N-ame of student;

Physics Teacher:

Physics A OCR

A2 Syllabus H55B replacing 7BB3

Module: G484 The l{ewtonian World

Topic: 4.2.1 Circular Motion

1.2"2 Gravitational Fields

o The se que stions have been taken lrom recent papers up to 20A9,

based on specification 3883.

. The o1d specification is a close but not perfect match and someparts of the new specification are not covered.

. Remember a separate fornuTaldata sheet is available in theexamination. You may need to look at this to answer thesequestions

eFig. 3.1 shows a i-otating fairground ride where a seat S of mass m is suspended by a light chain.\,Vhen the ride rotates at a constant speed v,, the chain makes an angie 6 with the veriical so thaithe seat is a distance rfrom the axis of rotation.

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itritr

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rtt\l\I\I\l\

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= -n*\\*t *.; I !-i

. /r.J '.\; i l\+-ly

Fig.3.1

(ai (i) On Fig. 3.1 draw and label arrows to represent the forces acting on the seat. l2l

(ii) By referring to the forces in (i), explain the condition necessary for the seat to move in ahorizontal circle.

n\\1.\V\ -J

...f......:-.".-....-...t.E,-...$....[......i...i..i.........,-..e.....:*.................'.1.....+...........i.....'.:'....+..........,..

. ! '-l.\ i '

......,:'.t-.-,i.;......F...,,.n......\...J..i..,.. . ............121{. \./(iii) Write do,+;n an algebraic expression for the magnitude Fof the resultant force on the seai

in terms of m, r and v.r:\, * --t i \n - ,r \l l-\l

?

r llilil lilit ililt ilril lllll lllli lllll lllll lllll llil llllrol I TEOO^O*

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3(b) (i) When the ride rotates, the seat is travelling in a circle of radius 5.0 m at a constant speed

of 4.2m s-1. Show that the angle d is about 20'.

Tstng = d (l)f \-/

fqo'g € a3 O

Tsi1(}- = f"*9 =-----4Tcs) @

erfiq

LJ

0 "* 3 @'.)'- o.S r-J-t t

o36fx

4} = I ?.B' AO fr,..qr{C f"- C.U*<ll4l

(ii) When a child occupies the seat during a ride al4.2ms-1, will the angle gremain at 20"

[Total: i0]

over

r firil ilil iltil lllll illilllil illl lllil lllll lill llil-841 758209.

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g{b} When the drum is rotated at one particular speed, a metal side panel of the machine casing

vibrates loudly. Explain why this happens.

/?\

,'.,,',',12]

(c) A fault develops in the motor, causing the coil to stop rotating. Magneiic flux from theelectromagnet of the motor still links with the now stationary coil. Fig.3.2 shows how the fluxlinkage of the coil varies with time.

DJ

flux linkage / oWb iurns

1

0

-l

_Z

n-J

Fig.3.2

(i) Using Fig. 3.2 staie a time at which the e.m.f. induced across the ends of the coii is

1 zeroa\

'....-.:...r.,."..,1..':....'.;r....... mS tfit

' i.,r'j"'

2 a maximum. . ..:",l.fr ..8.':'....d..-,-....,,..:.. *r Ui t21

(ii) Use the graph of Fig. 3.2 to calculate the peak value of the e.m.f. across the ends of thecoil.

:,i{i

.. :i

,'.--l

(. ".

ilffiffi llffi ililr ffill lffitffiilll lllll ilill ffiilr llil llil-959752309.

ra (a)

(b)

€

Explain why the acceleration due to gravity and the graviiational field strength at thesurface have the same value"

''{i !1*"!__

ti,,,,,*...=..1......i...'."',.r".".i-*..."..."f....;....,...\".J.........",,g."":..,..t."'.r.1.".*:."...,,1.+1.,....:.:.....-....'!,.

...j'.''x'...'.+..i.....-:h...;'....'..'..l.'.*"1'.'..t.',,,'.".".i":.'."*.."!:..:'",...''.(t). / \:_/A space probe, with its engines shut down, orbits Mars'at a constant distance of 3500 kmabove the centre of the planet in a time of 1 10 minutes"

ti) Calculate the speed of the space probe.

*tg''

(c)

d

I i."\y u

q tt- f{--. -(i

(

t\,. ::"'e,+r**''_ rir"

3

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F--L,

frc f*ntt- Fl4l

I ffiilt lliil ililt ilil ilil ffit llilt llili iliil lllt ifii

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(d) (i) Write down an algebraic expression tor g at the surface of a planet in terms of its mass Mand radius fr.

t1l

(ii) The acceleration due to gravity at the surface of Mars is 3.7 m s-2. Calculate the radius ofMars, in kilometres.

ri:- : {.r-l tc.*jbtc-t'r.. S.-}}.tfait:

* t. "'"q-1 f

,/: , . \tt , r r 4 )t; - v,k[]tc F_,-l[r;-F1 r I te

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I+ (a)

E

Define gravitationalfield strength at a point in a gravitationalfieid.

1 ffixn tql osmc$vJ.. 0rr. .u n*!... rnass(nk I-h^Ak pdM. l:.*u

spftcs'ln*bgA s.-l.1&icr* $ ElpianefJ- O .r1I

The gravitational field strength at the surface of a planet of uniform density is 40N kg-1.

A satellite of mass 1500 kg is launched from the sudace into a circular orbit around theplanet at a height of 1.5 x '105m with an orbital period of 4.5 x '103s.

The radius of the planet is 2.0 x 107m.

{i} Estimate the increase in potential energy of ihe satellite.

"U\ = /Sco n /Fo F /.5-Frcf O;- ?.o F /CIq

potentiat enersy = 9.:.Qf.(Pjt ttl

{iii Suggest with a reason whether your estimate in {i} is likely toie larger- or smallerthan the true value.

: llae*r- *$ S d{sr*nncA ,^ftrr= hs81* O

calculate the kinetic energy of the satellite in orbit. 6cfAa OAJ tJeJ

, ,f ldqu^n= &n( A= &rr h (a.o X/o"? t /.5r.ror) lcxc 5c{'4,r'

T- U *#',;- O--,,{filL

: J- I r fOY "^ as-[a*'iqt

f\nv\ e.sxtog?D

(b)

{iii)

v

E= L2 ,!ntsoo r

s-q F(o'r

kinetic enersy = .fft.nJ.A:...J t4l

ForExaminefs

Use

q

(c) Fig. 2.1shows how the gravitationat field strength

centre of the planet of radius 2'0 x 107 m'g varles with distance r from the

tg-.-..-9.::'"'

g/N kg-1

Fig- 2-1

(!) Use Fig' 2.1 to write down the value of g at a height of 4'0 x

surface.

107m above the

(r)\J=5..........i'l kg-1 [1]

-':\ -a-, ;: .ir' i:

(iii

(iii)

write down an algebraic expression for g at a distance r from ihe centre of the

planet. The planet can ne treated u, u p"oini mass of magnitu de M situated at its

The value of g at the surface is 40 N kg-1. Use this information and your answer to

(ii) to check, by a suitable calculation' your answer to (i)'

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[t-otal: 12]

For

ExaninefsUse

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(e

ln*a distant galaxy, the planet Odyssey is orbited by two small moons Scylla and Charybdis,labeiled 0, S, Frespectively in Fig. 4.1. The distances of the moons from the centre of the planetare Sfr anci 4R, where R is the radius of the planet.

(a)

(b)

Fig.4.1

Draw a gravitational fieid line of the planet passing through moon S. t1l

The radius R of the planet is 2.0x107m. The gravitational field strength g at its surface is40 N kg-1.

{i} Write down a formula for the gravitational field strength g at the surJace of the planet ofmass M.

-+, = qYIfR- O. .. . .rJ...:............."t..!.\......Y- ........t11

(ii) Use the data above to show that the gravitational field strength at S is 1 .6 N kg-1.

$r=qI ;]: *= {rY\\-' S)' €

u

l. a f{ L*. -t

no nnhrt- p'"d-nt u.r4-

@ arnu hura^d

te-,\[Fq 3 P ttu\of

Js L5R)'a?" t€) i qt'r

t'l >9-4oeJ

: q'0' n-2-}SKQr : 4Grt'=\r Js€'

aert-{i*-'?*

J3(iii) Show that the gravitational field strength at G is 2.5 N kg-t.

}: = $:-, *5* ; '*@ = ..e-'r r.llg

g, $ r)- Lr

i e nc *1-c(?r 6-ASq<r

r ffiilr ililr ililt ililt ililt ilffi ililt ilIilIilil ilil ilil

iI

rl(iv) Using an average value of g, estimate the increase AEin gravitational pcrential energy of

a small space vehicle of mass 3.0 x 103 kg when it moves-from the orbit of C to the orbitof S.

Owsaga 3 - &"S + t.b : &.<)S AAEe = fn$*,Ah

)-

= 3.C t\tt}3 h: l-:-?. t(3't

2.Ds x J'onlo-'}Od^--{a

o^E-

. . .!:l.I, r.g.]: .....J r3l

(c) Calculate the orbital period of S. Assume that the gravitational effects of the two moons oneach other are negligible in comparison to the gravitational force of o.

gravitational field strength at s= 1.6Nkg-r 3 = ca-nl'*i t:4-ha.l Aci'adius of orbit = 1 .0 x 1 0B m

q = vLdr ? 4n'(se/T[= {.!sf)-5-€ A *'rL

i^ == tI:(Lta)\.' = .Qfi li**)

t{'i

J ?*--,

: j +n L( IFa)

34'?'l r,o Y

period= . .. .5:.O...L.1.9.1....... - r+r

[Total:12]

n over

r ffiiil ilil fiilt ililt ililt illil ilfit tilt ililt tilt ilil

(+'

iLThis question is about gravitational fields. You may assume that all the mass of the Earth,or the Moon, can be considered as a point-mass at its centre.

(a) lt is possible to find the mass of a planet by measuring the gravitational field strengthat the sudace of ihe planet and knowing its radius.

{i) Define gravitationalfield strength, g.

: S{.t"ltlf,*H.}'.'.ru*[ , k ,""*....i:],.k*'1...*I,..+sf ....f.t.s1;**...\."f.........

. . .i, *-s F.*ll tf. .; .U).t jl L""""""' """"" til

{ii} Write down an expression for g at the surface of a planet in terms of its mass Mand radius R.

t1l

(iii) Show that the mass of the Earth is 6.0 x 1024kg.

radius of the Ear{h = 6400 km

.r - '-{- " ."r.'rltl-tq-

qe *1f;'a Ftl> \-/t- q,.

y, = '-'i'F I F \b.'b*t r.tf1"' '= ; ':"- F l: * "f*'

'-r *!r n0 n\alL {*"€-r{a tFf,:*ll Oj$r'uref' t1l

(b) (i) Use the data below to show the value of g at the Moon's surface is about1 .7 N kg-1.

mass of Eadh = 81 x mass of Moonradius of Earth = 3.7 x radius of Moon

:. -'A"t ( '-(

u\ti"\-' :-1}tr.\ *t

q. ^.{,qt

;: -t-''-l:: O*' r^L

A

\

it{

0&-

I

0l2l

= "i"tt n'f,1

? t-Lc"t 1' t

*l":' I\/t:?"

f\.-'l'-rtl F l'*t\,/'

,t*tl"^: h t:.^

i

ForExaminels

Use

I

I

i3

(ii) Explain why a high jumper who can clear a 2 m bar on Earth should be abie toclear a 7 rn bar on the Moon. Assume that the high jump on the Moon is inside a'space bubble' where Earth's atmospheric conditions exist.

..1..&l-.rg,-s,l*....t1..T.:':.....;..b...-r-...-.......--.s.l',*.*-..-::j.....:*j'&{d,,.h*.'-.!'.i.-}.-.'3.-:,}*.:i-:>

that the Moon moves in a circular orbit about the centre of the Earth. Estimate theperiod of this orbit to the nearest day.

mass of Earth = 6.0 x 1024kg1 dav -= 8.6 x 1oa s

n ) /-.\I : ,*W- .Yt (,*t- r -\'/\ *i

--. k'aA.

-- 'r l{ :"--

l*^vI -+{l i-3:

!

ForExaminer's

Use

o

tu110," -q.\ssl..k!:$1.!$s.l $e*e;r..in...{:c4o....i.:.1r.....'C. . ...... .. . ... t3l

a/M,- ,$' U, @ {;* RrQ'3/ t.> = /r 'Sotcl "ifiJoGt"A"" UetweEi the\centres of the t'artti a'no the Moon is 3.8 x 108m. Assume

I

no* E ,rlt*l'Q.r_' t-rr

* , ,-S g.

t *,r * '*d-'i - f't.""

gglr._o".1^_

.,"* p04f

*.*q9eA

-t-fF*F t d ": F{r*

*, '-.. rr ' qi

rs.t d fuf "= tt/t o 'r* 5f ^-

o

t -r e1-a , f l'*-* .'

t-? +f rr-'r)

CD O)

f". orYi " rh";utrr&

Q cactecl

[t'otal:13]

2824 JanA3 [Turn over

.300281 06-

I ilflll lllll ffiil ErtiI iltEt iilil itSti ffiilt Fiil itHtrLf

A binary star is a pair of stars which move in circular orbits around their common centre ofmass. For stars of equal mass, ihey move in the same circular orbit, shown by the dotiedline in Fig.2.1. ln this question, consider the stars to be point masses situated at theircentres at opposite ends of a diameter of the orbit.

Ctf,lnrx

;4

(a) (i)

{ii)

,..r,,..r*,....,..-......S...... O................ i2l

(b) Newton's law of gravitation applied to the situation of Fig. 2.1 may be expressed as

GM2

ffiFar

Examiner's

Use

0

-30028107-

I llffit ttilffiilt lilll ilill lllll lllil lffi llil llil,>

(c) (i) Show that the orbital period f of each star is related to its speed vby v = 2nR/7.

vt-&v ? &..( r- e*@*€*: !.

-7 T*Pou,,,

(ii) Show that the magnitude of the centripetal force required to keep each starmoving in its circular Path is

,_ 4n2MR

7z

F= Su" I(\

V=ry9 JeaIt

q" p= Ho

= -t1r+nift/snR \L\ ?"_/ E

TL 8-

F- qHL--+R'

(iii) Use equations from (b) and (ii) above to show that the mass of each star is given

by

g3M = 16n2

GT2

,::, +i7 zHA['{ =*yz

,-,--"t

-+3t--r I

1 day = 86400s

l't : 16 r nzr (sxo)?

= ry:"y97z^

*9n Y:'T?- q

t2)

l2l

H

(d) Binary stars separated by a distance of 1 x 1011 m have been observed with an orbital

period of 100 days. Calculate the mass of each star.

{rfd,lu.lr : 1319"= srp'oz

6,6-15{o-"t< [otr

a 3 -?b x{O 3'

n 66,{r")'O

&frto' t*

mass = 4.:.9-L(P- kg t2l

[Total: 13]

[Turn over

ffiFor

Examiner's

Use

J

ro(t

i&

This question is about orbits around the Sun.

(a) The gravitational force of the Sun, mass M provides the centripetal force which holds theEarth in a near circular orbit of radius F.

By considering the Earth as an isolated planet moving in a circular orbit show that its speed

v is given by the equation u = rp- .

F = i-i$-r,* (D f. ,-nu3

Rt€

"n;1 = vbr"-

gQV

n;*h-t,4Y tni'{*

{'** dNard: h'-S\t",

i'{"c\Ne"'\t1q\kr**., 'fl

.u !4q \dtt!-;ufr

i3I

(b) A space observatory to monitor activity on the surface of the Sun has been placed in a circularorbit, which is 1% smaller than the orbit of the Earth, as shown in Fig. 3.1 .

AA

)

"r^""/ iobservatory !

Earth's orbit

not to scaleI

Fig.3.1

Explain why the equation of part (a) predicts that the observatory should orbit the Sun in lessthan one year.

: ,ftf..m*Uo in L*)....*N,:*n.:l&. iJ -'r; r*re5&8,...g..q..il...i.$... *t"**A{S.[

ilq:-=.r O.'+

.1..*..t.....r.L *.**$l**$ {s,*;tr*.....*.+s.p-*-il....{..i.u....i.e.*e**:..r]t' .i]..,........

-:;;; ;;;;,; \; :.-"'.u*:n il: r; ;;*;,:; 0, : ,;,C.*,.-\ Gl\4.oc'-f .,-'-r- &-ct, .*,ro i" i^-c--'+*i=J \

I lilll ililt Iilil tilfl lllil ilffi ilil ililt lil] ilil lilt" 912303808 *

fro"rroo,

t-l(c) Fig.3.2 shows the special case where the Earth and observatory are positioned so that

orbit the Sun in exactl!, one year.

observatorynot to scale

fboth

F.

(i i)

(iii)

i9.3.2

(i) Explain why in this special ease the speed of the observatory must be less than thespeed of the Earth.

::t\^n-...orhih crrc*rq$ars*....i>...1*ss *^A^ thnL .""t L--kK =" v rn\lst pd te$ '[a,. :h$- $Ar$s e-rbilal jp-r"a (D

-An\".Ssld P*,t -tn^r.nr.o( Sun...i.r.. .snnsil,q,r

Y*ft:iA}h*s,ry Qez\nh OI-QJ*s,N.rr*".\'.s. .R4* I .$m-s\ar.\ \o... qaro.shqful^cr.' qf1

:so s*_il[ -d-.$Is$+p^ 8r*"d | ***nrt* S : 13]

$osilrto"r'r^4 a0v\qJ, lo plc-t( up $b'^AJ.y [rotar:l']t*-- doso-n,"hti tD

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tg

q (a) Fig.2.1 shows a graph of the variation of the gravitational field strength g of the Earthwith distance rfrom its centre.

r/ 106m

20.0

g/N kg'

Fig.2.1

(i) Define gravitationalfield sirength at a point.f\

: fnrcs Sss u.tur,t l$\9l$$ fuf *^qy- Foinf)

(ii)

(iii)

Write down an algebraic expression for the gravitational fieldsurface of the Earth in terms of its mass M, ils radius Fgravitational constant G.

$= -F. o

l{:

strength g at theand the universal

ht= 5 -?11 r r-o"tfCI rnaitc ftr &Nuua

(iv)

5.0 F

tet

Calculate the two distances from the centre of the Earth at which g = 0.098 N kg-t.Explain how you arrived at your answers.

qd-.L'' f')-

q-+ S so<-J

tao

5-+g

e+Jtoo.

SO n --+ \oror.s (oDt"= loora

"' --"- " " :" " " " " " " " "'l2l

(b) A spacecraft on a journey from the Earth to the Moon feels no resultant gravitational pullfrom the Earth and the Moon when it has travelled to a point 0.9 of the distance betweentheir centres. Calculate the mass of the Moon, using the value for the mass of the Earthin a(iii).

-,omass = ....7:.!f..XlA::....... kg tsl

flotal: 12]

9Is = t$* fr\R,' ff \J He- = t1*

4?3 R=?xR..

Ka6LI

b4.r,r

=b\

. l-q

= \g . o'" \q-€4*- 6l

= 7-+ x

It{.=$lx$^*o

Lr.to

[Turn over

For

ExaninefsUse

\sI O Fig.2.1 shows a ball at rest, hanging on a vertical thread from a fixed support, S.

O Vettcc.I ortor''rri rr opoa'\a-

dl:,qe-hc\^:

Fig.2.1

0 \CIlCIdlrsc( T/*$;*%\ U--r'

(a) On Fig.2.1 draw and labelarrows to represent the two forces acting on the ball. t2l

(b) Fig. 2.2 represents the ball moving in a circle about a vertical axis through S. OnFig.2.2 draw and label anows to represent the two forces acting on the ball. Explainhow they provide the force to make the ball move in a circular path.

S

ia

f

{c) The ball has a mass of 0.O20kg and moves in a circle of radius 0.050m at1.2 revolutions per second. Assume that the thread supporting the ball has negligiblemass. Calculate

(i)

V:the speed of the ball

C , Rrsuthanb{o*s*tveq sur-q ?}lensmrwe^(lrt lf^-rr\.,.s.,.'o...ho,.'}se.i{ilo..:L :. ( onm. o*ov..n I. *msinn. . o.{rvt d{.r., hori.*mdnarl. {nr"O . ...rle^k d.

-len sino... p.fa\r.t d*'r h.orgrrdmrt. {.orcAl"rl .. .. ...

hr^rqsrA cc^M. t.cjscJ'*.CI .... rsl

(ii) the magnitr4le of the force which keeps the ball moving in a circular path.

F = EI' $ og2g x o-3-F( = o'c,s J-f o'o so U

rorce =......g:.4..T.LO......* I3l

(d) Predict and explain the difference in the path of the ball when it is rotating at a higher

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I

l

I

I

l

r

l

Fs

0t,Usa

{erriJ0J ShO\^lA.

rY.s

Fig.2,2

l2l

fl-otal : 131

ffurn over