Post on 19-Dec-2021
Workshopig
a
I
I Ltp x2y3 It Byod jdi docI tdyyT di Lt B oilydoc By
3 25
Along x 5g doc 5dgii dI CLtMC5yYg3Gdy t Py 5g2dg
125 Lt B y dy t 25Bysdy25 52 6B y dy
y o I
Si di fo 25652 6B y dy25 FL t B
org
Q2Stokes Theorem TAT da ft DT
Iit
Convert it to cylindrical
5 25 sino cos05 sindOT
5z Sindsi cos018 Its cosofE
J di C 25sins lots 2 cosd sold f I DZ O S R
Si di R fo 2sin'dold LDx i o 5 I Co ily Co 4g I
5 I J Liye5 costs sin 08 sind5 toosold
4s sin I
Q3
i
3x't3yCartesian to Cylindrical
x scosol yessing 2 Z
D J _350501 3555435 cosittsintftp.T do fo5fYf23s2 sdsddd2
5x2iTxo3sBds
louf3sI 1207
Now ftp.t dE fosi da
Convert it to cylindricalE cos45 sin ofof
of sings cos OfE I
J s3 cos ft sin4015 t s osdsindfcosbtsinH.JO
Three sides to cylinder8
G Top dei sdsDIET da O
2 Bottom dei sdsdf IJ dot O
3 side do sold diesii dei s cos40 tsin 40 ddd2
Question 3
147 ut de f i do
it r3indrthr2coscetr2tandFl7
J rIgdCr2 r's in d
rsitnaFee since4 rEosa
rateof Catana
D J Cupsin d trY Goshsince
sinYet coshe since
Coste4 r
since
Question 4
Surface consists of two parts
G The icecream cone
r Rid O 2E A o o
doe R'sina.dd.deri
da CRsind RsindddddR4sin2ddddO
fu da R f sin'd 1 01
R Czaffaz sina.IQ ob
2itR4CE sinEs
tfICt 3Fsz
do
2 The core
A If I O 72T r o R
dersinddrdddudw 4sinclcosdrsdrdd
e.fr dro
Su doi fsl r3dr o
dot Bzer
Therefore
So.da tRI Iz Bz tfs
t 2rt3fsT
d
Question 3 Archimedes
Lets define the pyramid to have
vertices at
Como Ca a b Ca a b Ea a b Gaia b
Calculate for this surface
The force on the bottom surface is
p egb die dedge
E fpdoi faaf.aegbd.edu I
Question 5
E b l
4a'begE
The force on a tilted surface can be
found by considering the equation ofthe surface surface in diagram
Z hat x O
The normal to this surface is justthe gradient of the scalar function
2 Z C
ri p Cz IaEE EE
Normalising we have of at battab
K
B
dz2 Laxda da doc
ota dydlridyd.IAEbx.azdyftabd.caEbxraztbz
adydx.cat BI
Now in ywe are integrating between
the lines yx and y K
E HedaaL f
I f gbzxtadgd.cat boil
easab C z.cyd.cat boil
es f IcaE bail
2egbagCAI BI
On surfaces and the 2
component will be the same as
but the se or y component will
be different
E 2ebICaEtb5E 2ebICaEtbEE 2egeba Cai 551
Summing the forces on all five
surfaces we have
E 4a'begI 8e I3
Is a begI
s Gal beg Ivolume xegI as expected