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Published byJeremy Walton Modified over 9 years ago
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Angular Velocity: Sect. 1.15 Overview only. For details, see text! Consider a particle moving on arbitrary path in space: –At a given instant, it can be considered as moving in a plane, circular path about an axis Instantaneous Rotation Axis. In an infinitesimal time dt, the path can be represented as infinitesimal circular arc. As the particle moves in circular path, it has angular velocity: ω (dθ/dt) θ
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Consider a particle moving in an instantaneously circular path of radius R. (See Fig.): –Magnitude of Particle Angular Velocity: ω (dθ/dt) θ –Magnitude of Linear Velocity (linear speed): v = R(dθ/dt) = Rθ = Rω
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Particle moving in circular path, radius R. (Fig.): Angular Velocity: ω θ Linear Speed: v = Rω (1) Vector direction of ω normal to the plane of motion, in the direction of a right hand screw. (Fig.). Clearly : R = r sin(α) (2) (1) & (2) v = rωsin(α) So (for detailed proof, see text!): v = ω r
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Gradient (Del) Operator: Sect. 1.16 Overview only. For details, see text! The most important vector differential operator: grad A Vector which has components which are differential operators. Gradient operator. In Cartesian (rectangular) coordinates: ∑ i e i (∂/∂x i ) (1) NOTE! (For future use!) is much more complicated in cylindrical & spherical coordinates (see Appendix F)!!
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can operate directly on a scalar function ( gradient of Old Notation: = grad ): = ∑ i e i (∂ /∂x i ) A VECTOR! can operate in a scalar product with a vector A ( divergence of A; Old: A = div A ): A = ∑ i (∂A i /∂x i ) A SCALAR! can operate in a vector product with a vector A ( curl of A; Old: A = curl A ): ( A) i = ∑ j,k ε ijk (∂A k /∂x j ) A VECTOR! ( Older: A = rot A) Obviously, A = A(x,y,z)
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Physical interpretation of the gradient : (Fig) The text shows that has the properties: 1. It is surfaces of constant 2. It is in the direction of max change in 3. The directional derivative of for any direction n is n = (∂ /∂n) (x,y) Contour plot of (x,y)
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The Laplacian Operator The Laplacian is the dot product of with itself: 2 ; 2 ∑ i (∂ 2 /∂x i 2 ) A SCALAR! The Laplacian of a scalar function 2 ∑ i (∂ 2 /∂x i 2 )
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Integration of Vectors: Sect. 1.17 Overview only. For details, see text! Types of integrals of vector functions: A = A(x,y,z) = A(x 1,x 2,x 3 ) = (A 1,A 2,A 3 ) Volume Integral (volume V, differential volume element dv) (Fig.): ∫ V A dv ( ∫ V A 1 dv, ∫ V A 2 dv, ∫ V A 3 dv)
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Surface Integral (surface S, differential surface element da) (Fig.) ∫ S A n da, n Normal to surface S
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Line integral (path in space, differential path element ds) (Fig.): ∫ BC A ds ∫ BC ∑ i A i dx i
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Gauss’s Theorem or Divergence Theorem (for a closed surface S surrounding a volume V) See figure; n Normal to surface S ∫ S A n da = ∫ V A dv Physical Interpretation of A The net “amount” of A “flowing” in & out of closed surface S
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Stoke’s Theorem (for a closed loop C surrounding a surface S) See Figure; n Normal to surface S ∫ C A ds = ∫ S ( A) n da Physical Interpretation of A The net “amount” of “rotation” of A
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