Mecânica Fundamental
description
Transcript of Mecânica Fundamental
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Mecânica Fundamental
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Conceitos Fundamentais
• espaço.• tempo.
sistema de coordenadas.x, y, e z.
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Partícula ou ponto de massa
• tem massa mas não extensão espacial.
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Grandezas Físicas e Unidades
A unidade padrão de comprimento é o metro.m
A unidade padrão de massa é o quilograma.kg
A unidade padrão de tempo é o segundo.s
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Grandezas Escalares e Vetoriais
•Escalar.(densidade, volume e temperatura.)
•Vetores.(deslocamento espacial)
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Vetores
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Se A é o deslocamentode P1(x1, y1, z1) a P2(x2, y2, z2)
entãoAx = x2 - x1Ay = y2 - y1Az = z2 - z2
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Definições Formais e Regras
[Ax, Ay, Az] = [Bx, By, Bz]
Ax = Bx Ay = By Az = Bz
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Adição
= [soma 1º, soma 2º, soma 3º]
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Multiplicação por um Escalar
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Subtração de Vetores
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O Vetor Nulo
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A Lei Comutativa da Adição
Exemplo:
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A Lei Associativa
= [Ax + (Bx + Cx), Ay + (By + Cy), Az + (Bz + Cz)]
= [(Ax + Bx) + Cx, (Ay + By) + Cy, (Az + Bz) + Cz
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A Lei Distributiva
= c[Ax + Bx, Ay + By, Az + Bz]
= [c(Ax + Bx), c(Ay + By), c(Az + Bz)]
= [cAx + cBx, cAy + cBy, cAz + cBz]
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Módulo de um Vetor
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Vetores Unitários
= [Ax, Ay, Az]
= [Ax, 0, 0] + [0, Ay, 0] + [0, 0, Az]
= Ax [1, 0, 0] + Ay [0, 1, 0] + Az [0, 0, 1]
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Significado Geométrico das Operações Vetoriais
Igualdade de Vetores
A = BABBy
Ay
Ax
Bx
y
xO
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A
B
Ay
Ax
By
Bx
C
A
B
C = A+B = B+A
x
y
O
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O negativo de um vetor.
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A
A
A
3A
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O Produto Escalar
Assim:
= Ax(Bx + Cx) + Ay(By + Cy) + Az (Bz + Cz)
= AxBx + AyBy + AzBz + AxCx + AyCy + AzCz
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Definição alternativa do produto escalar.
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Exemplos do Produto Escalar
F
S
Módulo:
Ortonormalidade de uma base:
Trabalho:
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Lei dos Cossenos
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O Produto Vetorial
i j k
Ax Ay Az
Bx By Bz
i j
Ax Ay
Bx By
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Pode-se mostrar que:
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Pode-se mostrar que:
i
j k
i j k
1 0 0
0 1 0
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Interpretação Geométrica do Produto Vetorial
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Ortogonalidade do produto vetorial
= AxCx + AyCy + AzCz
= Ax (AyBz - AzBy) + Ay (AzBx - AxBz) + Az (AxBy - AyBx)= AxAyBz - AzBy Ax + AyAzBx - AxBz Ay + AzAxBy - AyBx Az
= 0
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= (2)(1) + (1)(−1) + (−1)(2)= 2 − 1 − 2= −1
(2) (1) (−1)
(1) (−1) (2)
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Ângulo entre A e B
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Torque ou Momento da Força
rF
P
O
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Cossenos diretores
Ex: seja n unitario de A
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Ex: encontrar o unitário perpendicular aos vetores
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Produtos Triplos
Comutando duas linhas
Prove que:
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Derivada de um Vetor
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Vetor Posição de uma Partícula
O x
y
z
ix
jy
kz
r
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O Vetor Velocidade
r
r+r
r
v
OP
P’
P’’
P’’’
P (4)
P(5)
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Vetor Aceleração
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Exemplo:
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Exemplo:
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Integração Vetorial
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Exemplo:
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v2v1
Velocidade Relativa
r2r1
r12
v12
O
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vrel
vrel
vrel
vrel
v0v0
v0
v0
v0
vrelvrel
vrel
vrel
v0v0
v0
v0
v0
v
v = 2 v0
v
v =0
v
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Derivadas de Produtos de Vetores
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+_
+
+
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Componentes Normal e Tangencial da Aceleração
C
S
n
n’
P’
P
’
’
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Esta apresentação foi desenvolvida por
Gustavo de Almeida Magalhães Sáfar
no Departamento de Física do Instituto de Ciências Exatas
da Universidade Federal de Minas Gerais.