost = = = l l 5 5 1l 36mgl P.E. gained = 3mg = mgl 18mgl 3mv + = gl 7gl v = v= gl 15 B1 (6) [13] GCE Mechanics M3 (6679) January 011 6
10 7. u v mg (a) 1 1 mgl mv mu ( cosα cosθ) = ( ) v = u + gl α θ cos cos * = (4) (b) 3 3 cos α = v = gl cosθ + u At top θ = 360 v = gl 1 + u 5 v > 0 gl + u >0 5 4gl u > 5 gl u > * 5 (4) GCE Mechanics M3 (6679) January 011 7
11 (c) Equation of motion along radius at lowest point: mv T1 mg = l 3 θ = 180 v = gl u 5 16gl v = + u 5 m 16gl T1 = + u + mg l 5 1mg mu = + 5 l At highest point: mv T + mg = l mu θ = 360 T = mg + mg 5 l mu 9mg T = l 5 T = 5T 1 1 mg mu 9 5 mu mg + = 5 l l 5 66mg 4mu = 5 l 33gl u = * 10 (9) [17] GCE Mechanics M3 (6679) January 011 8
12 Further copies of this publication are available from Edexcel Publications, Adamsway, Mansfield, Notts, NG18 4FN Telephone Fax Order Code UA06583 January 011 For more information on Edexcel qualifications, please visit Edexcel Limited. Registered in England and Wales no Registered Office: One90 High Holborn, London, WC1V 7BH
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