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Unformatted text preview: Curvature Vector: from Newton’s Second Law to Beam Deflection and to Relativity Shina Tan 1 Curvature Vector Any point r on a curve can be parameterized according to the length s from some starting point on the curve to point r . We define ˆ s = d r ds = r ( s ) . (1) Using the formula ds 2 = d r · d r , (2) we can check that ˆ s is a unit vector, because d r ds · d r ds = dr · d r ds 2 = ds 2 ds 2 = 1 . ˆ s is the local tangential direction on the curve. If ˆ s changes direction as you go along the line, the line is curved . We can define the local curvature vector c : c ≡ d ˆ s ds = r 00 ( s ) . (3) If the curve changes its local tangential direction by 1 ◦ per meter, its curvature is c = 1 ◦ 1m = ( π/ 180)radians 1m = 0 . 01745 / m . The local curvature vector c is always perpendicular to the local direction vector ˆ s : c · ˆ s = 0 . (4) Proof: c · ˆ s = d ˆ s ds · ˆ s = 1 2 d (ˆ s · ˆ s ) ds = 1 2 d 1 ds = 0 . Example 1– At any point on a circle of radius R , there is a local curvature vector whose magnitude is c = 1 /R , and whose direction is toward the center of the circle. This is obviously perpendicular to the local tangential direction vector. Remark – Unlike the local tangential direction vector which reverses direction if you choose a different end of the curve as the starting point of s , the local curvature vector c does not depend on the parameterization. For instance, no matter how you parameterize the length on a circle, the curvature vector always points toward the center of the circle. 1 Figure 1: Curvature vectors at various points on a parabola. Example 2– Find the local curvature of any planar curve y = y ( x ). Solution: the local direction on the curve can be represented by the angle, θ = arctan y ( x ). So c = dθ ds = d arctan y p 1 + y 2 dx = y 00 (1 + y 2 ) 3 / 2 . Thus the general formula for the curvature of a planar curve is c ( x ) = y 00 ( x ) 1 + y 2 ( x ) 3 / 2 . (5) Example 3– The local curvature of a parabola y = 1 2 kx 2 is c ( x ) = k (1+ k 2 x 2 ) 3 / 2 . The curvature is largest at the parabola’s apex, and diminishes along its wings. Obviously, the wings are “straighter” than the bottom. See Figure 1. Example 4– Find the curvature of a solenoid. Solution: A solenoid can be described by the parametric equations x = R cos αs R , y = R sin αs R , z = √ 1- α 2 s, where x 2 ( s ) + y 2 ( s ) + z 2 ( s ) = 1 so that s is the length parameter of the solenoid. Taking the second order derivatives of x ( s ), y ( s ), and z ( s ), we find c ( s ) =- α 2 R cos αs R ,- α 2 R sin αs R , , and c ( s ) = α 2 /R . The curvature vector points toward the axis of the solenoid, and is perpendicular to the cylindrical surface on which the solenoid lies. If α = 1, the solenoid degenerates into a circle with radius R , and c = 1 /R . If α < 1, the curvature is smaller than 1 /R ....
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This note was uploaded on 01/22/2012 for the course PHYS 3202 taught by Professor Tan during the Spring '11 term at Georgia Tech.
- Spring '11