Maths for as Level Physics

Maths For AS Level Physics

Maths for AS Level Physics

Teacher – S Broomsgrove & J Harris

Maths for AS Level Physics Aim -1-

by SBroomsgrove, Castle

Maths For AS Level Physics This is a short course to improve the maths skills that you will nee !or AS level physics" Objective By the en o! this short course you will be able to work out mathematical answers using the methos given an apply them in the conte#t o! physics" Resorces $ou will nee to print o% the appropriate & or ' pages !rom the intranet each week, or print them all at the start an keep them sa!e" For each topic you are provie with a help sheet" (n this help sheet you are given a etaile metho to work out answers" These are !ollowe by e#amples o! how the maths is use in physics when appropriate" $ou will then be aske to complete several )uestions, on the corresponing )uestion sheet" *ach sheet contains &+ marks" Resorces are fo!" o! the i!tra!et !"er# Share"$ Scie!ce – A level Physics – AS Level Maths %orse %o!te!ts ' Sta!"ar" (orm )' Sig!i*ca!t (igres a!" +ecimal Places ,' Rearra!gi!g -.atio!s /' 0ra1hs a!" 0ra"ie!ts 2' 3!its

(ee"bac4 ! you have any )ueries please re!er them to Mr Broomsgrove"

-2-



H-LPSH--T  Sta!"ar" (orm 5hat is sta!"ar" form a!" 6hy is it se"7 n science we o!ten nee to use very small an very large numbers" t is there!ore use!ul to be able to write these own in an abbreviate notation" This is known as stanar !orm" -8am1le  '+ +++ - '"+ # .+/ The number is written as a single integer multiplie by .+ to a power" The power is !oun by counting the number o! places a!ter the 0rst number" '+ +++ has !our 1eros there!ore it is '"+ #.+ to the power / written as a small number ne#t to the ten" -8am1le ) /23 +++ - /"23 # .+4 ! a!ter the 0rst number there are not 5ust 1eros these are written a!ter a ecimal point, as shown above, /"23 in this case" There!ore you can think about changing the number back as moving the ecimal place by the number shown as the power" /"23 /2"3 /23 /23+ /23++ /23+++ move. move& move' move/ move4 -8am1le , +"+++4++ - 4"+ # .+6/ ! the number is very small we still write the number as a single integer multiplie by .+ but to a negative power" The power is !oun by counting the number o! 1eros, or the number o! places the number is a!ter the ecimal point" +"+++4++ has !our 1eros, or the 4 is the !ourth ecimal place, there!ore it is 4 multiplie by .+ to the power o! 6/" -8am1le / +"+++++/4. - /"4. # .+62 As with large number i! there are not only 1eros present a!ter the 0rst number they are put a!ter the ecimal point in stanar !orm" 9o6 yo have a go::::: -3-



;3-ST Put each o! these numbers into stanar !orm" A B C 7

'++ +++ 4/++ .&. +++ 389 +++ +++

-8ercise ) = mar4 for each> Put each o! these numbers into stanar !orm A B C 7

+"++3 +"++++++89 +"++++/423 +"+29/'

-8ercise , = mar4 for each > :rite each o! these numbers in !ull, not in stanar !orm A 2 # .+2 B '"/4 # .+8 C 3"+ # .+6' 7 8"38& # .+64 -8ercise / = mar4 for each> Put each o! these numbers into stanar !orm A '43 +++ +++ +++ B +"++ +++ +++ +++ +++ /'& C /42 3'8 9.& 7 +"++2'/&. -8ercise 2 =) mar4s are a6ar"e" for each> :rite each o! these constants use in physics in stanar !orm A ;niversal
-4-



H-LPSH--T ) Sig!i*ca!t (igres a!" +ecimal Places 5hy "o 6e !ee" sig!i*ca!t *gres a!" "ecimal 1laces7 ! you were aske to measure the length o! your page with a ruler you woul probably suggest a length o! &8"/cm or something similar, this woul appear reasonable" $ou woul be unlikely to say that the length was &8"'9/23cm, this woul be unrealistic as you coul not possibly measure to that accuracy with a ruler" The same nees to be applie when calculating answers" ! you start !rom numbers o! & signi0cant 0gures you shoul only )uote your answer to & signi0cant 0gures as you cannot be more accurate than the in!ormation you are working with" 5hat is a sig!i*ca!t *gre or a "ecimal 1lace7 ! a number is )uote to ' signi0cant 0gures, abbreviate to '"s"!", it will show only three integers other than 1ero" -8am1le  .'/ &23 +++ - .'/ +++ +++ to '"s"!" :e roun the number to the nearest 'r signi0cant 0gure" The number above is nearer to .'/ +++ +++ than .'4 +++ +++ as the !ourth number was a &" -8am1le ) 892 3'& .'/ - 893 +++ +++ to '"s"!" =ere the number has been roune up as the !ourth number was a 3" ! a number is )uote to & ecimal places, abbreviate to "pl", it will show only & numbers a!ter the ecimal place, i! there are no numbers a!ter the ecimal place then 1eros are ae" -8am1le , &'/"42... - &'/"42 to & "pl" The same application as above works with rouning up or own" >ote however that the above number is shown to &"" pl" but is in this case 4"s"!" -8am1le / &.++ - &.++"++ to &""pl" ?or a!s6ers shol" !ever be .ote" to higher accracy tha! the i!formatio! yo starte" 6ith i! a calclatio!' -8am1le 2 +e!sity @ massvolme ! Mass - .+"+.48g an volume - &"'cm' There!ore ensity - .+"+.48?&"' - /"'4/3'8.' g cm6' -5-


Maths For AS Level Physics =owever the answer shoul only be )uote to &"s"! as the volume )uote was to &"s"!" So# "e!sity @ /', g cm $, 9o6 yo have a go: ;3-ST :rite each o! the !ollowing to the number o! signi0cant 0gures )uote in brackets" A &'/ 223 @'"s"! B .'/ +++ 9&8 @4"s"! C .&2 /42 +8+ @&"s"! 7 .&9"324 @/"s"!" -8ercise ) = mar4 each> :rite each o! the !ollowing to the correct number o! ecimal places, )uote in brackets" A &.'"/42 @. "pl" B /"423 989 @/ "pl" C .&++ @&"pl" 7 2'"&./ 423 '&@' "pl" -8ercise , = mar4 each> :rite true or !alse !or each one, are they to the state number o! signi0cant 0gures or ecimal places A &'"//& .& - &'"// @&""pl" B 824 +++ - 83+ +++ @&"s"! C 4&2"'89 328 - 4&2"'88 @' "pl" 7 .&2 /42 - .&3 +++ @'"s"! * '43"/'& ..3 - '43"4 @. "pl" F 32"89 - 33"+ @.""pl -8ercise / =, mar4s> Carry out the appropriate calculation an )uote the answer to the correct number o! signi0cant 0gures" 7ensity - mass?volume Mass6 .&"'2+&g

olume D /"8cm'

-8ercise 2 =, mar4s> Carry out the appropriate calculation an )uote the answer to the correct number o! ecimal places" Eesistance - oltage?current

-6-


Maths For AS Level Physics oltage - &/9"4'9 Current - '"44A H-LPSH--T / :e cant rely on the sun being in a convenient position, so another iea is neee" This iea is to measure the angle of elevation o! the tree !rom a known istance, say '+ m, an then to compare the ratio o! tree height to istance with the ratio o! the sies o! a right6 angle triangle having the same angle" Trigonometry is base on this iea, an the !act that we can calculate the ratios o! the sies o! a right6angle triangle !or any angle" These ratios are calle trigonometric ratios. Forty years ago the trigonometric ratios were publishe in books, but toay they can be found !rom calculators" The basic trigonometric ratios are the sine, the cosine an the tangent " They are o!ten abbreviate to sin @rea as Gsine, cos @rea as Gco1 an tan" @The tangent as a trigonometric ratio is i%erent !rom the geometric iea o! a tangent to a curve" These ratios are e0ne !or the angle θ@Gtheta by the rules Sin θ = Ο pposite/Hypotenuse Cos θ = Adjacent/Hypotenuse Tan θ = Opposite/Adjacent You may reca t!is "y remem"erin#

$OHCAHTOA

-%-


Maths For AS Level Physics H-LPSH--T 2 Rearra!gi!g -.atio!s 5hy "o 6e !ee" to be able to rearra!ge e.atio!s7 n science, incluing physics many e)uations are use to calculate i%erent )uantities" Although e)uations are o!ten learnt in one particular !ormat, it is essential to be able to rearrange them to 0n any one o! the )uantities within the e)uation" -8am1le  7ensity - MassHolume But what i! you know the volume an ensity an want to 0n the mass" :e move volume to the other sie o! the e)uation but instea o! iviing by volume we multiply" 7ensity # olume - Mass There!ore i! we wante to calculate volume olume - Mass?7ensity At
-&-



;3-ST Eearrange the !ollowing e)uations to show how you woul calculate the )uantity in bol a!ter each e)uation" T Eearrange the ieal gas e)uation to show the calculation neee to 0n each o! the !ollowing )uantities" eal gas e)uation P - nET A P D Pressure B  D olume C n D >umber o! moles 7 T D Temperature -8ercise , =) mar4s each> ;sing the e)uation Pressre @ (orceArea calculate A Force, when pressure - 'Pa an area -m .9m& B Pressure when, Force - 39/> an area - 89m& C Area when, pressure - /Pa an Force - 2/> 7 Area when, pressure - +"4Pa an Force - .3> -8ercise / =/ mar4s > Calculate the number o! charge carriers, n, using the e)uation  vAne, i!  - &+Amps A - +"++4m& e - ."2#.+6.8C v - +"' cm per min"

-'-



- 1( -



H-LPSH--T C 0ra1hs a!" 0ra"ie!ts 5he! "o 6e se gra1hs a!" gra"ie!ts7 ever 5ust 5oin the point to give a 5agge line" 0ra"ie!ts The graient o! a graph can be use to 0n the relationship between & measurable )uantities" A graient can be calculate by 0ning the change in the y a#is an iviing by the change in the # a#is" The -.atio! of a Straight Li!e 0ra1h *)uations o! straight lines are in the !orm y - m#  c @m an c are numbers" m is the graient o! the line an c is the y6intercept @where the graph crosses the y6a#is" -8am1le 

- 11 -


Maths For AS Level Physics The above graph has e)uation y - @/?'# D &" The graient change in y ? change in # - / ? ' @m" t cuts the y6a#is at 6&, an this is the constant in the e)uation, @c" (i!"i!g the gra"ie!t of a crve' To 0n the graient o! a curve, you must raw an accurate sketch o! the curve" At the point where you nee to know the graient, raw a tangent to the curve" A tangent is a straight line which touches the curve at one point only" $ou then 0n the graient o! this tangent" -8am1le ) Fin the graient o! the curve y - # at the point @', 8"

- 12 -



;3-ST Plot a graph o! the !ollowing ata" n the e#periment the strength o! the aci was change, the temperature, pressure, time an amount were kept constant throughout" The volume o! gas prouce was recore each time, this is a reOection o! the reaction rate" Strengt h o! aci @M olume o! gas prouce  @cm'

+".

+"&

+"/

+"9

."+

&"+

'"+

4"9

.&"'

&'"4

'+"'

48"/

*nsure you have 5oine the ata in the most appropriate way" -8ercise ) =D mar4s> For the above graph can you prouce an e)uation in the !orm y m# c, to show the mathematical relationship" ;sing your e)uation can you calculate the volume o! gas you woul e#pect to prouce i! the strength o! the aci wasI A +"4M B 4"+M

- 13 -



H-LPSH--T E 3!its 5hy "o 6e !ee" !its7 $ou may have one" (ne what A pencil A sweet >umbers are not meaning!ul unless we know what they are in re!erence to" %ha!gi!g !its n physics we o!ten work on very small or very large scales that woul not be appropriate !or say, a biologist" Those scales can be converte an the relationships between them are well ocumente" t is essential however that you can apply that in!ormation" There are also many specialist units, these are always relate to more well known units so that scientists can correspon, however again it is important that you unerstan an can work within those units an their conversions" The S< 3!its S units are the Sta!"ar" ano n 6.& .+ Pico p -8am1le  .+ km - .+ +++m k stans !or kilo meaning a thousan so there are .+ # .+++ metres" - 14 -


Maths For AS Level Physics -8am1le ) 8"9F - +"+++++89F  stans !or micro so there are 8"9 # .+62Faras +erivi!g 3!its Mathematical relationships can be use to 0n the units o! a )uantity" -8am1le , 7ensity - Mass?olume There!ore the units areI - kilograms?m' (r we can write this as kg m6'" By changing the ' to 6' we enote that it is actually ivie rather than times, there!ore the ivision sign is no longer neee" 9o6 yo have a go:::::

- 15 -



;3-ST :rite each o! the !ollowing amounts in base S units, without the pre0#es" A 44 grams B &'3 km C '"/ s 7 '/4cm -8ercise ) = mar4 each> :rite each o! these amounts with a suitable pre0#" A +"+++ ++3 983 m B &'/ /42 +++ kg C '/4 389 +++ +++ m 7 +"++ +++ +++89 s -8ercise , =/ mar4s> :ork out the units o! area" Pressure - Force ? area Force is measure in >ewtons an Pressure in Pascals" -8ercise / =/ mar4s> :ork out the units o! current" Current - charge?time Charge is measure in C an time is measure in s" -8ercise 2 =/ mar4s> :ork out the units o! speci0c heat capacity, c" ) - m c QT ) - energy @units in 5oules, J m - mass @units in grams, g QT - cha!ge in temperature @units o! Kelvin use !or temperature, K ?o 6ill !ee" to rearra!ge the e.atio! *rst:

- 16 -


Maths For AS Level Physics %o!gratlatio!s yo have com1lete" yor maths for AS Physics corse::::::::::::::::::::

- 1% -


Maths for as Level Physics

Recommend Documents