Question 4:
4 | cm
12cm
2
June 2008
◦
1 Acarispulledatconstantspeedalongahorizontalstraightroadbyaforceof200Ninclinedat35 to
the horizontal. Given that the work done by the force is 5000J, calculate the distance moved by the
car. [3]
2 A bullet of mass 9 grams passes horizontally through a fixed vertical board of thickness 3cm. The
speed of the bullet is reduced from 250ms
−1
to 150ms
−1
as it passes through the board. The board
exerts aconstantresistiveforceonthebullet. Calculatethemagnitudeofthis resistiveforce. [4]
3 Theresistanceto themotion of acarof mass600kg iskvN, wherevms
−1
is thecar’s speed andk is
a constant. The car ascends a hill of inclinationα, where sinα= 1. The power exerted by the car’s
10
engineis12000Wand thecarhasconstantspeed20ms
−1.
(i) Show that k = 0.6. [3]
Thepowerexerted bythecar’sengineisincreasedto16000W.
(ii) Calculatethemaximum speed ofthecarwhileascendingthehill. [3]
Thecarnowtravelsonhorizontalgroundand thepowerremains16000W.
(iii) Calculatetheacceleration ofthecaratan instantwhen itsspeedis32ms
−1.
[3]
4 A golfer hits a ball from a point O on horizontal ground with a velocity of 35ms
−1
at an angle ofθ
abovethehorizontal. ThehorizontalrangeoftheballisRmetres andthetimeofflightist seconds.
(i) Express t interms ofθ,andhenceshowthatR = 125sin2θ. [5]
Thegolferhitstheballso thatitlands110mfromO.
(ii) Calculatethetwopossiblevaluesoft. [5]
©OCR2008 4729/01Jun08
June 2008 3
5
0.01kg
0.02kg 10cm
3cm
0.5kg
Fig.1
A toy is constructed by attaching a small ball of mass 0.01kg to one end of a uniform rod of length
10cm whose other end is attached to the centre of the plane face of a uniformsolid hemisphere with
radius 3cm. The rod has mass 0.02kg, the hemisphere has mass 0.5kg and the rod is perpendicular
totheplanefaceofthehemisphere(seeFig.1).
(i) Showthatthedistancefromtheballtothecentreofmassofthetoyis10.7cm,correctto1decimal
place. [4]
(ii)
Fig.2
The toy lies on horizontal ground in a position such that the ball is touching the ground (see
Fig.2). Determine whether the toy is lying in equilibrium or whether it will move to a position
wheretherod isvertical. [4]
[Turnover
©OCR2008 4729/01Jun08
0.4m
5
June 2008
8 (i)
B 8cm C
6cm
A D
17cm
Fig.1
AuniformlaminaABCDisintheformofaright-angledtrapezium. AB = 6cm,BC = 8cmand
AD =17cm(seeFig.1). Takingx-andy-axesalongADandABrespectively,findthecoordinates
ofthecentreofmassofthelamina. [8]
(ii)
C
D
B
7cm
30°
A
Fig.2
Thelaminais smoothly pivoted atAanditrestsinaverticalplanein equilibriumagainstafixed
◦
smooth block of height 7cm. The mass of the lamina is 3kg. AD makes an angle of 30 with
the horizontal (see Fig.2). Calculate the magnitude of the force which the block exerts on the
lamina. [5]
©OCR2008 4729/01Jun08
2
Jan 2009
1 A stone is projected from a point on level ground with speed 20ms
−1
at an angle of elevation of
θ◦
abovethehorizontal. Whenthestoneisatitsgreatestheightitjustpassesoverthetopofatreethatis
17mhigh. Calculateθ. [4]
2
A 12 cm 30°
B
9 cm
15 cm
C
A uniform right-angled triangular lamina ABC with sides AB = 12cm, BC = 9cm and AC = 15cm
isfreely suspended fromahingeatitsvertex A. Thelaminahasmass2kg and isheld in equilibrium
◦
withABhorizontalbymeansofastringattachedtoB. Thestringisatanangleof30 tothehorizontal
(seediagram). Calculatethetensionin thestring. [4]
3
C
80 cm
B D
30 cm
G
140 cm 200 cm
F
30 cm
A E
80 cm
AdoorismodelledasalaminaABCDE consistingofauniformrectangularsectionABDEofweight
60N and a uniform semicircular section BCD of weight 10N and radius 40cm. AB is 200cm and
AE is 80cm. The door is freely hinged atF and G, where G is 30cm below Band F is 30cmabove
A(seediagram).
(i) Find the magnitudes and directions of the horizontal components of the forces on the door at
eachofF andG. [4]
(ii) CalculatethedistancefromAE tothecentreofmassofthedoor. [6]
©OCR2009 4729Jan09
0.24 m
0.4 m
2.8 m
3
June 2009
5 (i)
C
6 cm
B D
Fig. 1
Fig.1showsauniformlaminaBCDintheshapeofaquartercircleofradius6cm. Showthatthe
distanceofthecentreofmassofthelaminafromBis3.60cm,correctto3significantfigures.
[2]
A uniform rectangular lamina ABDE has dimensions AB = 12cm and AE = 6cm. A single plane
object is formed by attaching the rectangular lamina to the lamina BCD along BD (see Fig.2). The
massofABDE is3kgand themassofBCDis2kg.
C
6 cm
B D
12 cm
A E
6 cm
Fig. 2
(ii) Takingx-andy-axesalongAEandABrespectively,findthecoordinatesofthecentreofmassof
theobject. [7]
TheobjectisfreelysuspendedatC andrestsinequilibrium.
(iii) CalculatetheanglethatAC makeswiththevertical. [2]
Turnover
©OCR2009 4729Jun09
9 m
2
Jan 2010
1 Findtheaveragepowerexertedbyaclimberofmass75kgwhenclimbingaverticaldistanceof40m
in2minutes. [3]
2 Asmallsphereofmass0.2kgisdroppedfromrestataheightof3mabovehorizontalground. Itfalls
vertically, hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height
of1.8mabovethe ground.
(i) Calculatethemagnitudeoftheimpulsewhichthe groundexertsonthesphere. [5]
(ii) Calculatethecoefficientofrestitutionbetweenthesphereandtheground. [2]
3
0.3 m 0.7 m
0.8 m
0.2 kg 0.3 kg
Fig. 1
A uniform conical shell has mass 0.2kg, height 0.3m and base diameter 0.8m. A uniform hollow
cylinderhasmass0.3kg,length0.7manddiameter0.8m. Theconicalshellisattachedtothecylinder,
withthe circumferenceofitsbasecoincidingwithoneendofthecylinder(seeFig.1).
(i) Showthatthedistanceofthecentreofmassofthecombinedobjectfromthevertexoftheconical
shellis0.47m. [4]
80°
Fig. 2
The combined object is freely suspended from its vertex and is held with its axis horizontal. This is
achievedbymeansofawireattachedtoapointonthecircumferenceofthebaseoftheconicalshell.
◦
Thewiremakesanangleof 80 withtheslantedgeoftheconicalshell(seeFig.2).
(ii) Calculatethetensioninthewire. [4]
©OCR2010 4729Jan10
3
Jan 2010
4 A car of mass 700kg is moving along a horizontal road against a constant resistance to motion of
400N.Ataninstantwhenthecaristravellingat12ms
−1
itsaccelerationis0.5ms
−2.
(i) Findthedrivingforce ofthecaratthisinstant. [2]
(ii) Findthepoweratthisinstant. [2]
Themaximumsteadyspeedofthecaronahorizontalroadis35ms
−1.
(iii) Findthemaximumpower ofthecar. [2]
Thecarnowmovesatmaximumpoweragainstthesameresistanceupaslopeofconstantangleθ◦
to
the horizontal. Themaximumsteadyspeeduptheslopeis12ms
−1.
(iv) Findθ. [4]
5 Two spheres of the same radius with masses 2kg and 3kg are moving directly towards each other
on a smooth horizontal plane with speeds 8ms
−1
and 4ms
−1
respectively. The spheres collide and
the kinetic energy lost is 81J. Calculate the speed and direction of motion of each sphere after the
collision. [12]
6
30 m s–1
P 40 m s–1
V
1
250 m
q
1
O A
AparticlePisprojectedwithspeedV ms −1 atanangleofelevationθ fromapointOonhorizontal
1 1
ground. When P is vertically above a point A on the ground its height is 250m and its velocity
componentsare40ms
−1
horizontallyand30ms
−1
verticallyupwards(seediagram).
(i) ShowthatV = 86.0andθ =62.3 ◦ ,correctto3significantfigures. [5]
1 1
At the instant when P is vertically above A, a second particle Q is projected from O with speed
V ms −1 atanangleof elevationθ. PandQhitthegroundatthesametimeandatthesameplace.
2 2
(ii) Calculatethetotaltimeof flightofPandthetotaltime offlightof Q. [4]
(iii) CalculatetherangeoftheparticlesandhencecalculateV andθ. [8]
2 2
Turnover
©OCR2010 4729Jan10
Jan 2010
4
7
5 m
O
3 m
w
P
0.2 kg
Fig. 1
AparticlePofmass0.2kgismovingonthesmoothinnersurfaceofafixedhollowhemispherewhich
hascentreOandradius5m. Pmoveswithconstantangularspeedωinahorizontalcircleatavertical
distanceof3mbelowthelevelofO(seeFig.1).
(i) Calculatethemagnitudeoftheforceexertedbythehemisphere onP. [3]
(ii) Calculateω. [4]
5 m
O
3 m
P
0.1 kg
Fig. 2
AlightinextensiblestringisnowattachedtoP. Thestringpassesthroughasmallsmoothholeatthe
lowest point of the hemisphere and a particle of mass 0.1kg hangs in equilibrium at the end of the
string. Pmovesinthesamehorizontalcircleasbefore(seeFig.2).
(iii) CalculatethenewangularspeedofP. [8]
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whoseworkisusedinthispaper. Toavoidtheissueofdisclosureofanswer-relatedinformationtocandidates,allcopyrightacknowledgementsarereproducedintheOCRCopyright
AcknowledgementsBooklet. Thisisproducedforeachseriesofexaminations,isgiventoallschoolsthatreceiveassessmentmaterialandisfreelyavailabletodownloadfromourpublic
website(www.ocr.org.uk)aftertheliveexaminationseries.
IfOCRhasunwittinglyfailedtocorrectlyacknowledgeorclearanythird-partycontentinthisassessmentmaterial,OCRwillbehappytocorrectitsmistakeattheearliestpossibleopportunity.
ForqueriesorfurtherinformationpleasecontacttheCopyrightTeam,FirstFloor,9HillsRoad,CambridgeCB21GE.
OCRispartoftheCambridgeAssessmentGroup;CambridgeAssessmentisthebrandnameofUniversityofCambridgeLocalExaminationsSyndicate(UCLES),whichisitselfadepartment
oftheUniversityofCambridge.
©OCR2010 4729Jan10
2
June 2010
1 Aparticleisprojectedhorizontallywithaspeedof7ms
−1fromapoint10mabovehorizontalground.
The particle moves freely under gravity. Calculate the speed and direction of motion of the particle
attheinstantithitsthe ground. [6]
2 (i)
A O 6 cm C
B
Fig. 1
A uniform piece of wire, ABC, forms a semicircular arc of radius 6cm. O is the mid-point of
AC (see Fig.1). Show that the distance from O to the centre of mass of the wire is 3.82cm,
correctto3significantfigures. [2]
(ii)
A
B
D
3 grams
5 grams
C
Fig. 2
Two semicircular pieces of wire, ABC and ADC, are joined together at their ends to form a
circularhoopofradius6cm. ThemassofABCis3gramsandthemassofADCis5grams. The
hoopisfreelysuspendedfromA(seeFig.2). CalculatetheanglewhichthediameterAC makes
withthevertical,givingyouranswercorrecttothenearestdegree. [5]
3 The maximum power produced by the engine of a small aeroplane of mass 2 tonnes is 128kW. Air
resistance opposes the motion directly and the lift force is perpendicular to the direction of motion.
Themagnitudeoftheairresistanceisproportionaltothesquareofthespeedandthemaximumsteady
speedinlevelflightis80ms
−1.
(i) Calculatethemagnitudeoftheairresistancewhenthespeedis60ms
−1.
[5]
◦
Theaeroplaneisclimbingata constantangleof2 tothehorizontal.
(ii) Findthemaximumaccelerationataninstantwhenthe speedof theaeroplaneis60ms
−1.
[4]
©OCR2010 4729Jun10
3
June 2010
4
T
20°
G
A B
2.5 m 1.5 m
Anon-uniformbeamABoflength4mandmass5kghasitscentreofmassatthepointGofthebeam
where AG = 2.5m. The beam isfreely suspended from itsend A and is held ina horizontal position
◦
by means of a wire attached to the end B. The wire makes an angle of 20 with the vertical and the
tensionisTN(seediagram).
(i) CalculateT. [3]
(ii) CalculatethemagnitudeandthedirectionoftheforceactingonthebeamatA. [7]
5
45°
l
w
m
One end of a light inextensible string of length l is attached to the vertex of a smooth cone of semi-
◦
verticalangle45 . Theconeisfixedtothegroundwithitsaxisvertical. Theotherendofthestringis
attached to a particle of mass m which rotates in a horizontal circle in contact with the outer surface
of the cone. The angular speed of the particle is ω(see diagram). The tension in the string is T and
the contactforcebetweentheconeandtheparticleisR.
(i) By resolving√horizontally and vertically, find two equations involving T and R and hence show
thatT = 1m( 2g+lω2). [6]
2
(ii) Whenthestringhaslength0.8m,calculatethegreatestvalueofωforwhichtheparticleremains
incontactwiththecone. [4]
[Questions6and7areprintedoverleaf.]
Turnover
©OCR2010 4729Jun10
4
June 2010
6 AparticleAofmass2mismovingwithspeeduonasmoothhorizontalsurfacewhenitcollideswith
astationaryparticleBofmassm. Afterthecollisionthe speedofAisv,thespeedofBis3v andthe
particlesmoveinthesamedirection.
(i) Findvintermsofu. [3]
(ii) ShowthatthecoefficientofrestitutionbetweenAandBis 4. [2]
5
B subsequently hits a vertical wall which is perpendicular to the direction of motion. As a result of
the impact, Bloses 3 ofitskineticenergy.
4
(iii) ShowthatthespeedofBafterhittingthewallis 3u. [4]
5
(iv) B then hits A. Calculate the speeds of A and B, in terms of u, after this collision and state their
directionsofmotion. [8]
7
B
5 m 5 m
30° 30°
A C
Asmallballofmass0.2kgisprojectedwithspeed11ms
−1
upalineofgreatestslopeofarooffrom
a point A at the bottom of the roof. The ball remains in contact with the roof and moves up the line
ofgreatestslopetothetopoftheroofatB. Theroofisroughandthe coefficientoffrictionis 1. The
2
◦
distanceABis5mandABisinclinedat30 tothehorizontal(see diagram).
(i) ShowthatthespeedoftheballwhenitreachesBis5.44ms
−1,correctto2decimalplaces.
[6]
The ball leavesthe roof at B andmoves freelyunder gravity. The point C is at the lower edge of the
◦
roof. Thedistance BC is5mandBC isinclinedat30 tothehorizontal.
(ii) Determinewhetheror nottheballhitstheroofbetweenBandC. [7]
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OCRiscommittedtoseekingpermissiontoreproduceallthird-partycontentthatitusesinitsassessmentmaterials. OCRhasattemptedtoidentifyandcontactallcopyrightholders
whoseworkisusedinthispaper. Toavoidtheissueofdisclosureofanswer-relatedinformationtocandidates,allcopyrightacknowledgementsarereproducedintheOCRCopyright
AcknowledgementsBooklet. Thisisproducedforeachseriesofexaminations,isgiventoallschoolsthatreceiveassessmentmaterialandisfreelyavailabletodownloadfromourpublic
website(www.ocr.org.uk)aftertheliveexaminationseries.
IfOCRhasunwittinglyfailedtocorrectlyacknowledgeorclearanythird-partycontentinthisassessmentmaterial,OCRwillbehappytocorrectitsmistakeattheearliestpossibleopportunity.
ForqueriesorfurtherinformationpleasecontacttheCopyrightTeam,FirstFloor,9HillsRoad,CambridgeCB21GE.
OCRispartoftheCambridgeAssessmentGroup;CambridgeAssessmentisthebrandnameofUniversityofCambridgeLocalExaminationsSyndicate(UCLES),whichisitselfadepartment
oftheUniversityofCambridge.
©OCR2010 4729Jun10
2
Jan 2011
1
A B
0.6 m
D C
A uniform square frame ABCD has sides of length 0.6m. The side AD is removed from the frame,
andtheopenframeABCDisattachedatAtoafixedpoint(seediagram).
(i) Calculatethedistance ofthecentreof massoftheopenframefromA. [5]
TheopenframerotatesaboutAintheplaneABCDwithangularspeed3rads
−1.
(ii) Calculatethespeedofthecentreof massoftheopenframe. [2]
2 The resistance to the motion of a car is kv
3
2N, where vms
−1
is the car’s speed and k is a constant.
The power exerted by the car’s engine is 15000W, and the car has constant speed 25ms
−1
along a
horizontalroad.
(i) Showthatk = 4.8. [3]
With the engine operating at a much lower power, the car descends a hill of inclination α, where
sinα= 1. Ataninstantwhenthespeedofthe caris16ms −1,itsaccelerationis0.3ms −2.
15
(ii) Giventhatthemassof thecaris700kg,calculatethepoweroftheengine. [5]
3
A
30°
0.5 m
B
60°
P
A particle P of mass0.4kg isattached toone endof each of twolightinextensible strings whichare
both taut. The other end of the longer string is attached to a fixed point A, and the other end of the
shorterstringisattachedtoafixedpointB,whichisverticallybelowA. ThestringAPmakesanangle
◦ ◦
of 30 with the vertical and is 0.5m long. The string BP makes an angle of 60 with the vertical. P
moveswithconstantangularspeedinahorizontalcirclewithcentreverticallybelowB(seediagram).
ThetensioninthestringAPistwicethe tensioninthestringBP. Calculate
(i) thetensionineachstring, [4]
(ii) theangularspeedof P. [4]
©OCR2011 4729 Jan11
3
Jan 2011
◦
4 A block of mass 25kg is dragged 30m up a slope inclined at 5 to the horizontal by a rope inclined
◦
at 20 to the slope. The tension in the rope is 100N and the resistance to the motion of the block is
70N.Theblockisinitiallyatrest. Calculate
(i) theworkdonebythetensionintherope, [2]
(ii) thechangeinthepotentialenergyoftheblock, [2]
(iii) thespeedoftheblockafterithasmoved30muptheslope. [4]
5 A uniform solid is made of a hemisphere with centre O and radius 0.6m, and a cylinder of radius
0.6m and height 0.6m. The plane face of the hemisphere and a plane face of the cylinder coincide.
(Theformulaforthevolumeofasphereis 4πr3.)
3
(i) Showthatthedistanceofthecentre ofmassof thesolidfromO is0.09m. [5]
(ii)
2 N
0.6 m
O
0.6 m
45°
The solid is placed with the curved surface of the hemisphere on a rough horizontal surface
◦
and the axis inclined at 45 to the horizontal. The equilibrium of the solid is maintained by a
horizontal force of 2N applied to the highest point on the circumference of its plane face (see
diagram). Calculate
(a) themassofthesolid, [4]
(b) thesetofpossiblevaluesof thecoefficientoffrictionbetweenthesurface andthesolid.
[3]
[Questions6and7areprintedoverleaf.]
Turnover
©OCR2011 4729 Jan11
4
Jan 2011
6 A small ball B is projected with speed 14ms
−1
at an angle of elevation 30
◦
from a point O on a
horizontalplane,andmovesfreelyundergravity.
(i) Calculatetheheightof Babovetheplanewhenmovinghorizontally. [2]
Bhasmass0.4kg. AttheinstantwhenBismovinghorizontallyitreceivesanimpulseofmagnitude
INsinitsdirectionof motionwhichimmediatelyincreasesthespeedofBto15ms
−1.
(ii) CalculateI. [3]
For theinstantwhenBreturnstotheplane,calculate
(iii) thespeedanddirectionofmotionofB, [4]
(iv) thetimeofflight,andthedistanceofBfromO. [5]
7 Three small smooth spheres A, B and C of masses 0.2kg, 0.7kg and mkg respectively are free to
moveinastraightlineonasmoothhorizontaltable. InitiallyBandC arestationaryandAismoving
with velocity 1.8ms
−1
directly towards B. The coefficient of restitution for the collision between A
andBise. ImmediatelyafterthiscollisionthespeedofAisgreaterthanthespeedofB.
(i) Calculatethesetofpossiblevaluesofe. [9]
It is now given that the speed of B immediately after the collision with A is 0.75ms
−1.
B continues
its motion and strikes C directly in a perfectly elastic collision. B has speed 0.25ms
−1
immediately
afteritscollisionwithC.
(ii) Calculatethetwopossiblevaluesofm. [6]
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OCRiscommittedtoseekingpermissiontoreproduceallthird-partycontentthatitusesinitsassessmentmaterials. OCRhasattemptedtoidentifyandcontactallcopyrightholders
whoseworkisusedinthispaper. Toavoidtheissueofdisclosureofanswer-relatedinformationtocandidates,allcopyrightacknowledgementsarereproducedintheOCRCopyright
AcknowledgementsBooklet.Thisisproducedforeachseriesofexaminationsandisfreelyavailabletodownloadfromourpublicwebsite(www.ocr.org.uk)aftertheliveexaminationseries.
IfOCRhasunwittinglyfailedtocorrectlyacknowledgeorclearanythird-partycontentinthisassessmentmaterial,OCRwillbehappytocorrectitsmistakeattheearliestpossibleopportunity.
ForqueriesorfurtherinformationpleasecontacttheCopyrightTeam,FirstFloor,9HillsRoad,CambridgeCB21GE.
OCRispartoftheCambridgeAssessmentGroup;CambridgeAssessmentisthebrandnameofUniversityofCambridgeLocalExaminationsSyndicate(UCLES),whichisitselfadepartment
oftheUniversityofCambridge.
©OCR2011 4729 Jan11
2
June 2011
1
A sledge with itsloadhasmass 70kg. Itmovesdowna slope andthe resistance tothe motionof the
sledge is 90N. The speed of the sledge is controlled by the constant tension in a light rope, which is
attachedtothe sledge and paralleltothe slope (see diagram). While travelling 20m down the slope,
thespeedofthesledgedecreasesfrom2.1ms
−1
to1.4ms
−1
anditdescendsaverticaldistanceof3m.
(i) Calculatethechangeinenergyofthesledgeanditsload. [4]
(ii) Calculatethetensionintherope. [3]
◦
2 A car of mass 1250kgtravelsalong astraight roadinclined at 2 to the horizontal. The resistance to
themotionofthecariskvN,wherevms
−1
isthespeedofthecarandk isaconstant. Thecartravels
ataconstantspeedof25ms
−1uptheslopeandtheengineofthecarworksataconstantrateof21kW.
(i) Calculatethevalueofk. [4]
(ii) Calculatetheconstantspeedofthe carona horizontalroad. [3]
3 A uniform lamina ABCDE consists of a square ACDE and an equilateral triangle ABC which are
joinedalongtheir commonedgeAC toformapentagonwhosesidesare each8cminlength.
(i) Calculatethedistance ofthecentreof massofthelaminafromAC. [5]
(ii) The lamina is freely suspended from A and hangs in equilibrium. Calculate the angle that AC
makeswiththevertical. [2]
4 TwosmallspheresAandBaremovingtowardseachotheralongastraightlineonasmoothhorizontal
surface. A has speed 3ms
−1
and B has speed 1.5ms
−1
before they collide directly. The direction of
motion of B is reversed in the collision. The speeds of A and B after the collision are 2ms
−1
and
2.9ms
−1
respectively.
(i) (a) ShowthatthedirectionofmotionofAisunchangedbythecollision. [2]
(b) Calculate thecoefficientofrestitutionbetweenAandB. [2]
ThemassofBis0.2kg.
(ii) FindthemassofA. [3]
Bcontinuestomoveat2.9ms
−1
andstrikesaverticalwallatrightangles. Thewallexertsanimpulse
ofmagnitude0.68NsonB.
(iii) CalculatethecoefficientofrestitutionbetweenBandthewall. [4]
©OCR2011 4729 Jun11
3
June 2011
5 A particle is projected with speed 7ms
−1
at an angle of elevation of 30
◦
from a point O and moves
freely under gravity. The horizontal and vertically upwards displacements of the particle from O at
anysubsequenttimetsarexmandymrespectively.
(i) Expressx andyintermsof t andhencefindtheequationofthetrajectoryofthe particle. [4]
(ii) Calculatethevaluesof xwheny= 0.6. [4]
(iii) Findthedirectionof motionofthe particlewheny =0.6andtheparticle isrising. [4]
6
0.2 m
Q
0.12 m
P
30°
Fig. 1
A container is constructed from a hollow cylindrical shell and a hollow cone which are joined along
◦
theircircumferences. Thecylindricalshellhasradius0.2m,andtheconehassemi-verticalangle30 .
Two identical small spheres P and Q move independently in horizontal circles on the smooth inner
surfaceofthecontainer(seeFig.1). Eachspherehasmass0.3kg.
(i) Pmovesinacircleofradius0.12mandisincontactwithonlytheconicalpartofthecontainer.
CalculatetheangularspeedofP. [5]
(ii)
Q
Fig. 2
Q moveswithspeed2.1ms
−1
andisincontactwithboththecylindricalandconicalsurfacesof
thecontainer(seeFig.2). Calculatethemagnitudeoftheforcewhichthecylindricalshellexerts
onthesphere. [4]
(iii) CalculatethedifferencebetweenthemechanicalenergyofPandofQ. [5]
[Question7isprintedoverleaf.]
Turnover
©OCR2011 4729 Jun11
June 2011 4
7
FN
Q
0.8 m
60°
V P
Fig. 1
◦
A uniform solid cone of height 0.8m and semi-vertical angle 60 lies with its curved surface on a
horizontal plane. The point P on the circumference of the base is in contact with the plane. V is
the vertex of the cone and PQ is a diameter of its base. The weight of the cone is 550N. A force of
magnitudeFNandlineofactionPQisappliedtothebaseofthecone(seeFig.1). Theconetopples
aboutV withoutsliding.
(i) Calculatetheleastpossiblevalueof F. [4]
T N
Q
0.8 m
60°
V P
Fig. 2
The force ofmagnitudeFN isremovedandanincreasingforceofmagnitude TNactingupwardsin
the vertical plane of symmetry of the cone and perpendicular to PQ is applied to the cone at Q (see
Fig.2). Thecoefficientoffrictionbetweenthecone andthehorizontalplaneisµ.
(ii) GiventhattheconeslidesbeforeittopplesaboutP,calculatethegreatestpossiblevalueforµ.
[10]
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OCRiscommittedtoseekingpermissiontoreproduceallthird-partycontentthatitusesinitsassessmentmaterials. OCRhasattemptedtoidentifyandcontactallcopyrightholders
whoseworkisusedinthispaper. Toavoidtheissueofdisclosureofanswer-relatedinformationtocandidates,allcopyrightacknowledgementsarereproducedintheOCRCopyright
AcknowledgementsBooklet.Thisisproducedforeachseriesofexaminationsandisfreelyavailabletodownloadfromourpublicwebsite(www.ocr.org.uk)aftertheliveexaminationseries.
IfOCRhasunwittinglyfailedtocorrectlyacknowledgeorclearanythird-partycontentinthisassessmentmaterial,OCRwillbehappytocorrectitsmistakeattheearliestpossibleopportunity.
ForqueriesorfurtherinformationpleasecontacttheCopyrightTeam,FirstFloor,9HillsRoad,CambridgeCB21GE.
OCRispartoftheCambridgeAssessmentGroup;CambridgeAssessmentisthebrandnameofUniversityofCambridgeLocalExaminationsSyndicate(UCLES),whichisitselfadepartment
oftheUniversityofCambridge.
©OCR2011 4729 Jun11
2
Jan 2012
1 A particle P is projected with speed 40 m s−1 at an angle of 35° above the horizontal from a point O. For the
instant 3 s after projection, calculate the magnitude and direction of the velocity of P. [5]
2
(cid:95) (cid:95)
r cm
Fig. 1
A child’s toy is a uniform solid consisting of a hemisphere of radius r cm joined to a cone of base radius
r cm. The curved surface of the cone makes an angle α with its base. The two shapes are joined at the plane
faces with their circumferences coinciding (see Fig. 1). The distance of the centre of mass of the toy above
the common circular plane face is x cm.
[The volume of a sphere is 4 πr3 and the volume of a cone is 1 πr2h.]
3 3
r(tan2 α − 3)
(i) Show that x = . [4]
8 + 4 tan α
The toy is placed on a horizontal surface with the hemisphere in contact with the surface. The toy is released
from rest from the position in which the common plane circular face is vertical (see Fig. 2).
Fig. 2
(ii) Find the set of values of α such that the toy moves to the upright position. [3]
© OCR 2012 4729 Jan12
3
Jan 2012
3
B
P
1.6 m
60°
A
A uniform rod AB of mass 10 kg and length 2.4 m rests with A on rough horizontal ground. The rod makes
an angle of 60° with the horizontal and is supported by a fixed smooth peg P. The distance AP is 1.6 m (see
diagram).
(i) Calculate the magnitude of the force exerted by the peg on the rod. [3]
(ii) Find the least value of the coefficient of friction between the rod and the ground needed to maintain
equilibrium. [5]
4 A particle P of mass 0.2 kg is attached to one end of a light inextensible string of length 1.2 m. The other end
of the string is fixed at a point A which is 0.6 m above a smooth horizontal table. P moves on the table in a
circular path whose centre O is vertically below A.
(i) Given that the angular speed of P is 2.5 rad s−1, find
(a) the tension in the string, [4]
(b) the normal reaction between the particle and the table. [3]
(ii) Find the greatest possible speed of P, given that the particle remains in contact with the table. [5]
5 A car of mass 1500 kg travels up a line of greatest slope of a straight road inclined at 5° to the horizontal.
The power of the car’s engine is constant and equal to 25 kW and the resistance to the motion of the car is
constant and equal to 750 N. The car passes through point A with speed 10 m s−1.
(i) Find the acceleration of the car at A. [5]
The car later passes through a point B with speed 20 m s−1. The car takes 28 s to travel from A to B.
(ii) Find the distance AB. [7]
[Questions 6 and 7 are printed overleaf.]
Turn over
© OCR 2012 4729 Jan12
Jan 2012 4
6 A small ball of mass 0.5 kg is held at a height of 3.136 m above a horizontal floor. The ball is released from
rest and rebounds from the floor. The coefficient of restitution between the ball and floor is e.
(i) Find in terms of e the speed of the ball immediately after the impact with the floor and the impulse that
the floor exerts on the ball. [4]
The ball continues to bounce until it eventually comes to rest.
(ii) Show that the time between the first bounce and the second bounce is 1.6e. [2]
(iii) Write down, in terms of e, the time between
(a) the second bounce and the third bounce,
(b) the third bounce and the fourth bounce. [2]
(iv) Given that the time from the ball being released until it comes to rest is 5 s, find the value of e. [5]
7 A particle P is projected horizontally with speed 15 m s−1 from the top of a vertical cliff. At the same instant a
particle Q is projected from the bottom of the cliff, with speed 25 m s−1 at an angle of θ° above the horizontal.
P and Q move in the same vertical plane. The height of the cliff is 60 m and the ground at the bottom of the
cliff is horizontal.
(i) Given that the particles hit the ground simultaneously, find the value of θ and find also the distance
between the points of impact with the ground. [6]
(ii) Given instead that the particles collide, find the value of θ, and determine whether Q is rising or falling
immediately before this collision. [9]
Copyright Information
OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders
whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright
Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible
opportunity.
For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a
department of the University of Cambridge.
© OCR 2012 4729 Jan12
June 2012
2
1 A particle, of mass 0.8 kg, moves along a smooth horizontal surface. It hits a vertical wall, which is at right
angles to the direction of motion of the particle, and rebounds. The speed of the particle as it hits the wall is
4 m s−1 and the coefficient of restitution between the particle and the wall is 0.3. Find
(i) the impulse that the wall exerts on the particle, [3]
(ii) the kinetic energy lost in the impact. [2]
2 A car of mass 1600 kg moves along a straight horizontal road. The resistance to the motion of the car has
constant magnitude 800 N and the car’s engine is working at a constant rate of 20 kW.
(i) Find the acceleration of the car at an instant when the car’s speed is 20 m s−1. [4]
The car now moves up a hill inclined at 4° to the horizontal. The car’s engine continues to work at 20 kW
and the magnitude of the resistance to motion remains at 800 N.
(ii) Find the greatest steady speed at which the car can move up the hill. [4]
3
4 m
Q
A B
30° 1.5 m
P
A uniform beam AB of mass 15 kg and length 4 m is freely hinged to a vertical wall at A. The beam is held in
equilibrium in a horizontal position by a light rod PQ of length 1.5 m. P is fixed to the wall vertically below
A and PQ makes an angle of 30° with the vertical (see diagram). The force exerted on the beam at Q by the
rod is in the direction PQ. Find
(i) the magnitude of the force exerted on the beam at Q, [3]
(ii) the magnitude and direction of the force exerted on the beam at A. [6]
© OCR 2012 4729 Jun12
June 2012
3
4 A boy throws a small ball at a vertical wall. The ball is thrown horizontally, from a point O, at a speed of
14.4 m s−1 and it hits the wall at a point which is 0.2 m below the level of O.
(i) Find the horizontal distance from O to the wall. [4]
The boy now moves so that he is 6 m from the wall. He throws the ball at an angle of 15° above the
horizontal. The ball again hits the wall at a point which is 0.2 m below the level from which it was thrown.
(ii) Find the speed at which the ball was thrown. [6]
5 A particle P, of mass 2 kg, is attached to fixed points A and B by light inextensible strings, each of length
2 m. A and B are 3.2 m apart with A vertically above B. The particle P moves in a horizontal circle with
centre at the mid-point of AB.
(i) Find the tension in each string when the angular speed of P is 4 rad s−1. [7]
(ii) Find the least possible speed of P. [6]
6 Three particles A, B and C are in a straight line on a smooth horizontal surface. The particles have masses
0.2 kg, 0.4 kg and 0.6 kg respectively. B is at rest. A is projected towards B with a speed of 1.8 m s−1 and
collides with B. The coefficient of restitution between A and B is 1.
3
(i) Show that the speed of B after the collision is 0.8 m s−1 and find the speed of A after the collision. [6]
C is moving with speed 0.2 m s−1 in the same direction as B. Particle B subsequently collides with C. The
coefficient of restitution between B and C is e.
(ii) Find the set of values for e such that B does not collide again with A. [7]
[Question 7 is printed overleaf.]
Turn over
© OCR 2012 4729 Jun12
June 2012
4
7
(a + 5) cm
D C
5 cm
A 5 cm B
The diagram shows the cross-section through the centre of mass of a uniform solid prism. The cross-section
is a trapezium ABCD with AB and CD perpendicular to AD. The lengths of AB and AD are each 5 cm and the
length of CD is (a + 5) cm.
(i) Show the distance of the centre of mass of the prism from AD is
a2 + 15a + 75
cm. [5]
3(a + 10)
The prism is placed with the face containing AB in contact with a horizontal surface.
(ii) Find the greatest value of a for which the prism does not topple. [3]
The prism is now placed on an inclined plane which makes an angle θ° with the horizontal. AB lies along a
line of greatest slope with B higher than A.
(iii) Using the value for a found in part (ii), and assuming the prism does not slip down the plane, find the
greatest value of θ for which the prism does not topple. [6]
Copyright Information
OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders
whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright
Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible
opportunity.
For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a
department of the University of Cambridge.
© OCR 2012 4729 Jun12
Jan 2013
2
1 A block is being pushed in a straight line along horizontal ground by a force of 18 N inclined at 15° below
the horizontal. The block moves a distance of 6 m in 5 s with constant speed. Find
(i) the work done by the force, [3]
(ii) the power with which the force is working. [2]
2 A car of mass 1500 kg travels along a straight horizontal road. The resistance to the motion of the car is
kv2 1 N, where v m s–1 is the speed of the car and k is a constant. At the instant when the engine produces a
power of 15 000 W, the car has speed 15 m s–1 and is accelerating at 0.4 m s–2.
(i) Find the value of k. [4]
It is given that the greatest steady speed of the car on this road is 30 m s–1.
(ii) Find the greatest power that the engine can produce. [3]
3 A particle A is released from rest from the top of a smooth plane, which makes an angle of 30° with the
horizontal. The particle A collides 2 s later with a particle B, which is moving up a line of greatest slope of
the plane. The coefficient of restitution between the particles is 0.4 and the speed of B immediately before
the collision is 2 m s–1. B has velocity 1 m s–1 down the plane immediately after the collision. Find
(i) the speed of A immediately after the collision, [4]
(ii) the distance A moves up the plane after the collision. [2]
The masses of A and B are 0.5 kg and m kg, respectively.
(iii) Find the value of m. [3]
4
D 6 cm C
6 cm
A B
A uniform square lamina ABCD of side 6 cm has a semicircular piece, with AB as diameter, removed (see
diagram).
(i) Find the distance of the centre of mass of the remaining shape from CD. [6]
The remaining shape is suspended from a fixed point by a string attached at C and hangs in equilibrium.
(ii) Find the angle between CD and the vertical. [2]
© OCR 2013 4729/01 Jan13
Jan 2013 3
5
B
1.5 m
P
2.5 m
60°
A
A uniform rod AB, of mass 3 kg and length 4 m, is in limiting equilibrium with A on rough horizontal
ground. The rod is at an angle of 60° to the horizontal and is supported by a small smooth peg P, such that
the distance AP is 2.5 m (see diagram). Find
(i) the force acting on the rod at P, [3]
(ii) the coefficient of friction between the ground and the rod. [5]
6 A particle of mass 0.5 kg is held at rest at a point P, which is at the bottom of an inclined plane. The particle
is given an impulse of 1.8 N s directed up a line of greatest slope of the plane.
(i) Find the speed at which the particle starts to move. [2]
The particle subsequently moves up the plane to a point Q, which is 0.3 m above the level of P.
(ii) Given that the plane is smooth, find the speed of the particle at Q. [4]
It is given instead that the plane is rough. The particle is now projected up the plane from P with initial
speed 3 m s–1, and comes to rest at a point R which is 0.2 m above the level of P.
(iii) Given that the plane is inclined at 30° to the horizontal, find the magnitude of the frictional force on the
particle. [4]
© OCR 2013 4729/01 Jan13 Turn over
6 m
4 | m
June 2013
2
1 A and B are two points on a line of greatest slope of a smooth inclined plane, with B a vertical distance of
8 m below the level of A. A particle of mass 0.75 kg is projected down the plane from A with a speed of
2 m s−1. Find
(i) the loss in potential energy of the particle as it moves from A to B, [2]
(ii) the speed of the particle when it reaches B. [4]
2 The power developed by the engine of a car as it travels at a constant speed of 32 m s−1 on a horizontal road
is 20 kW.
(i) Calculate the resistance to the motion of the car. [3]
The car, of mass 1500 kg, now travels down a straight road inclined at 2° to the horizontal. The resistance to
the motion of the car is unchanged.
(ii) Find the power produced by the engine of the car when the car has speed 32 m s−1 and is accelerating at
0.1 m s−2. [4]
3
A
C
θ°
4
c
m
P N
B
A uniform semicircular arc ACB is freely pivoted at A. The arc has mass 0.3 kg and is held in equilibrium by
a force of magnitude P N applied at B. The line of action of this force lies in the same plane as the arc, and is
perpendicular to AB. The diameter AB has length 4 cm and makes an angle of i° with the downward vertical
(see diagram).
(i) Given that i= 0, find the magnitude of the force acting on the arc at A. [6]
(ii) Given instead that i= 30, find the value of P. [4]
© OCR 2013 4729/01 Jun13
June 2013
3
4 A solid uniform cone has height 8 cm, base radius 5 cm and mass 4 kg. A uniform conical shell has height
10 cm, base radius 5 cm and mass 0.4 kg. The two shapes are joined together so that the circumferences of
their circular bases coincide.
(i) Find the distance of the centre of mass of the shape from the common circular base. [4]
8 cm 10 cm
The object is suspended with a string attached to the vertex of the cone and another string attached to the
vertex of the conical shell. The object is in equilibrium with the strings vertical and the axis of symmetry of
the object horizontal (see diagram).
(ii) Find the tension in each string. [4]
© OCR 2013 4729/01 Jun13 Turn over
June 2013
4
5 A vertical hollow cylinder of radius 0.4 m is rotating about its axis. A particle P is in contact with the
rough inner surface of the cylinder. The cylinder and P rotate with the same constant angular speed. The
coefficient of friction between P and the cylinder is n.
(i) Given that the angular speed of the cylinder is 7 rad s−1 and P is on the point of moving downwards,
find the value of n. [5]
The particle is now attached to one end of a light inextensible string of length 0.5 m. The other end is fixed
to a point A on the axis of the cylinder (see diagram).
A
0.5 m
P
0.4 m
(ii) Find the angular speed for which the contact force between P and the cylinder becomes zero. [5]
6
4 m s–1 u m s–1
A B
0.2 kg m kg
The masses of two particles A and B are 0.2 kg and m kg respectively. The particles are moving with constant
speeds 4 m s−1 and u m s−1 in the same horizontal line and in the same direction (see diagram). The two
particles collide and the coefficient of restitution between the particles is e. After the collision, A and B
continue in the same direction with speeds 4(1- e+ e 2 ) m s−1 and 4 m s−1 respectively.
(i) Find u and m in terms of e. [6]
(ii) Find the value of e for which the speed of A after the collision is least and find, in this case, the total
loss in kinetic energy due to the collision. [5]
(iii) Find the possible values of e for which the magnitude of the impulse that B exerts on A is 0.192 N s.
[4]
© OCR 2013 4729/01 Jun13
June 2013
5
7
B
u m s–1
P
θ° 70°
O A
The diagram shows a surface consisting of a horizontal part OA and a plane AB inclined at an angle of 70°
to the horizontal. A particle is projected from the point O with speed u m s−1 at an angle of i° above the
horizontal OA. The particle hits the plane AB at the point P, with speed 14 m s−1 and at right angles to the
plane, 1.4 s after projection.
(i) Show that the value of u is 15.9, correct to 3 significant figures, and find the value of i. [7]
(ii) Find the height of P above the level of A. [3]
The particle rebounds with speed v m s−1. The particle next lands at A.
(iii) Find the value of v. [5]
(iv) Find the coefficient of restitution between the particle and the plane at P. [1]
© OCR 2013 4729/01 Jun13
June 2014
2
1 A football is kicked from horizontal ground with speed 20 m s−1 at an angle of i° above the horizontal. The
greatest height the football reaches above ground level is 2.44 m. By modelling the football as a particle and
ignoring air resistance, find
(i) the value of i, [2]
(ii) the range of the football. [2]
2 A uniform solid cylinder of height 12 cm and radius r cm is in equilibrium on a rough inclined plane with
one of its circular faces in contact with the plane.
(i) The cylinder is on the point of toppling when the angle of inclination of the plane to the horizontal is
21°. Find r. [3]
The cylinder is now placed on a different inclined plane with one of its circular faces in contact with the
plane. This plane is also inclined at 21° to the horizontal. The coefficient of friction between this plane and
the cylinder is µ.
(ii) The cylinder slides down this plane but does not topple. Find an inequality for µ. [2]
3
C
13 cm
B D
8 cm
A 10 cm E
A uniform lamina ABCDE consists of a rectangle ABDE and an isosceles triangle BCD joined along their
common edge. AB=DE =8cm , AE=BD=10cm and BC =CD=13cm (see diagram).
(i) Find the distance of the centre of mass of the lamina from AE. [5]
(ii) The lamina is freely suspended from B and hangs in equilibrium. Calculate the angle that BD makes
with the vertical. [3]
© OCR 2014 4729/01 Jun14
June 2014
3
4
R
θ
P Q
6 cm 14 cm
A uniform rod PQ has weight 18 N and length 20 cm. The end P rests against a rough vertical wall. A
particle of weight 3 N is attached to the rod at a point 6 cm from P. The rod is held in a horizontal position,
perpendicular to the wall, by a light inextensible string attached to the rod at Q and to a point R on the wall
vertically above P, as shown in the diagram. The string is inclined at an angle i to the horizontal, where
sin i= 3 . The system is in limiting equilibrium.
5
(i) Find the tension in the string. [3]
(ii) Find the magnitude of the force exerted by the wall on the rod. [4]
(iii) Find the coefficient of friction between the wall and the rod. [2]
5 (i) A car of mass 800 kg is moving at a constant speed of 20 m s −1 on a straight road down a hill inclined
at an angle a to the horizontal. The engine of the car works at a constant rate of 10 kW and there is a
resistance to motion of 1300 N. Show that sin a = 5 . [4]
49
(ii) The car now travels up the same hill and its engine now works at a constant rate of 20 kW. The resistance
to motion remains 1300 N. The car starts from rest and its speed is 8 m s−1 after it has travelled a
distance of 22.1 m. Calculate the time taken by the car to travel this distance. [5]
6 Two small spheres A and B, of masses 2m kg and 3m kg respectively, are moving in opposite directions
along the same straight line towards each other on a smooth horizontal surface. A has speed 4 m s−1 and
B has speed 2 m s−1 before they collide. The coefficient of restitution between A and B is 0.4.
(i) Find the speed of each sphere after the collision. [6]
(ii) Find, in terms of m, the loss of kinetic energy during the collision. [4]
(iii) Given that the magnitude of the impulse exerted on A by B during the collision is 2.52 N s, find m. [3]
© OCR 2014 4729/01 Jun14 Turn over
June 2014
4
7
A
30°
B
45°
C 0.5 m P
A small smooth ring P of mass 0.4 kg is threaded onto a light inextensible string fixed at A and B as shown
in the diagram, with A vertically above B. The string is inclined to the vertical at angles of 30° and 45° at A
and B respectively. P moves in a horizontal circle of radius 0.5 m about a point C vertically below B.
(i) Calculate the tension in the string. [3]
(ii) Calculate the speed of P. [3]
The end of the string at B is moved so both ends of the string are now fixed at A.
(iii) Show that, when the string is taut, AP is now 0.854 m correct to 3 significant figures. [2]
P moves in a horizontal circle with angular speed 3.46 rad s
−1.
(iv) Find the tension in the string and the angle that the string now makes with the vertical. [4]
8 A child is trying to throw a small stone to hit a target painted on a vertical wall. The child and the wall are
on horizontal ground. The child is standing a horizontal distance of 8 m from the base of the wall. The child
throws the stone from a height of 1 m with speed 12 m s−1 at an angle of 20° above the horizontal.
(i) Find the direction of motion of the stone when it hits the wall. [6]
The child now throws the stone with a speed of V m s −1 from the same initial position and still at an angle of
20° above the horizontal. This time the stone hits the target which is 2.5 m above the ground.
(ii) Find V. [6]
Copyright Information
OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders
whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright
Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible
opportunity.
For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a
department of the University of Cambridge.
© OCR 2014 4729/01 Jun14