In: Other Topics

Submitted By jeepking96
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ME 2330 By K. L. Ting 5. Tangential and Normal components Let R(s) = x(s) i + y(s) j represent a curve and s is the independent parameter representing the arc length of the curve measured from a reference point. At s and s’, a particle moves from P to P’ and the displacement is dR (Fig. 1). If P and P’ is infinitesimally separated, i.e. ds= s’-s → 0, l dR l = ds, dR/ds = ut.

ut is a unit vector (called unit tangent) tangent to the curve at P (Fig. 2). At P and P’, let ut and ut‘ be the unit vectors in the tangent direction. The lines normal to ut and ut‘ intersect at O (Fig. 2). With ds → 0, point O is the center of curvature and ds/dθ = ρ where ρ, called radius of curvature, is the length OP. In Fig. 3, one may observe that dut is normal to ut and since ut and ut’ are unit vectors, dut/dθ is a unit vector, designated as un. un is the unit vector (called unit normal) in the direction from P to O. That is, dut/ds =(dut/dθ)(dθ/ds) = (1/ρ) un or dut/ds = k un

where k (= 1/ρ = dθ/ds) is the curvature at P.

Fig. 1

Fig. 2

Fig. 3

With a Cartesian coordinate system, the radius of curvature at (x, y) can be calculated as

ρ = [1 + (dy/dx)2]3/2/ ld2y/dx2l or ρ=

where the dots on x and y represent the first and second derivatives with respect to any independent parameter, such as t or s. In a circle, the center of curvature is the center of the circle and the radius of curvature is the radius of the circle.

Let a particle move along a path R(s). Differentiating R with respect to time yields V = dR/dt = (dR/ds)(ds/dt) = ut where it again yields a = dV/dt = where = ut )dt = ut + (dut /ds)(ds/dt) = ut + ( 2/ρ)un. at represents the tangential acceleration, which is caused by the is the tangential velocity or the speed of the particle. Differentiating

change of the speed, and 2/ρ = v2/ρ the normal acceleration of the particle. The normal acceleration,…...

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