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D–d
Cone
L
Inclinação
tg
α
2
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D–d
2L
D–d
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60 – 40
80
60 – 40
2 x 80
60 – 40
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=
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80
20
160
20
160
=
=
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1
4
1
8
1
8
ou
1: 4
ou
1: 8
= 0, 125
Recorrendo a uma tabela de tangentes, verifica-se, que a 0,125 corresponde um
ângulo de 7º 10’, pelo que o ângulo do vértice do cone, será de 14º 20’.
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I
+J(&>6+=+..+3(6M 4/6 >6+=&
/:
>)0'
0+0( N>+1N>(&( 6(0'
&+0'
.*
R)4'
+
CENFIM
12
I
/1/4+&/1'
6L/0/*
+6L/&
M (6 J(&
/M +)>1+)0/+.:
/13+.
I
5+),+&>6 )S6(&
/.'
6=1(.0( 0'
5'
.-(.
$
( (E(4>*
+&>6+=+..+3(6
(0'
&
+0'
.*
R)4'
+
I
#I
>L*
&
+'
&0(
M +N>+)*
'
0+0(
(*
(&
6'
)+&/4/6=&
'
6()*
/+*
/&
)(+&
M =+&
++4+L+&/4/)(
@[
%I
)4/)*
&
+ .( +6(0'
0+0( %M9 66
[
$M 66
.*
+L(1(4(&>6+&
(3&
+0( *
&
V..'
6=1(.
+ %M9 66 0( 4/6=&
'
6()*
/0/4/)(
+ $M 66
X
N =
X
10,2 x
X
10
6,8
4/&
&
(.=/)0(6 $0'
5'
.-(.
X
4/&
&
(.=/)0(6
0'
5'
.-(.
= 15 divisões
8I \M =/&
*
+)*
/M )(4(..7&
'
/0(.1/4+&+(.=(&
+*
&
+).5(&
.+1M 0( 6+'
. # 0'
5'
.-(.
2
5
5- :85,
:
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:
<7
:+7
,
5:7
,
5 !"
.*
( 6<*
/0/ <M ./6()*
(M =&
7*
'
4/M )/ *
/&
)(+6()*
/ 0( =(,+.4/6 3&
+)0( 4/)'
4'
0+0( (
0( =/>4/4/6=&
'
6()*
/
(&
6'
*
( *
/&
)(+&
M .2 4/6 +5+),/ 6+)>+1M *
+)*
/ .>=(&
:
B
4'
(. 42)'
4+. (E*
(&
'
/&
(.M 4/6/
'
)*
(&
'
/&
(.6+.M 0( 4/6=&
'
6()*
/'
3>+1 +/4>&
./0/4+&
&
/
+&
++(E(4>,;/M =/0(6 0+&.(
I +./
2&
6>1+ K
CENFIM
4+./.K
4/)( <0(:
'
)'
0/=(1/..(>.0'
R6(*
&
/. ( 0 ( =(1/.(>4/6=&
'
6()*
/
tg α =
D - d
2L
0/)0( .( *
'
&
+α
13
R)3>1/ α
>.*
'
:
'
4+,;/ 0+ :
2&
6>1+
C
R)3>1/ 0( '
)41'
)+,;/ 0+ 4/&
&
(0'
,+ 0/ 4+&
&
/
=/&
*
+:
(&
&
+6()*
+.DM <'
3>+1 +/R)3>1/α]C
.(6'R)3>1/)/5<&
*
'
4( 0/4/)(D
(6 .(M 4/6 (:
('
*
/M .(3>)0/+ '
3
tg α =
D-d
2
L
D-d
=
2L
'
3
E(6=1/)>6<&
'
4/K
+14>1+&/R)3>1/0( '
)41'
)+,;/0+4/&
&
(0'
,+0/4+&
&
/=/&
*
+:
(&
&
+6()*
+.M =+&
+(E(4>*
+&
/4/)( 0+ '
3#
'
3#
CENFIM
14
tg α =
e
D–d
2L
=
60 - 25
2 x 20
= 0,875
α = 41º 10’ (ver tabela de tangentes)
/*
+K .(3>)0//.4+./.M +4/&
&
(0'
,+0/4+&
&
/M +/3'
&
+&
M =/0( *
/6+&57&
'
+./&
'
()*
+,-(.
C'
3. % ( 8D
'
3%
I +./
Fórmula 2:
'
3 8
4/)( <0(:
'
)'
0/=(1+.>+4/)'
4'
0+0( 4M /0'
R6(*
&
/
tg α =
c
( /4/6=&
'
6()*
/
donde se tira α
2
E(6=1/)>6<&
'
4/K
(*
(&
6'
)+&/ R)3>1/ 0( '
)41'
)+,;/ + 0+&Q4/&
&
(0'
,+0/4+&
&
/=/&
*
+:
(&
&
+6()*
+.M =+&
+
*
/&
)(+&/4/)( &
(=&
(.()*
+0/)+ '
3 9
/)'
4'
0+0(
$Z [ $M $
tg α =
c
2
=
0,10
2
= 0, 05
α = 2º 52’ (tabela das tangentes)
CENFIM
15
4 2
5
*+ 5 : 5,
:,
+6+'
/&=+&
*
( 0/.4+./.M /.4/)(.0( +4/=1+6()*
/M *
V6 >6+=(N>()+4/)'
4'
0+0(
(.*
(.4+./.M 0(5(6 +=1'
4+&.( :
2&
6>1+.&
7='
0+.M N>( 0;/M 0'
&
(4*
+6()*
(M /R)3>1/0(
'
)41'
)+,;/M (6 3&
+>. ( .>L0'
5'
.-(. 0(4'
6+'
. 0/ 3&
+>M 4/6 >6+ +=&
/E'
6+,;/
.>:
'
4'
()*
(
;/:
2&
6>1+.=&
7*
'
4+.M N>( 0'
.=().+6 /&
(4>&
./+>6+*
+L(1+0( *
+)3()*
(.
.*
+ =&
7*
'
4+M .2 < +4('
*
75(1 =+&
+ /.4/)(.N>( );/ >1*
&
+=+..+6
C
R)3>1/)/5<&
*
'
4( 4/&
&
(.=/)0()*
( [ $I
0+4/&
&
(0'
,+M 0( $I
% Z 0( 4/)'
4'
0+0(
U
DM 0(*
(&
6'
)+)0/>6 R)3>1/0( '
)41'
)+,;/
^M )/67E'
6/C
*
+)3()*
( 4/&
&
(.=/)0()*
( [ $M 9D
4'
6+0(.*
(.5+1/&
(.M /(&
&
/M N>( +>6()*
+4/6 /R)3>1/M *
/&
)+ .( 6>'
*
/'
6=/&
*
+)*
(
+&
++.>+(E(4>,;/M =/0(6 0+&.(
I +./
4+./.K
4/)( <0(:
'
)'
0/=(1/..(>.0'
R6(*
&
/.( =(1/.(>4/6=&
'
6()*
/
α=
Fórmula 1
=1'
4+,;/
D-d
L
x 28,65
(α - ângulo de inclinação)
(*
(&
6'
)+& / R)3>1/ 0( '
)41'
)+,;/ 0+ 4/&
&
(0'
,+ 0/ 4+&
&
/ =/&
*
+
:
(&
&
+6()*
+.M =+&
+(E(4>*
+&/4/)( '
)*
(&
'
/&0+ '
3?
'
3?
CENFIM
>6( 0'
&
'
3'
0/=+&
+L+'
E/+*
+4+)0/Q0'
&
('
*
+0+=(&
:
>&
+,;/
16
28 - 23
α = 28,65 x
7º +
20
16 x 60’
= 7º 9’
100
I +./
= 28,65 x 0,25 = 7º, 16 (ler 7º e 16 centésimos do grau) ou
4/)( <0(:
'
)'
0/=(1+.>+4/)'
4'
0+0(
α = c x 28,65
Fórmula 2 :
=1'
4+,;/
(α - ângulo de inclinação)
+0/ >6 4/)( 4/6 >6+ 4/)'
4'
0+0( 0(
# Z
C'
3
$DM 0(*
(&
6'
)+&/
R)3>1/0( '
)41'
)+,;/0+4/&
&
(0'
,+0/4+&
&
/
/)'
4'
0+0(
# Z [ $M #
α = c x 28,65 = 0,15 x 28,65 = 4º,29 (4º e 29 centésimos do grau)
α = 4º +
29 x 60’
100
= 4º 17´
'
3 $
>.*
'
:
'
4+,;/ 0+ :
2&
6>1+
+&
+ R)3>1/. =(N>()/.M +*
< $IM =/0( .(M .(6 (&
&
/
+=&
(4'
75(1M 4/):
>)0'
&/ 4/6=&
'
6()*
/ 0/ +&
4/ +LM 4/6 + *
+)3()*
( 0/ R)3>1/ α C '
3
D
CENFIM
17
(P
+>6+4'
&
4>):
(&
V)4'
+0( &
+'
/
[
.(>4/6=&
'
6()*
/6(0( π
'
3
>6 R)3>1/ +/ 4()*
&
/M αIM 4/&
&
(.=/)0( >6
+&
4/+L0( 4/6=&
'
6()*
/E
x =
2πR
x αº
360º
360 x x
α=
2πR
α=
360 x x
=
2π
360
( por ser R = 1)
x x = 57,29 x x
2π
M =/'
.N>( +*
<4(&
4+0( $IM .( =/0( 4/):
>)0'
&/+&
4/( +*
+)3()*
( 0/R)3>1/ αM =/0(
(.4&
(5(&.(K
α = 57,29 x tg α
α = 57,29
α = 57,29
/*
+K
D–d
2L
c
2
ou
=
=
tg α =
28,65
D–d
2L
D-d
L
=
c
2
ou ainda
28,65 c
4/6=&
'
6()*
/ 0( >6 +&
4/ 4/&
&
(.=/)0()*
( + >6 R)3>1/ 0(
$M 8 # C
.>=/.*
//
[ D ( /4/6=&
'
6()*
/0+*
+)3()*
('
3>+1 +$M 8%
$IM < '
3>+1 +
M />.(P
+M >6+
0'
:
(&
(),+0(
CENFIM
18
$M 8%
$M 8 # [ $M$$ 9
4/&
&
(.=/)0()*
( +>6 (&
&
/0( 4(&
4+0( %U
M =&
+*
'
4+6()*
( 0(.=&
(J75(1 K (&
&
/N>( <M 4/6
(:
('
*
/M '
):
(&
'
/& Q. =/..'
L'
1'
0+0(. 0( +=&
(4'
+,;/ 0+ /&
'
()*
+,;/ 0/ 4+&
&
/ =/&
*
+
:
(&
&
+6()*
+.
+&
+ α [ #I $U
M /(&
&
/);/>1*
&
+=+..+ U
2
7: !"
:
!-
8; 7
.*
( =&
/4(../ =(&
6'
*
( / *
/&
)(+6()*
/ +>*
/67*
'
4/M =(1/ 0(.1/4+6()*
/ 0/ 4+&
&
/
1/)3'
*
>0'
)+1 ( /0+(E(4>,;/0( 4/)(.0( &
+J/75('
.4/6=&
'
6()*
/.
(6M 4/6/ 0(.5+)*
+3(6M *
/&
)(+& 4/)(. (E*
(&
'
/&
(. 0( =/>4+ 4/)'
4'
0+0(M =/'
./
0(.+1'
)O+6()*
/0/4+L(,/*
( 625(1 );/5+'
M 3(&
+16()*
(M +1<6 0/. $66
(1/ :
+4*
/M 0+ .>P
('
,;/ 0+ =(,+ .(&0(:
('
*
>/.+ C
0(5'
0/ +/ 0(.+1'
)O+6()*
/DM /.=/)*
/.
42)'
4/.0( 4()*
&
+3(6M ./:
&
(6 0(:
/&
6+,-(.M / N>( '
6=1'
4+ 4/&
&
(4,-(.4/).*
+)*
(.M )/
&
(+=(&
*
/0/4+L(,/*
( 625(1
<7
:+7
7
,
5A
5-
(P
+ M /0(.+1'
)O+6()*
/+0+&+/4/)*
&
+=/)*
/ C'
3
D
('
E/ 0/ 4/)( <M *
(/&
'
4+6()*
(M =(1/ 6()/.M + &
(4*
+ 0(:
'
)'
0+ =(1+.(E*
&
(6'
0+0(.
6+3(&
+*
&
'
JO/&
'
J/)*
+1M !M <=+&
+1(1++/('
E/'
)'
4'
+1
/&
*
+)*
/M + &
(4*
+
0(5( :
+J(&
M 4/6 / ('
E/ '
)'
4'
+1
(
M 0/.=/)*
/.
M >6 R)3>1/ α M '
3>+1 +/ .(6'
('
E/)/5<&
*
'
4( 0/4/)( αU
'
3
CENFIM
19
0(.1/4+6()*
/
M 0/4/)*
&
+=/)*
/M 4+14>1+ .( =(&
=()0'
4>1+&
6()*
(+
/*
&
'
R)3>1/&
(4*
R)3>1/[
]K
.() α
[
[
.() α
6+.M =+&
+ /.=(N>()/.R)3>1/.C
+*
< $IDM =/0( .( 4/):
>)0'
&+ *
+)3()*
( 4/6 / .()/
0/R)3>1/ α C
/>M /N>( </6(.6/M
(
'
3
DM 0( 6/0/N>( K
x = L . tg α
Ora, tg α =
D–d
2 l
c
=
(fórmulas conhecidas)
2
'
3
/&4/).(3>'
)*
(M .(&
7:
74'
1 4/)O(4(&/ 0(.1/4+6()*
/+'
6=&
'
6'
&+/ 4/)*
&
+=/)*
/M .(P
+
N>+)0/ .( 0;/ +. 0'
6().-(. 0/ 4/)(M .(P
+ N>+)0/ < 0+0+ + 4/)'
4'
0+0( (M L(6
()*
()0'
0/M /4/6=&
'
6()*
/*
/*
+1
I +./
Fórmula 1 :
I +./
Fórmula 2 :
CENFIM
0+=(,+
4/)( <0(:
'
)'
0/=(1+..>+.0'
6().-(.
x =
D–d
2l
x L
4/)( <0(:
'
)'
0/=(1+.>+4/)'
4'
0+0(
x =
c
2
x L
20
=1'
4+,-(.
I E(6=1/
+14>1+&/0(.+1'
)O+6()*
/0/4/)*
&
+=/)*
/M =+&
+*
/&
)(+&>6 4/)(
4/6 +..(3>'
)*
(.0'
6().-(.K
[
9 66
0[
# 66
[
$$66
1[
# 66
=1'
4+)0/+:
2&
6>1+ K
x =
I
E(6=1/
28 – 25
2 x 125
x 300 = 0,012 x 300 = 3,6 mm
+0/ >6 4/)(M 4/6 + 4/)'
4'
0+0( 0( #Z C
$M$#DM 4+14>1+& /
0(.+1'
)O+6()*
/ 0/ 4/)*
&
+=/)*
/M .+L()0/ N>( / 4/6=&
'
6()*
/*
/*
+1 0+ =(,+ < 0(
$$
66
=1'
4+)0/+:
2&
6>1+ K
x=
7
,
B !"
<,
:
0, 05
2
7
,
5A
x 200 = 0,025 x 200 = 5 mm
5-
13>).4+L(,/*
(.625('
. (.*
;/6>)'
0/.0>6+3&
+0>+,;/.'
*
>+0+)+L+.(M 4/6 )2)'
/
=/& 4'
6+ /> 0>6 6/.*
&
+0/& 4/6 =/)*
('
&
/M 4/1/4+0/ ./L&
( / ('
E/ 0/ =+&
+:
>./ 0(
0(.+1'
)O+6()*
/M =(&
6'
*
'
)0/1(&+6(0'
0+M 0'
&
(4*
+6()*
(M (6 :
&
+4,;/0( 6'
1B
6(*
&
/
.*
/+>6()*
+M 4(&
*
+6()*
(M +=&
(4'
.;/0+&
(3>1+,;/
/ ()*
+)*
/M =/0(6 =_&.( +13>6+. &
(.(&
5+. K =+&
+&
(+1'
J+& / 0(.+1'
)O+6()*
/M <
=&
(4'
./0(.+=(&
*
+&/=(&
)/N>( '
6/L'
1'
J+/ +L(,/*
(M )/L+&
&
+6()*
/
/& 4+>.+ 0+ =/..B
5(1 :
/13+M ()*
&
( +. 3>'
+. 0/ L+&
&
+6()*
/M + L+.( =/0( :
'
4+&
'
6/L'
1'
J+0+M 4/6 >6+1'
3('
&
+/L1'
N>'
0+0( (6 &
(1+,;/Q.>+=&
'
6('
&
+=/.'
,;/
+N>'&
(.>1*
+ >6 0(.5'
/ 0/ 4/)*
&
+=/)*
/ (M =/&4/).(3>'
)*
(M >6 (&
&
/ )/ R)3>1/ 0/
4/)(
/)*
>0/M (.*
( 4/)*
&
+*
(6=/ );/ < 0( &
(4(+&
M .( + 3>'
+ 0/
+L(,/*
( 625(1 :
/&
=&
'
.67*
'
4+
CENFIM
21
2
,
)9+
5:7
,
5<; 7 '
+
1 )9+
5 2
6 4(&
*
/. *
/&
)/.M =/0( +0+=*
+&.( >6 +=+&
(1O/M =+&
+*
/&
)(+6()*
/ 42)'
4/M N>(
+4/6=+)O++67N>'
)+
\>*
'
1'
J+0/0+.(3>'
)*
(:
/&
6+ C '
3
(=/'
.0( 0(.+=(&
*
+&/=+&
+:
>./
4+&
&
/
DK
M N>( .>P
('
*
++=/&
4+0/=+&
+:
>./*
&
+).5(&
.+1M +/
! C
=+&
+ 1'
L(&
*
+& / 4+&
&
/ < )(4(..7&
'
/M (6 4(&
*
/. 4+./.M 0(.6/)*
+&
4/6=1(*
+6()*
(M /:
>./0( 4/6+)0/DM :
'
E+ .( /*
'
&
+)*
( 0( +4/=1+6()*
/
6 .(3>'
0+M :
'
E+ .( + *
+L(1+ 3&
+0>+0+
M +/ L+&
&
+6()*
/
M +/4+&
&
/!
M =/&'
)*
(&
6<0'
/ 0/.
L&
+,/.0( .>=/&
*
(C
4/)./1+.D
/L&
( +*
+L(1+
M (.*
74/1/4+0+>6+&
<3>+0( 3>'
+ M )+N>+1M .( 0(.1/4+/'
)0'
4+0/&
M ./1'
07&
'
/4/6 /*
'
&
+)*
(
'
)41'
)+,;/0+&
<3>+ M =/0( 5+&
'
+&.(3>)0//R)3>1/ αM +:
/&
6+&4/6 +=(,+
'
3
+15/ )/ 4+./ 0( >6 *
/&
)/ 6>)'
0/ 0( 0>=1/ 4+&
&
/M 4/6 3>'
+./&
*
/3/)+'
.M '
.*
/ <M (6
R)3>1/&
(4*
/M 4/6/P
7);/.( 0'
.=-( 0/4+&
&
/*
&
+).5(&
.+1M =+&
+(:
(4*
>+&=+..+3().(6
=&
/:
>)0'
0+0(M 4/1/4+ .( +4/&
&
(0'
,+=/&
*
+:
(&
&
+6()*
+.M =(&
=()0'
4>1+&+/('
E/0/*
/&
)/
..'
6M /4+&
&
/=&
'
)4'
=+1M 0(.1/4+)0/ .(M +&
&
+.*
+4/6 (1( /'
)0'
4+0/&
'
)0'
4+0/& M =/&.>+ 5(JM .(3>'
)0/ / 4+6'
)O/ N>( 1O( '
6=-( + &
<3>+ '
)41'
)+0+ M
0(.1/4+M *
&
+).5(&
.+16()*
(M /4+&
&
/! (M +/6(.6/*
(6=/M +:
(&
&
+6()*
+
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22
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2
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a = h1 x tg
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= 0, 1562
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= 8º 53’
= 90º - 8º 53’ = 81º 7’
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3º)
x = r x cotg
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a = h1 x tg
= 28 x 0,1562 = 4, 373 mm
5º)
X = d + 2a + 2 x + d’
= 20 + (2 x 4,675) + 8
6º)
=
= 4 x 1,1688 = 4, 675 mm
37,35 mm
X1 = d + 2a + 2 x + d’
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(.(),+
'
15+M
CENFIM
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&
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+
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+0(
(&
&
+1O('
&
/.
).'
)/ <4)'
4/ &
/:
'
..'
/)+1
61