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1 M Centres of Mss - Rigid bodies nd composites. Figure A continer is formed by removing right circulr solid cone of height l from uniform solid right circulr cylinder of height 6l. The centre O of the plne fce of the cone coincides with the centre of plne fce of the cylinder nd the is of the cone coincides with the is of the cylinder, s shown in Figure. The cylinder hs rdius l nd the bse of the cone hs rdius l. () Find the distnce of the centre of mss of the continer from O. (6) Figure The continer is plced on plne which is inclined t n ngle θ to the horizontl. The open fce is uppermost, s shown in Figure. The plne is sufficiently rough to prevent the continer from sliding. The continer is on the point of toppling. (b) Find the vlue of θ. () (Totl 0 mrks) Edecel Internl Review

2 M Centres of Mss - Rigid bodies nd composites. Figure A bowl B consists of uniform solid hemisphere, of rdius r nd centre O, from which is removed solid hemisphere, of rdius r nd centre O, s shown in Figure. 65 () Show tht the distnce of the centre of mss of B from O is r. 5 (5) Edecel Internl Review

3 M Centres of Mss - Rigid bodies nd composites Figure The bowl B hs mss M. A prticle of mss km is ttched to point P on the outer rim of B. The system is plced with point C on its outer curved surfce in contct with horizontl plne. The system is in equilibrium with P, O nd C in the sme verticl plne. The line OP mkes n ngle θ with the horizontl s shown in Figure. Given tht tn θ =, 5 (b) find the ect vlue of k. (5) (Totl 0 mrks). [The centre of mss of uniform hollow cone of height h is h bove the bse on the line from the centre of the bse to the verte.] A mrker for the route of chrity wlk consists of uniform hollow cone fied on to uniform solid cylindricl ring, s shown in the digrm bove. The hollow cone hs bse rdius r, height 9h nd mss m. The solid cylindricl ring hs outer rdius r, height h nd mss m. The mrker stnds with its bse on horizontl surfce. Edecel Internl Review

4 M Centres of Mss - Rigid bodies nd composites () Find, in terms of h, the distnce of the centre of mss of the mrker from the horizontl surfce. (5) When the mrker stnds on plne inclined t rctn to the horizontl it is on the point of toppling over. The coefficient of friction between the mrker nd the plne is lrge enough to be certin tht the mrker will not slip. (b) Find h in terms of r. () (Totl 8 mrks). The finite region bounded by the -is, the curve y =, the line = nd the line =, is rotted through one complete revolution bout the -is to form uniform solid of revolution. () Show tht the volume of the solid is π. () (b) Find the coordintes of the centre of mss of the solid. (5) (Totl 9 mrks) Edecel Internl Review

5 M Centres of Mss - Rigid bodies nd composites 5. Figure The region R is bounded by prt of the curve with eqution y =, the positive -is nd the positive y-is, s shown in Figure. The unit of length on both es is one metre. A uniform solid S is formed by rotting R through 60 bout the -is. () Show tht the centre of mss of S is 8 5 m from O. (0) Figure Figure shows cross section of uniform solid P consisting of two components, solid cylinder C nd the solid S. The cylinder C hs rdius m nd length l metres. One end of C coincides with the plne circulr fce of S. The point A is on the circumference of the circulr fce common to C nd S. When the solid P is freely suspended from A, the solid P hngs with its is of symmetry horizontl. (b) Find the vlue of l. () (Totl mrks) Edecel Internl Review 5

6 M Centres of Mss - Rigid bodies nd composites 6. 6 O Figure A uniform solid hemisphere, of rdius 6 nd centre O, hs solid hemisphere of rdius, nd centre O, removed to form bowl B s shown in Figure. 0 () show tht the centre of mss of B is from O. (5) 6 O 6 Figure The bowl B is fied to plne fce of uniform solid cylinder mde from the sme mteril s B. The cylinder hs rdius nd height 6 nd the combined solid S hs n is of symmetry which psses through O, s shown in Figure. 0 (b) Show tht the centre of mss of S is 6 from O. () Edecel Internl Review 6

7 M Centres of Mss - Rigid bodies nd composites The plne surfce of the cylindricl bse of S is plced on rough plne inclined t to the horizontl. The plne is sufficiently rough to prevent slipping. (c) Determine whether or not S will topple. () (Totl mrks) 7. An open continer C is modelled s thin uniform hollow cylinder of rdius h nd height h with bse but no lid. The centre of the bse is O. () Show tht the distnce of the centre of mss of C from O is h. (5) The continer is filled with uniform liquid. Given tht the mss of the continer is M nd the mss of the liquid is M, (b) find the distnce of the centre of mss of the filled continer from O. (5) (Totl 0 mrks) 8. Figure y R O The shded region R is bounded by the curve with eqution y =, the -is nd the lines = nd =, s shown in Figure. The unit of length on ech is is m. A uniform solid S hs the shpe mde by rotting R through 60 bout the -is. Edecel Internl Review 7

8 M Centres of Mss - Rigid bodies nd composites () Show tht the centre of mss of S is 7 m from its lrger plne fce. (6) Figure H T S A sporting trophy T is uniform solid hemisphere H joined to the solid S. The hemisphere hs rdius m nd its plne fce coincides with the lrger plne fce of S, s shown in Figure. Both H nd S re mde of the sme mteril. (b) Find the distnce of the centre of mss of T from its plne fce. (7) (Totl mrks) 9. A uniform solid is formed by rotting the region enclosed between the curve with eqution y =, the -is nd the line =, through one complete revolution bout the -is. Find the distnce of the centre of mss of the solid from the origin O. (Totl 5 mrks) Edecel Internl Review 8

9 M Centres of Mss - Rigid bodies nd composites 0. A bowl consists of uniform solid metl hemisphere, of rdius nd centre O, from which is removed the solid hemisphere of rdius with the sme centre O. 5 () Show tht the distnce of the centre of mss of the bowl from O is. (5) The bowl is fied with its plne fce uppermost nd horizontl. It is now filled with liquid. The mss of the bowl is M nd the mss of the liquid is km, where k is constnt. Given tht the 7 distnce of the centre of mss of the bowl nd liquid together from O is, 8 (b) find the vlue of k. (5) (Totl 0 mrks). H C h r O A body consists of uniform solid circulr cylinder C, together with uniform solid hemisphere H which is ttched to C. The plne fce of H coincides with the upper plne fce of C, s shown in the figure bove. The cylinder C hs bse rdius r, height h nd mss M. The mss of H is M. The point O is the centre of the bse of C. () Show tht the distnce of the centre of mss of the body from O is h + r. 0 (5) Edecel Internl Review 9

10 M Centres of Mss - Rigid bodies nd composites The body is plced with its plne fce on rough plne which is inclined t n ngle α to the horizontl, where tnα =. The plne is sufficiently rough to prevent slipping. Given tht the body is on the point of toppling, (b) find h in terms of r. () (Totl 9 mrks). A closed continer C consists of thin uniform hollow hemisphericl bowl of rdius, together with lid. The lid is thin uniform circulr disc, lso of rdius. The centre O of the disc coincides with the centre of the hemisphericl bowl. The bowl nd its lid re mde of the sme mteril. () Show tht the centre of mss of C is t distnce from O. () The continer C hs mss M. A prticle of mss M is ttched to the continer t point P on the circumference of the lid. The continer is then plced with point of its curved surfce in contct with horizontl plne. The continer rests in equilibrium with P, O nd the point of contct in the sme verticl plne. (b) Find, to the nerest degree, the ngle mde by the line PO with the horizontl. (5) (Totl 9 mrks) Edecel Internl Review 0

11 M Centres of Mss - Rigid bodies nd composites. O r A B r A child s toy consists of uniform solid hemisphere, of mss M nd bse rdius r, joined to uniform solid right circulr cone of mss m, where m < M. The cone hs verte O, bse rdius r nd height r. Its plne fce, with dimeter AB, coincides with the plne fce of the hemisphere, s shown in the digrm bove. () Show tht the distnce of the centre of mss of the toy from AB is ( M m) r. 8( M + m) (5) The toy is plced with OA on horizontl surfce. The toy is relesed from rest nd does not remin in equilibrium. (b) Show tht M > 6m. () (Totl 9 mrks) Edecel Internl Review

12 M Centres of Mss - Rigid bodies nd composites. r O r A r A toy is formed by joining uniform solid right circulr cone, of bse rdius r nd height r, to uniform solid cylinder, lso of rdius r nd height r. The cone nd the cylinder re mde from the sme mteril, nd the plne fce of the cone coincides with plne fce of the cylinder, s shown in the digrm bove. The centre of this plne fce is O. () Find the distnce of the centre of mss of the toy from O. (5) The point A lies on the edge of the plne fce of the cylinder which forms the bse of the toy. The toy is suspended from A nd hngs in equilibrium. (b) Find, in degrees to one deciml plce, the ngle between the is of symmetry of the toy nd the verticl. () The toy is plced with the curved surfce of the cone on horizontl ground. (c) Determine whether the toy will topple. () (Totl mrks) Edecel Internl Review

13 M Centres of Mss - Rigid bodies nd composites 5. O A uniform solid cylinder hs rdius nd height. A hemisphere of rdius is removed from the cylinder. The plne fce of the hemisphere coincides with the upper plne fce of the cylinder, nd the centre O of the hemisphere is lso the centre of this plne fce, s shown in the digrm bove. The remining solid is S. () Find the distnce of the centre of mss of S from O. (6) The lower plne fce of S rests in equilibrium on desk lid which is inclined t n ngle θ to the horizontl. Assuming tht the lid is sufficiently rough to prevent S from slipping, nd tht S is on the point of toppling when θ = α, (b) find the vlue of α. () Given insted tht the coefficient of friction between S nd the lid is 0.8, nd tht S is on the point of sliding down the lid when θ = β, (c) find the vlue of β. () (Totl mrks) Edecel Internl Review

14 M Centres of Mss - Rigid bodies nd composites 6. Figure y R O The shded region R is bounded by prt of the curve with eqution y = ( ), the -is nd the y-is, s shown in Fig.. The unit of length on both es is cm. A uniform solid S is mde by rotting R through 60 bout the -is. Using integrtion, () clculte the volume of the solid S, leving your nswer in terms of π, () (b) show tht the centre of mss of S is cm from its plne fce. Figure 8 cm (7) cm C S A B A tool is modelled s hving two components, solid uniform cylinder C nd the solid S. The dimeter of C is cm nd the length of C is 8 cm. One end of C coincides with the plne fce of S. The components re mde of different mterils. The weight of C is 0W newtons nd the weight of S is W newtons. The tool lies in equilibrium with its is of symmetry horizontl on two smooth supports A nd B, which re t the ends of the cylinder, s shown in Fig.. (c) Find the mgnitude of the force of the support A on the tool. (5) (Totl 6 mrks) Edecel Internl Review

15 M Centres of Mss - Rigid bodies nd composites 7. r h O A child s toy consists of uniform solid hemisphere ttched to uniform solid cylinder. The plne fce of the hemisphere coincides with the plne fce of the cylinder, s shown in the digrm bove. The cylinder nd the hemisphere ech hve rdius r, nd the height of the cylinder is h. The mteril of the hemisphere is 6 times s dense s the mteril of the cylinder. The toy rests in equilibrium on horizontl plne with the cylinder bove the hemisphere nd the is of the cylinder verticl. () Show tht the distnce d of the centre of mss of the toy from its lowest point O is given by d = h + hr + 5r ( h + r). (7) When the toy is plced with ny point of the curved surfce of the hemisphere resting on the plne it will remin in equilibrium. (b) Find h in terms of r. () (Totl 0 mrks) Edecel Internl Review 5

16 M Centres of Mss - Rigid bodies nd composites 8. () Show, by integrtion, tht the centre of mss of uniform right cone, of rdius nd height h, is distnce h from the verte of the cone. A h h A uniform right cone C, of rdius nd height h, hs verte A. A solid S is formed by removing from C nother cone, of rdius nd height h, with the sme is s C. The plne fces of the two cones coincide, s shown in the digrm bove. (b) Find the distnce of the centre of mss of S from A. (Totl 7 mrks) Edecel Internl Review 6

17 M Centres of Mss - Rigid bodies nd composites. () cone continer cylinder mss rtio πl 68πl π l M A 68 7 dist from O l l B Moments: l+ 68 = 7 l M Aft l 5 = = l ccept.l A (b) GX = 6l seen M l tnθ = 6l M A 7 = 9 θ = =.8 or 5 A [0]. () Mss rtios s 8 B 9 S 7 nything in correct rtio B r r B r + 9 = 7 r M Aft 8 65 = r * A 5 5 Edecel Internl Review 7

18 M Centres of Mss - Rigid bodies nd composites (b) Mg sinθ = kmg r cosθ M A=A leding to k = M A 5 8 [0]. () Object Mss c of m bove bse B(rtio msses) Cone m h+h Bse m h B(distnces) Mrker m d m 5h + m h = m d MAft d = h A (b) r d = 6r = h MAft A [8] Edecel Internl Review 8

19 M Centres of Mss - Rigid bodies nd composites. () Volume = = π y d π d MA π Aft 6 = π + = π * A (b) πρ = ρ π y d = ρ π d MA π = π Aft 6 5 = + = or wrt 0.6 A y = 0 by symmetry B [9] 5. () y d ( ) d = (6 8 + = ) d 5 8 = 6 + M A = M A y d = ( ) d = (6 8 + ) d 6 = 8 + M A = M A 6 0 y d 5 5 = = = M A 0 y d 6 8 Edecel Internl Review 9

20 M Centres of Mss - Rigid bodies nd composites l (b) = ( r l) A π M 56 5 π = π 6l l A ft 5 8 Leding to l = ccept ect equivlents or wrt.5 M A [] 6. () = + Mss π : MA 7 6 C of M from O: M Moment: 6 6 = M 80 0 = = * 08 A cso 5 (b) + = S B Mss π 6 88 : + = B C of M: = y M Moments : = y Acso (c) y y = 0 * M tnθ = tnθ = 0... M 6 θ =.9... A so criticl ngle =.9 if θ = it will NOT topple. A [] Edecel Internl Review 0

21 M Centres of Mss - Rigid bodies nd composites 7. () Bse Cylinder Continer Mss rtios πh πh πh Rtio of : : B y 0 h y B h π h y = πh MA Leding to y = h * cso A 5 (b) Liquid Continer Totl Mss rtios M M M Rtio of : : B h h y y B h h M y = M + M MA 5 y = h A 5 [0] 8. () Moments: π y d = V or y d y d = M 7 y d d ( = ) = = 96 (either) MA y d = d ( ) = 8 (both) A 9 9 Solving to find (= ) required dist = = m (*) M Acso 6 (b) H S T Mss 7π (ρ) π, ( ρ ) 96 H + S 5 = (ρ) π = (ρ) π B, M 9 5 Dist of CM from bse m m 6 7 BB 9 7π 5 5 Moments: = ( ρ ) π + ( ρ) = ( ρ ) π MA 9 = m or m (wrt) 0 A 7 Allow distnces to be found from different bse line if necessry [] Edecel Internl Review

22 M Centres of Mss - Rigid bodies nd composites 9. Use of ( ) y d = ( π ) π y d M d = d = A = A Using limits 0 nd = M 8 = A 5 [5] 0. () Smll Hemisphere Bowl Lrge Hemisphere Mss rtios 7 π π 8 Anything in the rtio : 7 : 8 6 π B = 8 M A 6 8 Leding to B 5 = cso A 5 (b) Bowl Liquid Bowl nd Liquid Mss rtios M km (k + )M B M + km = ( k + ) M M A 6 8 Leding to k = A 5 7 B [0]. () 5M = M h + M h + r 8 5 = h + h + r = 7 h + r M A(, 0) Edecel Internl Review

23 M Centres of Mss - Rigid bodies nd composites h + r = cso M A 5 0 (b) r tn α = 0r = h + r Leding to h = 6 7 r M A M A [9]. () Bowl Lid C Mss rtio nything in rtio : : B y 0 y B M(O) = y M y = (*) csoa (b) R y O A P Mg Mg M(A) Mg sinθ = Mg cos θ tnθ = M A=A M θ 56 coa 5 [9] Edecel Internl Review

24 M Centres of Mss - Rigid bodies nd composites Methods involving the loction of the combined centre of mss of C nd P. G is the centre of mss of C; G is the combined centre of mss of C nd P. First Alterntive C P C nd P Mss rtios y 0 y 0 Finding both coordintes of G M = y y = 9 A = = A O G ` 9 Verticl P tnθ = = 9 M θ 56 co A 5 Edecel Internl Review

25 M Centres of Mss - Rigid bodies nd composites Second lterntive O N P G G ` Verticl GG : G P = M : M = : OG =, OP = By similr tringles ON = OP = NG = OG = 9 M A A ON tnθ = = = NG 9 M θ 56 co A 5. () r r ; B; B 8 r r m. + M. = (m + M) M A 8 r( M m) = (*) A 5 8( M + m) Edecel Internl Review 5

26 M Centres of Mss - Rigid bodies nd composites (b) B D r C r A O No equil m CD M r( M m) r 8( M + m) M A 9(M m) > 8(M + m) M > 6m (*) A c.s.o. [9]. () Cylinder Cone Toy (6πr ) (πr ) (8πr ) mss rtio B dist. from O r ( )r B ( r) r = M A 5r = A 5 M for cler ttempt t Σm = Σm correct no. of terms. If distnces not mesured from O, BBMA vilble. (b) G A AG verticl, seen or implied M r tn θ = r M A θ = 7 5 ( d.p.) A second M for use of tn Edecel Internl Review 6

27 M Centres of Mss - Rigid bodies nd composites (c) X O V OX r sim s: = (= tn α) r r M 9r OX = A < OX M won t topple A c.s.o Note tht second M is independent, for the generl ide. [] 5. () Cylinder Hemisphere S Msses (ρ)π() ( ) (ρ) π (ρ)( [6π ] [8] [] [6] [M for ttempt t C, H nd S = C H msses] 6 π ) MA Distnces of CM from O 8 BB or lower fce Moments eqution: 6π (¾ ) π ( 5 = 6 (0.797) 6 ) = π M 8 A 6 Edecel Internl Review 7

28 M Centres of Mss - Rigid bodies nd composites (b) G A α X α G bove A seen or implied or mg sinα (GX) = mg cosα (AX) AX tn α = = XG [GX = 5 6 M M 5 8 =, tn α = ] α = 70.6 A 6 5 (c) Finding F nd R : R = mg cos β, F = mg sin β M Using F = µr nd finding tnβ [= 0.8] M β = 8.7 A [] 6. () V = y π d [= π ( ) d ] M ( ) d = 5 ( ) 5 or [ ] M A [M requires ttempt to squre nd integrte] V = 8π 5 A (b) Using y π d = [ π ( ) d M Correct strtegy to integrte [e.g. substitution, epnd, by prts] M [e.g. π ( u ) du ; 5 π ( ) d ]; u u 8 = π + or 5 6 π M A 8π Limits used correctly [correct vlues = ] 5 A ft V c (ρ) = π ( ρ) y d (seen nywhere) M = cm (*) no incorrect working seen A 7 Edecel Internl Review 8

29 M Centres of Mss - Rigid bodies nd composites (c) Moments eqution to find C of M of tool: e.g. W = 0W W ( ) M A A 5 (my be implied by net line) [ = from plne edge of S] 8 Moments bout B: 8R A = 0W W( ) R A = 59W (.9 W or.9w) A 5 [Moments bout other points: M A, Complete method to find R A ; using R A + R B = W with moments eqution M A ft; A s scheme] [6] 7. () Cylinder hlf-sphere toy πr hρ πr 6ρ πr hρ + πr 6ρ M A h 5r + r 8 πr hρ( h + r) + πr ρ 8 5r = (πr hρ + πr ρ)d d B B M A d = h + rh + 5r ( h + r) (*) A 7 (b) d = r, h + rh + 5r = r(h + r) M, M h = r A [0] Edecel Internl Review 9

30 M Centres of Mss - Rigid bodies nd composites 8. () A h Rdius of element = h B π h = = π h = h π h h 0 M M A 6 (b) Volume of lrge cone = h = V π Volume of smll cone = h π = V Volume of S = 9 V Volume V V 9 V 9 7 M A CM from A h h h + B B V 7h 7 h V 9 = V 8 9 M A 5h = 7 A 7 [] Edecel Internl Review 0

31 M Centres of Mss - Rigid bodies nd composites. This ws routine centre of mss problem requiring mss nd ssocited known centre of mss for stndrd volumes, combined in n pproprite moments eqution. A few mistkes occurred when cndidtes tried to write down their moments eqution without detiling ech prt in tble. However the mjor difficulty ws for those who couldn t produce correct volume for the cone nd occsionlly even the cylinder. The cone hd multiples of nd used with πr h nd the cylinder becme πr h. Some cndidtes filed to introduce the different vlue for r s l nd l before cncelling it hence giving n incorrect mss rtio. In prt (b), the condition for tipping with G bove the bottom point on the continer, ws used by ll who ttempted this prt. Recognising the use of 6l (their ) ws crucil to finding correct trigonometric rtio, nd those tht did mde few mistkes finding θ correctly. As epected few used l insted of l in the numertor for their epression for tnθ.. The mjority of cndidtes could complete, or nerly complete, prt () successfully. The given nswer did llow some cndidtes to correct their rithmetic errors. Those who reduced their volumes to rtio (8:9:7) before constructing the moments eqution produced clerer nd more strightforwrd solutions thn those who worked with the originl volume formule. A few cndidtes remembered their GCSE work on the rtio of volumes of similr solids nd stted directly tht the rtio of the two hemispheres ws 8:7. Prt (b) ws done very quickly nd esily by the most ble cndidtes but proved difficult for the mjority. Mny cndidtes proceeded stright from Q() to Q. Of those who ttempted prt (b), there ws firly even split between those who took moments bout the verticl through O for the two weights (or mentlly cncelled g nd used msses) nd those who found the coordintes of the centre of mss of the composite body. Some of the ltter group found only one coordinte nd others mde n error in their clcultions. Some of the former group produced moments eqution with trigonometricl rtios tht cncelled. Since this mde the given informtion tht tn θ = redundnt, they should hve been lerted to their mistke. 5. This question produced mny completely correct solutions. Some cndidtes however ignored the informtion provided bout the msses of the two prts of the route mrker nd clculted their own msses by using volumes insted, usully ssuming both sections to be solids. Almost ll could produce vlid, if not correct, moments eqution nd so gined some mrks. The most frequently seen error in prt (b) ws to hve the required frction upside down resulting in h = r. Some cndidtes lost the finl mrk here through giving r in terms of h insted of nswering the question sked. Edecel Internl Review

32 M Centres of Mss - Rigid bodies nd composites. Solutions to this question were spoilt by poor integrtion; d ws seen s 5,, nd even ln( ). Mny mrks were lost by cndidtes who were determined to rrive t the given nswer of π even though their working could not support this result. A lrge number of good cndidtes lost the finl mrk in prt (b) s they completely forgot tht coordintes hd been requested nd so y-coordinte ws needed s well s the -coordinte. Other errors in prt (b) included finding the centre of mss of lmin, ignoring the volume found in (), nd vrious further integrtion mistkes. There ws some confusion over the significnce of the lower limit for being ; some cndidtes seemed to think it ws necessry to subtrct from their result. 5. Prt () ws stndrd solution quoted correctly using two integrls. The mjority could hndle the pure mths successfully. This prt could be worked correctly without including π in either integrl s it clerly cncels. However, this led some cndidtes to omit π when clculting the mss/volume of S in (b). Mny cndidtes seemed to not know wht pproch to tke in (b) nd so mde no ttempt t ll. Of those who ttempted this prt, most took moments bout the join but every possible mistke concerning volumes nd distnces ws seen. Some took moments bout the centre of the plne fce of the cylinder, not lwys remembering to reclculte the lengths involved or use the volume of the totl solid. 6. Centres of mss were well understood nd, prt from the initil formultions for volumes of hemispheres, lrge number of cndidtes hd fully correct solutions to prts () nd (b). A few did not know the formul for the volume of sphere nd few forgot to divide by two. The ltter group obtined correct nswer for () (but scored only /5). However, their error ws eposed in (b) s they could not obtin the given nswer here. A few went bck nd found nd corrected their error; others gve up or fudged the nswer. Weker cndidtes sometimes omitted prt (c). Severl methods for prt (c) were seen the most common were finding the criticl ngle nd finding the point t which the verticl through the centre of mss cut the plne, showing tht the distnce from the is of symmetry ws less thn. A smll number hd the plne fce of the hemisphere on the plne nd few thought tht the nswer to (b) ws the distnce of the centre of mss from the plne. Edecel Internl Review

33 M Centres of Mss - Rigid bodies nd composites 7. This question ws, on the whole, well done with gret mny cndidtes gining full mrks. Prt () cused more problems thn prt (b) but mny cndidtes who hd difficulty with () went on to complete (b) successfully. There were significnt number of very wek ttempts t () which tried to involve integrtion, often pprently trying to prove the result for the centre of mss of cone. Among successful solutions, the most populr method ws to tret the continer s cylinder without ends nd combine this with one circulr end. However the lterntive involving the removl of lid from continer closed t both ends ws lso quite common nd ws usully successful. Problems with (b) were rre nd tended to rise only where cndidtes ttempted to consider the curved surfce, bse nd liquid s three seprte items, ignoring the given M nd often tking the msses s πh, πh nd πh. Some, however, completed this method successfully by using the mss of the liquid s πh. Geometricl solutions to either prt of the question were uncommon. 8. The principles here were generlly well known. Mistkes occurred in prt () with the ccurcy of the integrtion concerned, severl dropping fctors involving the frctions. And in prt (b), number of cndidtes filed to relise tht the integrl they hd clculted in prt () ws not necessrily the mss of the solid (especilly when they hd cncelled out fctor erlier). 9. This question proved to be, for the gret mjority, test of memory. Those who could remember the correct formul for the centre of mss of solid of revolution lmost lwys gined full mrks nd those who could not gined very little. Very few cndidtes used the ide of breking the solid up into elementry discs either s method of demonstrting the formul or of checking tht they hd remembered the formul correctly. 0. Prt () ws very well done nd full mrks were common. The best nd clerest solutions reduced the msses to the rtio 8 : : 7 before strting the clcultion for the centre of mss nd it my be good policy to encourge cndidtes to remove densities, rdii nd πs before they write down their moments eqution. Prt (b) proved more difficult but mny completely correct solutions were seen. A common error ws to fil to mke the connection with prt () nd tke the centre of mss of the bowl s being 8 from the surfce of the liquid. Another source of error ws including volumes in the mss rtios which, in this prt, re simply : k : k +.. This provided very esy nine mrks for mny nd lmost ll knew wht they were trying to do. The most frequent mistke in () ws to ssume tht volumes needed to be used for the cylinder nd hemisphere nd in (b) the frction ws often upside down. Where solutions did go wrong, there ws gin widespred fiddling. Even the volume versions sometimes ended up being rerrnged, fctorised or cncelled into the required epression. Edecel Internl Review

34 M Centres of Mss - Rigid bodies nd composites. Prt () ws generlly well done nd the mjority of those who could not obtin the printed nswer usully filed through lck of knowledge of the formul for the surfce re of the sphere. This formul is given in the Formul Booklet nd cndidtes in Mechnics re epected to be fmilir with, or know where to look for, Pure (or Core) Mthemtics formule which re pproprite to Mechnics module. Prt (b) ws generlly poorly done. Mny cndidtes were unble to visulise the sitution nd, fter few ttempts to drw digrm, bndoned the question. The simplest solution is to tke moments bout the point of contct with the floor or bout the centre of the plne fce of the hemisphere but this ws rrely seen. The most successful cndidtes tended to be those who worked out the position of the centre of mss of the hemisphere nd prticle combined, even though this involved more work.. The first prt ws generlly well done, lthough some cndidtes tried to introduce volumes into their clcultions but there were very few fully correct solutions to prt (b), where good digrm ws essentil.. This question ws generlly very well nswered with mny fully correct solutions to prts () nd (b). A plesing number of cndidtes lso were successful in presenting logicl solution to prt (c). Only the wekest cndidtes found the question inccessible nd scored fewer thn hlf mrks. The most common error in prt () ws inconsistent use of r - some cndidtes being confused with rdius s used in the volume formule nd the use of r in the question. This led to inconsistent lgebric equtions tht proved impossible to simplify nd solve. The most common error in prt (b) ws finding the complement of the required ngle or not stting the nswer to the required degree of ccurcy. Prt (c) proved difficult for mny cndidtes - common errors were using r/ s the rdius of the bse of the cone/cylinder nd ssuming the slnt length of the cone to be r. Correct results for two ngles were often clculted but the reltionship between them ws not sufficiently eplined for full mrks to be wrded. The most stisfctory solutions involved proving tht the line of ction of the weight pssed through the slnt side of the cone or tht the sum of the bse ngle of the cone nd the ngle from O to the edge of the cone/cylinder interfce to G ws less thn 90 degrees. 5. The vst mjority of cndidtes knew wht ws required in this question, which ws generlly source of high mrks. Edecel Internl Review

35 M Centres of Mss - Rigid bodies nd composites 6. In prt () the mjority of cndidtes were ble to recognise nd use the formul for the volume of revolution. The most common pproch to finding ( ) d ws to multiply out the brckets, which will hve tken up precious time nd ws often incorrect. Problems rose with the nture of the given eqution, with mny cndidtes not squring the squred fctor, so tht π ( ) d ws often seen, even fter π y d hd previously been quoted. Also, there were s mny cndidtes who used 0 nd s the limits of integrtion s there were who used the correct vlues of 0 nd ; presumbly triggered by the given sttement tht The unit of length on both es is cm. If this ws the only mistke, cndidtes only lost two mrks for this misunderstnding. Although there were mny very good solutions to this question, prt (b) did cuse problems. It my be tht t this stge weker cndidtes were rushed nd did not red the question well enough s errors such s using y d, not squring y, nd poor integrtion were common. In prt (c) mny cndidtes mde the question more unwieldy by working with their own weights rther thn using the given weights for S nd C, nd it ws surprising to see so mny cndidtes finding the centre of mss of the tool on route to nswering the question. However, most cndidtes were ble to gin some credit in this prt. 7. The correct method ws well known in the first prt nd the mjority were ble to obtin the printed nswer. Common errors were to omit or interchnge the densities, use n incorrect volume formul or to mesure distnce from wrong point. There ws less success with prt (b) where significnt number of cndidtes did not know where to strt. 8. No Report vilble for this question. Edecel Internl Review 5