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GHAPTER, , , , Differential Calculus, , CURVATURE AND RADIUS OF CURVATURE, , Consider the two circles shown in the figure., It is obvious that the ways in which the two, circles bend or ‘curve’ at the point P are not, the same. The smaller circle ‘curves’ or, changes its direction more rapidly than the, bigger circle, In other words the smaller circle is said to have greater curvature than the, other. This concept of curvature which holds, good for any curve is formally defined as, follows:, , Definition of curvature, , y, ayay, , y yr Ay, oO x, , , , Let P and Q be any two close points on a plane curve, Let the arcual distances ofP, and Q measured from a fixed point A on the given curve be s and s + As, so that PQ, , (the arcual length of PQ) is As., , Scanned by CamScanner

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Differential Calculus 327, , cet the tangents at P and 4 his |, ine in the plane of the cute ay mn angles wand w+ Aw with a fixed ', Then the angle between the tangents at P ni, Thus for a change of Ay in the areual, tangent to the curve changes by Ay,, Ay., Hence ay 8 the aver:, , And Q = Ay., length of the curve, the direction of the, , ay, rate T ha ‘ ., Be rate of bending of the curve (or average rate of change, , of direction of the tangent to the curve, curvature of the are PQ,, , , Ay) dy |, “ iim, as ag 8 the rate of bending of the curve with respect to arcual, distance at P or the curvature of the curve, by k., , For example, let us find the curvature of a, circle of radius at any point on it,, , Let the arcual distances of points on the, circle be measured from A, the lowest point, of the circle and let the tangent at A be chosen as the x-axis. Let AP = s and let the, tangent at P make an angle y with the, X-axis., , Then s=aACP, , =a [-- the angle between CA and CP equals the angle between the, tespective perpendiculars AT and PT.], , 1 . —, in the arcual interval PQ) or average, , at the point P. The curvature is denoted, , , , ale ale, , or Ss, , yw, dy, “ds., , Thus the curvature of a circle at any point on it equals the reciprocal of its, , radius. Equivalently, the radius of a circle equals the reciprocal of the curvature at, , any point on it. It is this property of the circle that has led to the definition of radius, , of curvature., , Radius of curvature of a curve at any point on it is defined as the reciprocal of the, , as., , dy", , Note: To find k or p of a curve at any point on it, we should know the relation, , between s and w for that curve, which is not easily derivable in most cases., Generally curves will be defined by means of their cartesian or parametric equa, tions. Hence formulas for p in terms of cartesian or parametric co-ordinates are, , Necessary, which are derived below: ;, , Some Basic Results: Let P (x, y) and Q (x + Ax, y+ Ay) be any two close points on, , curve y = f(x). Let AP =s and AQ=s + As where A is a fixed point on the curve., , Let the chord PQ make an angle O with the x-axis., , curvature of the curve at that point and denoted by p. Thus p = + =, , Scanned by CamScanner

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Differential Calculus B29, , L(y)”, EF tin? yy | \de |, Dy ay, da? dx?, , Note: As curvature (and hence radius of curvature) of a curve at any point is, , independent of the choice of x and y-axin, x and y can be interchanged in the, formula for p derived above, Thus p is also given by, , OY aan, This formula will be of use, when a infinite at a point,, ax, , Formula for Radius of Curvature in Parametric, Co-ordinates, , Let the parametric equations of the curve, x=af(t) and y=g(t)., , ‘ dx , dy, Then fee £2(0) and jag net)., , at, i, , Now, , , , Scanned by CamScanner