Study Guide Block 2: Vector Calculus Unit 2: Tangential and Normal Vectors 1.
Lecture 2.020 Study Guide
Block 2: Vector Calculus
Unit 2: Tangential and Normal Vectors
2.
Read Thomas, S e c t i o n s 14.3, 1 4 . 4 , 14.5
(Note: i f you a r e n o t t o o
f a m i l i a r w i t h t h e s e v e c t o r s , you may f i n d i t h e l p f u l t o view t h e
l e c t u r e a second time, a f t e r you have read t h e a p p r o p r i a t e s e c t i o n s
of Thomas.)
3.
Exercises:
2.2.1(L1
A p a r t i c l e moves i n t h e p l a n e according t o t h e equation of motion:
a.
Find t h e t a n g e n t v e c t o r t o t h e curve a t any p o i n t ( x ( t ), y ( t ) ) .
b.
How f a r does t h e p a r t i c l e t r a v e l between t = 0 and t = 6 1
A p a r t i c l e moves according t o t h e equation of motion:
a.
Find t h e p o s i t i o n , v e l o c i t y , speed, and a c c e l e r a t i o n of t h e p a r t i c l e a t t = 4.
b.
How f a r does t h e p a r t i c l e t r a v e l between t = 0 and t = 4 1
c.
Find a u n i t t a n g e n t v e c t o r t o t h e curve t = 4.
2.2.3(L)
a.
Show t h a t i f $ =
-+
b.
c.
9+
g!f,then
ds L-+
K(=)
N where
-b
t h e a c c e l e r a t i o n , a , i s given by
P
8
= 0 and
K
=
l$l
.
+
-+
Use t h e .expressions f o r a and v i n p a r t a. t o f i n d an expression
-+
-b
f o r v x a. From t h i s deduce an expression f o r K i n terms of f and
Use b. t o f i n d t h e c u r v a t u r e of t h e p a t h followed by t h e p a r t i c l e
which moves according t o t h e equation i n Exercise 2.2.2
Use p a r t a. t o Exercise 2 . 2 . 3
t o show t h a t t h e a c c e l e r a t i o n of a
p a r t i c l e i s always normal t o i t s p a t h of motion
i t s speed i s c o n s t a n t .
2.2.2
a t t = 4.
if
only
if
2.
Study Guide
Block 2: Vector C a l c u l u s
U n i t 2: T a n g e n t i a l and Normal V e c t o r s
2.2.5tLI
Show t h a t i f t h e e q u a t i o n o f a c u r v e i s g i v e n i n t h e C a r t e s i a n
form y = f ( x ) , t h e c u r v a t u r e ,
K,
a s d e f i n e d i n E x e r c i s e 2.2.3
a.
can b e computed from t h e " r e c i p e "
a . U s e t h e r e s u l t of E x e r c i s e 2.2.5
t o f i n d t h e c u r v a t u r e of y
-
e
2x
a t t h e p o i n t (0 , I ) .
e2 t +
3,
i t s p a t h o f motion i n C a r t e s i a n c o o r d i n a t e s i s y = eZX( t h i s
b. I f a p a r t i c l e moves a c c o r d i n g t o t h e e q u a t i o n
s h o u l d be e a s i l y checked by t h e r e a d e r ) .
8
= t
+
U s e t h i s f a c t together
w i t h t h e r e s u l t o f E x e r c i s e 2.2.3 b. t o s o l v e p a r t a. by a d i f f e r e n t method.
2.2.7
a. Find t h e c u r v a t u r e o f y = a x
+
b, where a and b a r e given c o n s t a n t s .
b. Find t h e c u r v a t u r e o f y =
-
where a i s a p o s i t i v e c o n s t a n t .
2.2.8(L)
A p a r t i c l e moves a c c o r d i n g t o t h e e q u a t i o n
a . Find i t s normal and t a n g e n t i a l components of a c c e l e r a t i o n a t any
t i m e t.
b. Find t h e c u r v a t u r e o f t h e p a t h o f motion b o t h from t h e r e s u l t s of
p a r t a . and from E x e r c i s e 2.2.3 b.
Study Guide
Block 2: Vector Calculus
Unit 2: Tangential and Normal Vectors
., . ,
Compute t h e t a n g e n t i a l and normal compane.nts o f a c c e l e r a t i o n o f +
a p a r t i c l e which moves according t o t h e equation, 8 = t3 f + s i n t j . I
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