10/7/15 1 Physics 131

■ Theme Music: Steve Dorff Every which way but loose ■ Cartoon: Bob Thaves Frank & Ernest

October 7, 2015 Physics 131 Prof. E. F. Redish

Vectors

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!r = xi + yj

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Review of Vectors (2-dimensional coordinates)

■ We have 2 directions to specify. We must – Choose a reference point (origin) – Pick 2 perpendicular axes (x and y) – Choose a scale

■ We specify our x and y directions by drawing little arrows of unit length in their positive direction.

■ A force vector is written ji ˆ ,ˆ

( )yxyx FFjFiFF ,ˆˆ =+=

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Adding Forces

■ We define the sum of two vectors as if they were successive displacements.

21 FFF

+= 1F 2FF

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Adding Vectors: Methods

■ There are 3 mathematical ways to add vectors

head parallelogram add components to tail rule (may use trig)

Trig review ■  The ratios of a triangle’s sides only depend on θ.

–  sin(θ) = opposite/hypotenuse –  cos(θ) = adjacent/hypotenuse –  tan(θ) = opposite/adjacent.

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Useful examples

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60o 60o

30o 30o

90o 90o

1 1

2 2

3 √

90o

1

1

2 √ 45o

45o

sin 30!( ) = 12

cos 30!( ) = 32

tan 30!( ) = 13

sin 60!( ) = 32

cos 60!( ) = 12

tan 60!( ) = 3

sin 45!( ) = 12

cos 45!( ) = 12

tan 45!( ) = 1

Vectors with trig – by components

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A = Axi + Ay j

= Acos(θ )( ) i + Asin(θ )( ) jB = Bxi + By j

= −Bsin(φ)( ) i + Bcos(φ)( ) jA +B = ?