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Unformatted text preview: 1 Chapter 11: Angular momentum Chapter 11: Angular momentum The Vector Product and Torque The Vector Product and Torque c τ = r Fsin φ b τ = r x F c The torque is the vector (or cross) product of the position vector and the force vector c The torque vector lies in a direction perpendicular to the plane formed by the position vector and the force vector Earlier we saw where the product of two vectors was a scalar. This was called the dot product Here we see that the product of two vectors can be another vector c The vector product of two vectors is also called the cross product 2 The Vector Product Defined The Vector Product Defined c Given two vectors, A and B c The vector (cross) product of A and B is defined as a third vector, C = A x B c C is read as “ A cross B ” c The magnitude of C is AB sin θ b θ is the angle between A and B c The quantity AB sin θ is equal to the area of the parallelogram formed by A and B c The direction of C is perpendicular to the plane formed by A and B c The best way to determine this direction is to use the righthand rule Properties of the Vector Product Properties of the Vector Product c The vector product is not commutative. The order in which the vectors are multiplied is important c To account for order, remember A x B =  B x A c If A is parallel to B ( θ = 0 o or 180 o ), then A x B = 0 b Therefore A x A = 0 c If A is perpendicular to B , then  A x B  = AB c The vector product obeys the distributive law b A x ( B + C ) = A x B + A x C c The derivative of the cross product with respect to some variable such as t is where it is important to preserve the multiplicative order of A and B ( ) d d d dt dt dt × = × + × A B A B B A 3 Vector Products of Unit Vectors Vector Products of Unit Vectors ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ × = × = × = × = − × = × = − × = × = − × = i i j j k k i j j i k j k k j i k i i k j c Signs are interchangeable in cross products b A x ( B ) =  A x B b ( ) j i j i ˆ ˆ ˆ ˆ × − = − × c The cross product can be expressed as c Expanding the determinants gives ˆ ˆ ˆ ˆ ˆ ˆ y z x y x z x y z y z x y x z x y z A A A A A A A A A B B B B B B B B B × = = − + i j k A B i j k ( ) ( ) ( ) ˆ ˆ ˆ y z z y x z z x x y y x A B A B A B A B A B A B × = − − − + − A B i j k Using Determinants Using Determinants Torque Vector Example Torque Vector Example c Given the force c τ = ? m ) ˆ 00 . 5 ˆ 00 . 4 ( N ) ˆ 00 . 3 ˆ 00 . 2 ( j i r j i F + = + = ˆ ˆ ˆ ˆ [(4.00 5.00 )N] [(2.00 3.00 )m] ˆ ˆ ˆ ˆ [(4.00)(2.00) (4.00)(3.00) ˆ ˆ ˆ ˆ (5.00)(2.00) (5.00)(3.00) ˆ 2.0 N m τ = × = + × + = × + × + × + × = ⋅ r F i j i j i i i j j i i j k 4 Angular Momentum Angular Momentum c Consider a particle of mass m located at the vector position r and moving with linear momentum p ( ) p r p r p r × = × + × = ∑ dt d dt d dt d τ dt d p r F r × = × = ∑ ∑ τ v v p...
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 Spring '08
 bose
 Physics, Angular Momentum, Force, Momentum

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