g484 Physics Newtonian World Notes Ocr Robbie Peck
g484 Physics Newtonian World Notes Ocr Robbie Peck
g484 Physics Newtonian World Notes Ocr Robbie Peck
b
= )*x 1+2 = @*2gms
-1
0
a
= )*x*+@ + ?*xv = 3* + ?*v2gms
-1
v=3=? ms
-1
*+)$)*x1*
-3
'$1+2'
2
= *+*3@A
*+)$)*x1*
-3
'$*+@'
2
+ *+)$)*x1*
-3
'$*+()'
2
= *+*2A
*+*3@ B *+*2 C collision is inelastic as momentum is conse%ve# by 2inetic ene%gy
isn6t+
Ci%cula% Motion
Angular displacement$ speed$ acceleration and intro to centripetal force
8he angular displacement is the angle in %a#ians %otate# by the %a#ius DE f%om
some %efe%ence osition+ 8his is #enote# by F+
# F=#t = Angular Speed = +
!en #G=#t = #
2
F=#t
2
= Angular Acceleration
2e kno3 s = r 4
5s,5t = r*54,5t+
v & r+& 2H%8 2(ere v is spee5 !n5 !ngu#!r spee5 !n5 s !rc
#engt(
8he osition vecto% at a given time is!
r = %cos$Gt'i + %sin$Gt'j
v = -%Gsin$Gt'i + %Gcos$Gt'j
a = -%G
2
cos$Gt'i +-%G
2
sin$Gt'j
4y fin#ing lal0 we fin# that a & r+
,
an# f%om above v=%G0 so
:o fo% a a%ticle moving in a ci%cle of %a#ius % with unifo%m angula% see# G0
8he velocity is tangential to the ci%cle
8he speed a%oun# the ci%cle is v&r+& 2H%8
8he acceleration acts towa%#s the centre of the ci%cle+
8he magnitu#e of the acceleration is r+
,
o% v
,
-r!nn
:ince "=ma0 the %esultant fo%ce is acting towa%#s the cent%e0 the centripedal
force. 8his coul# be gravity0 tension0 friction etc+ F & ma & mv
,
-r
Examle!
1 metal #isc of #iamete% 12cm is sun at a %ate of ?,* %evolutions e% minute+
Iesc%ibe how the see# of a oint on a #isc #een#s on %+ "in# the maximum
see#+
v = r C v J r
v = 6x10
-2
x 16H K3ms
-1
----------------------------------------------------------------------------------------------------------------
-------------------
"in# the o%bital see# an# cent%ietal accele%ation of the Ea%th as it o%bits %oun#
the :un at a mean #istance of 1+) x 1*
11
m+
v = 2H%8 = 2 x i x 1+)x1*
11
= 3@)+2)x2?x@*x@* = 2>,@)ms
-1
a = v
2
=% = $2>,@)
2
'=$1+)x1*
11
' = )+>x1*
-3
ms
-2
--------------------------------------------------------------------------------------------------------------------------
---------------------
)*g bung attache# to %oe an# swung in ho%iLontal ci%cle of %a#ius 3*cm+ C%oss-
sectional a%ea of ?x1*
-,
m
2
an# b%ea2ing st%ess )x1*
>
Ea+ :ee# inc%ease# until sna+
"in# max see#+
/= ",! re7uire5 centripet!# "orce = 5x10
9
x *-x10
-8
+= 200N
F = mv
2
,r =% 200= 0.05*v
2
+,*30x10
-2
+=% v = 3-.6-... ms
-1
.ravitational fields
8he mass of an ob<ect c%eates a g%avitational fiel# in the sace a%oun#
the ob<ect+ Can be mae# out using fiel# lines+ 8he #i%ection of the fiel#
is the #i%ection in which the small mass woul# move+
Newton6s law of g%avitation states wo point masses attract eac! ot!er wit! a
force directly proportional to t!e product of masses and inversely
proportional to t!e s"uare of t!eir separation.
"o%ce - %o#uct of masses
"o%ce - :ea%ation
-2
" - Mm=%
2
an# using the M%avitational constant0 M $@+@(x1*
-11
Nm
2
2g
-2
'
F & /.(m-r
,
Examle!
1 ?**2g satellite o%bits the Ea%th at a #istance of 1+)x1*
?
2m f%om the cent%e of
the Ea%th+ 8he mass of the Ea%th is @x1*
2?
2g+ "in# the g%avitational fo%ce acting
on the satellite an# the accele%ation an# o%bital see# of the satellite+
F = -@+@(3** N 1*
-11
x?**x@x1*
2?
=$1+)x1*
?
'
2
= (1*N
"=mv
2
=% so (1* = ?**xv
2
=1+)x1*
?
s v =1
"=ma so (1* = ?**a so a=1+(() ms
-2
a= vO2=% s 1+(() = v
2
=1+)x1*
?
=1@3ms
-1
.ravitational Field Strengt! at a oint in sace is the g%avitational fo%ce
exe%ience# e% unit mass on a small ob<ect lace# at that oint+
g & F-m 0 this is a mo%e gene%al fo%m of P=mg
1lying this to the g%avitational fiel# st%ength of a she%ical ob<ect we get
g & 0/.(m-r
,
1-m
g& /.(-r
,
Qsing these e5uations0 it is ossible to #e%ive any othe% e5uations you nee#!
Now it is also ossible to #e%ive
Phe%e 8 is the o%bital e%io#0 M the lanet mass0 M the
constant an# % the %a#ius+
1n# Rele%6s thi%# law states !e s"uare of t!e period of a planet is directly
proportional to t!e cube of its distance from t!e sun. i+e+ 8
2
- %
3
+
"o% communications0 satellites a%e useful in geostationary o%bit+ !e period of t!e
satellite is 2 day and is only possible above t!e e"uator.
Qsing the above e5uation0 the only un2nown is %+ 8he%efo%e it is easy to wo%2 out % to
be a%oximately 3)2**2m above the su%face of the Ea%th+
Examle!
:atu%n is a giant gaseous lanet with su%face g%avitational fiel# st%ength at its
su%face of 1*+?N2g
-1
+ 8he %a#ius of the lanet is @+*?x1*
(
2m+ Iete%mine the
mass an# mean #ensity of the lanet+
" = -MM=%
2
so 1*+? = --@+@(3** N 1*
-11
x M=$@+*?x1*
(
'
2
=S m=)+@>x1*
2@
2g
p=m=v = )+@>x1*
2@
=$?=3 x i $@+*?x1*
(
'
3
' = @1@+) 2gm
-3
----------------------------------------------------------------------------------------------------------------
------------------
Auite% has moons Eu%oa an# Callisto+ Eu%oa o%bits at #istance @+(x1*
)
2m
f%om Auite%6s cent%e with e%io# of 3+)) #ays an# Callisto o%bits at #istance 5
with o%bital e%io# 1@+( #ays+
8sing 9ep#er:s t(ir5 #!3 ;
2
Jr
3
so ;
2
=kr
3
so *3.55+
2
*6.7x10
5
+
-3
=k=-.19x10
-17
so 16.7
2
=-.19x10
-17
xr
3
=% r = 1881067km
:imle 9a%monic Motion
3ecap$ 4isplacement$ 5elocity and Acceleration
<isp#!cement x = 5ist!nce move5 >$ t(e osci##!tor in ! speci"ie5 5irection "rom
e7ui#i>rium position.
?mp#itu5e- ? = m!gnitu5e o" m!ximum 5isp#!cement
/erio5- ;- time "or one comp#ete osci##!tion
Fre7uenc$- " = num>er o" osci##!tions per unit time =% " = ;
-1
/(!se <i""erence- @ = Fr!ction o" !n osci##!tion t3o 3!ves !re out o" s$nc( in r!5i!ns
A body executes simple !armonic motion w!en its acceleration is directly
proportional to its displacement from its e"uilibrium position$ and is
directed towards t!e e"uilibrium position.
a - $x' the constant of %oo%tionality=2 Hf
a & /0,6f1
,
x
8he 5uantity 2Hf is the angular fre"uency + of the oscillato%+
:ince f=1=80 +& ,6-
8he #islacement x of a simle ha%monic oscillato% is given by
x & a cos0,6ft1 or x & a sin0,6ft1
8he velocity is e5ual to the g%a#ient of the #islacement against time g%ah+ Iitto
accele%ation g%ah+ Ehase #iffe%ence of 6-, between each+
/t follows that hase #iffe%ence between a an# x is i+ 8his is f%om a - $x'
8he maximum see# of an oscillato% occu%s when it t%avels th%ough the e5uilib%ium
osition $x=*'+
5
max
& 0, 6f1A
Examle!
1 mass is attache# to the en# of a s%ing+ 8he mass is ulle# #own an# %elease#
an# e%fo%ms simle ha%monic motion+ /n 1*s0 2) oscillations+ 1mlitu#e is ,cm+
"in# a
max
an# #%aw g%ah of accele%ation against #islacement+
)!ximum !cce#er!tion occurs 3(ere t(e 5isp#!cement is m!ximum. 2.5 AB so
!=-*262+)'
2
x,x1*
-2
=1>+(3ms
-1
+
From a & /0,6f1
,
x0 a is %oo%tional to x so a negative st%aight line+
*nergy of a Simple 7armonic 8scillator
1n oscillato% has maximum see# as it t%avels th%ough the e5uilib%ium osition an# is
momenta%ily at %est when it #islacement e5uals the amlitu#e+
8he 2inetic ene%gy of the oscillato% is Le%o at the ext%emes of the motion but the
oscillato% has maximum otential ene%gy at these oints+
8he 2inetic ene%gy of the oscillato% is maximum at the e5uilib%ium osition0 but the
oscillato% has minimum otential ene%gy at these oints+
4amped (otion is the %esult of f%iction an# ma2es the x against t g%ah loo2 li2e
sin$x'=x
3esonance is when fo%ce# oscillations occu% when an ob<ect is
fo%ce# to vib%ate at the f%e5uency if an exte%nal sou%ce+
*? >us engine m!kes p!rts suc( !s t(e se!ts o" t(e >us vi>r!te
!t t(e "re7uenc$ o" t(e engine'
Phen the fre"uency of the engine matches the natural fre"uency of the seat0 the
seat resonates 8he amplitude of the vibrations becomes much la%ge%+
1t .esonance!
Natural fre"uency of a forced oscillator is e5ual to the forcing fre"uency+
8he forced oscillator has maximum amplitude
8he forced oscillator absorbs maximum energy f%om the external source+
8he #eg%ee of damping affects both %esonant f%e5uency an# the amplitude of
the fo%ce# oscillato%; g%eate% #eg%ee of #aming slightly reduces resonant
fre"uency+
.esonance can be useful in mic%owaves o% nuisance in ca%s+
Examle!
1 en#ulum bob of mass @*g executes simle ha%monic motion with amlitu#e
12cm+ Ee%io# *+)2s+ "in# T
max
an# the maximum change in E
E
+
1
m!x
= *2 C"+? = *2 C0.52
-1
+*12x10
-2
+=1.-5ms
1
D
k m!x
= D
/ m!x
D
9 m!x
= 0.5 x 0.06 x 1.-5
2
= 0.063E
:oli#s0 Li5ui#s an# Mases
9inetic (odel
/n Solid the molecules vib%ate randomly about thei% e"uilibrium positions+
Closely ac2e# an# exe%t electrical forces on each othe%+
/n Li"uid the molecules have translational kinetic energy an# move
%an#omly because of collisions with othe% molecules+ 8he mean sea%ation is
g%eate% than in soli#s+
/n .as the molecules have random motion an# translational kinetic
energy+ Mean 2inetic ene%gy inc%eases with teme%atu%e+ Mean sea%ation is
g%eate% than in othe% states an# #een#s on the %essu%e+
Molecules exe%t negligible elect%ical fo%ces on each othe%
excet in collisions+
3andom motion of ai% molecules can be obse%ve# using
:rownian (otion $mathematical mo#el' of the smoke
molecules in a smoke cell whe%e they move in a <e%2y %an#om
way because of collisions with high see# molecules of ai%+
1 light is shone on a smo2e cell full of smo2e an# loo2e# un#e%
th%ough a mic%oscoe+
1 Mas in a containe% exe%ts %essu%e on the containe% walls because of molecula%
collisions with the containe% walls+
Phen a single molecule colli#es elastically with the wall0 the change in momentum is
,mv+ "%om Newton6s secon# law0 force& rate of c!ange of momentum =S "=
2mv=t whe%e t is the time between successive collisions on the wall+
E%essu%e on the wall = total fo%ce = a%ea
Examle!
Qse the 2inetic mo#el to exlain the change to the %essu%e exe%te# by gas in
containe% whne teme%atu%e inc%ease# an# volume #ec%ease#+
F = delta p over delta t F 2(en t(e temper!ture is incre!se5 t(e mo#ecu#es
(!ve more energ$ so move !t ! "!ster ve#ocit$. Aence momentum is incre!se5
!n5 time t!ken "or successive co##isions is #ess. Aence t(e "orce on t(e 3!## is
gre!ter. Aence t(e pressure is gre!ter
2(en vo#ume is 5ecre!se5 t(e time t!ken "or successive co##isions is #o3er so
t(e "orce on t(e 3!## is gre!ter so t(e pressure is gre!ter.
*nergy )!anges
%nternal *nergy of a substance is the sum of the random distribution of
kinetic an# potential ene%gies of all the atoms o% molecules+
1ll molecules have 2inetic ene%gy $vib%ations an# t%anslational motion' an# otential ene%gy
$elect%ical att%action between the molecules'+
/nte%nal ene%gy #een#s on the teme%atu%e of the substance+ 1t *R0 all molecules sto
moving an# vib%ating+ 8he inte%nal ene%gy is otential only+ 1s teme%atu%e is inc%ease#0
inte%nal ene%gy of a substance inc%eases #ue to molecules gaining 2inetic an# otential
ene%gy+
(elting
:oli# state to li5ui# state
:oiling
Li5ui# state to gaseous state
Iu%ing these two stes when teme%atu%e is increased
Mean :ea%ation inc%eases
Elect%ical otential ene%gy inc%eases
Mean 2inetic ene%gy of molecules %emains the same $bon#s a%e b%o2en but
molecules #o not move any faste%'
/nte%nal ene%gy inc%eases
*vaporation
Li5ui# state to gas state
"ast moving molecules escae f%om the su%face leaving the slowe% molecules0
cooling the substance+
Dccu%s at 1LL teme%atu%es of a li5ui#
.ate inc%ease# by blowing ove% su%face an# heating+
1 to 40 C to I0 E to "
/n soli# state+
1s tem inc%eases0 internal
energy inc%eases mainly
#ue to mo%e vibrations
4 to C0 I to E
Melting= 4oiling
Ene%gy use# to b%ea2
molecula% bon#s
Mean sea%ation inc%eases
%nternal energy inc%eases mainly #ue to inc%ease# electrical potential
energy+ $8em is constant'
Elect%ical otential ene%gy of a gas is Le%o $which is maximum because elect%ical
otential ene%gy is negative fo% lowe% teme%atu%es'+
Phen a hot ob<ect is in contact with a coole% ob<ect0 the%e is a net heat t%ansfe% f%om
the hot ob<ect to cool ob<ect+ Eventually0 both ob<ects# %each the same te%me%atu%e
an# a%e in t!ermal e"uilibrium+
8he #iffe%ence between #eg%ees an# Relvin is lus=minus ,;<+
!ermal =roperties of materials
8he specific !eat capacity of a substance is t!e energy re"uired per unit
mass of t!e substance to raise its temperature by 2 9+
c & *-m7t
Examle!
1 bea2e% contains ,*g of wate% at a teme%atu%e 2*
o
c+ 8he glass bulb of 2?P
filament lam is imme%se# in the wate%+ 8he teme%atu%e #oubles afte% )
minutes+ 8he secific heat caacity of wate% is ?2**A2g
-1o
C
-1
+ 1ssuming no loss of
heat0 fin# the efficiency of the lam as a heate%+
D=mc7t= *+*,x?2**x2*= @(2*A of ene%gy
2?P = 2?As
-1
= (2**A of ene%gy
@(2=(2* x 1** = >3U efficiency
1n exe%iment to #ete%mine the heat caacity of a soli#
o% li5ui#+
1n elect%ical heate% heats the substance+ /t6s
teme%atu%e is measu%e# using a the%momete%+
1mmete% an# voltmete% connecte# an# insulation to
%event heat loss+
Measu%e m an# V
*
of the substance+
:ta%t stowatch0 tu%n on0 an# measu%e / an# T+
1fte% a while0 sto stowatch an# tu%n off+
Measu%e V
i
an# t f%om stowatch
c= T/tm$Vf-Vi'
8he%e is no teme%atu%e change when a substance changes state+ 8otal kinetic
energy of the molecules %emains constant but thei% electrical potential energy
increases as molecula% bon#s a%e b%o2en+ 8e%m use# fo% ene%gy sulie# to change
the state is latent heat+ $latent means hi##en'
Latent !eat of fusion is the ene%gy sulie# to MEL8 a soli# substance to li5ui#
Latent !eat of vaporisation is ene%gy sulie# to 4D/L a li5ui# substance to gas
%deal gases
Dne mole of any substance has @+*2x1*
23
a%ticles+
8ne mole is t!e amount of substance t!at !as t!e same number of particles
as t!ere are atoms in 2,g of carbon 2, isotope.
Examle!
M% of 9e is ? g mol
-1
+ Iete%mine the mass of a helium atom+
Dne mole is @+*2x1*
23
atoms+ :o *+**?=$@+*2x1*
23
' 2g = @+@?x1*
-2(
2g
:oyles Law states!
!e pressure exerted by a fixed amount of gas is inversely proportional to
its volume$ provided its temperature is constant.
J 1T so T=constant
"o% a gas at constant teme%atu%e0 the %o#uct of %essu%e an# volume %emains
constant+ Ioubling the %essu%e will halve the volume+ i+e+ 1T1=2T2
8he above alies to i#eal gases $90 9e0 D a%e i#eal gases at s+t+ but as tem lowe%s
o% %essu%e ve%y high0 some gases #ea%t f%om thei% i#eal behaviou% because
elect%ical fo%ces between atoms a%en6t negligible'
Exe%iments show
J8 *"or const!nt vo#ume+ an# TJ8 *"or const!nt pressure+
Combining the above with 4oyle6s law we have
T8=constant so 1T181=2T282=$const!nt+
8he constant #een#s on the numbe% of moles $n' an# the mola% gas constant $.' so!
T=n.8
8his is the i#eal gas e5uation+
Examle!
Dxygen consists of D
2
molecules+ 8he mola% mass of oxygen is 32g mol
-1
+ "o%
1@*g of oxygen0 fin# numbe% of moles0 molecules an# the volume at 2*
o
C an#
1+*1x1*
)
Ea+
160,32=5 mo#s 5 x 6.02x10
23
= 3.01x10
2-
p1 = n&; =% 1.01x10
5
x1 =
5x,+31x$2*+1(3'
T= *+12m
3
ranslational kinetic energy
/nc%easing the temperature of the gas ma2es the atoms t%avel faste% an# thei%
means translational kinetic energy increases+ 8he mean kinetic energy of the
atoms is related to the t!ermodynamic temperature of the gas+
(ean translational kinetic energy of atoms J T
8he inte%nal ene%gy of the gas is almost enti%ely e5ual to the total 2inetic ene%gy of
the atoms0 so
%nternal energy J
"o% in#ivi#ual atoms0 the mean t%anslational
2inetic ene%gy E is given by
E= 3228
Phe%e 2 is the 4oltLmann constant+ 8he mean
2inetic ene%gy is e5ual to *+)mv
2
+ 1t a given
teme%atu%e0 the gas atoms have a %an#om
motion an# t%avel at #iffe%ent see#s+ 9oweve%0
the mean 2inetic ene%gy of the atoms %emains
constant at a given teme%atu%e+
Examle!
8he oute% su%face of a sta% ha# 9 an# 9e atoms+ 8em is ,***R+ M%$9'=1
M%$9e'=?+ Io hy#%ogen an# helium have the same mean 2inetic ene%gyW Calculate
mean see# of 9 atoms+ Iete%mine %atio of mean see# of hy#%ogen atoms ove%
mean see# of helium atoms+
3228= E=E2=12mv2 so ; J D
k
. So $es. 1 = root*3k;,m+ 10
-3
,*6.02x10
23
+=1.66x10
-
27
kg e!c( ($5rogen !tom. 1= root**3x1.38x10
-23
x8000+,*1.66x10
-27
++ = 1.- x10
-
ms
-
1
. 1 Jm so root*-+ = 2 =2.