AUGUST 8 , 2024
Recall
ID
(rectangular)
coordinates
jx
Y
-
w
Let
=
X
r
From
FORMULAyIj
E
[
Letr' be the position vector
(X , Y)
T
magnitude of the position vector
Pythagorean Theorem
be the
,
X+ y
r =
Now, , how do
*
Y
From trigonometry
sint
=
Y
-
cost
> X = rcost
So
,
,
recall
say
SOHCAHTOA
=
r
using
dividers by its
magnitude r
rcosti + rsinj
=
r
of t and re
in terms
we express
y in terms of f and r ?
(normalizing r)
=*
; y
we
,
&
>
↓FORMAT rsinj
=
r
costi + sintj
=FORMA+ inG]
Differentiating
jT
Y
with
respect to t , we then get
Now , how do
-
r
1
#
-
-
dod (costi
+
=
(X , Y)
sinY ,
In
a
we
?
expressi and j in terms of
matrix form :
()) (ssi ()
W
=
E
[
1
and
EOSMECOSY
T
=
X
~
Y
() (s
=
We then
have the following :
Y FORSUA-inE
=
-
Assignment : Find
#
.
the
J FORAA SE
=
following formulas :
1
d
For T :
-
we
know that
by FORMULA ,
↑ is indirectly dependent on t
> at
costi + sinEj
any given timet , the particle
We then use
= du
It
I
dEdt
&
=
.
may also be time-dependent
depends on E, where
found
on a distinct o with respect to the X-axis
be
night
CHAIN RULE
dudt
=
We can shorten
do into
> time
(cost + sintj)
of
E
(sinE + costj)j
>
↓
=
since
,
FORMULA
X
(E
d = (FORMULA
=
-
sinGi + costj)
derivative