CURVATURE AND RADIUS OF CURVATURE

SOME INTRODUCTION AND TYPES OF QUESTIONS ON RADIUS OF CURVATURE AND CURVATURE.......
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1. MATHS PRESENTATION BY- ABHISHEK RAO ROLL NO - 191882 BRANCH- B.TECH CSE CENTRAL UNIVERSITY OF HARYANA CURVATURE AND RADIUS OF CUVATURE

2. CURVATURE AND RADIUS OF CURVATURE Curvature is an important property of curve. Curvature measures the degree of sharpness of bending of a curve at that point on the curve. The rate of change of the direction of the tangent line, at a point on the curve, with respect to the arc length s along the curve is called the curvature of the curve.

3. Let A be a fixed point on the curve from which the arc length is measured. Let the arc lengths of the points P and Q on the curve be s and Δ s respectively. Then, Δ s is the length of the arc PQ. Let the tangents at the points P and Q on the curve make angles α and α + Δα respectively with the positive direction of the x-axis. Then, Δα is the angle between the tangents at the points P and Q on the curve. The curvature of the curve at the point P is defined as

4. Radius of curvature:- Let a curve be given. As discussed above, the angle between the tangents at the points P and Q on the curve is given by Δα . Let the normals at the points P and Q intersect at M. Then, the angle between the normals PM and QM is Δα that is angle PMQ= Δα . Join the points P and Q. As Q →P, the point of intersection M of the normals moves and takes a fixed position C in the limit. The point C is called the centre of curvature. The length PC is called the radius of curvature. Radius of curvature is the reciprocal of curvature and it is denoted by ρ.

5. Radius of curvature for cartesian curve: y=f(x) Q. Find the radius of curvature at the origin, for the curve Ans. Radius of curvature for paramatric form: x=f(t), y=g(t) Q. Find radius of curvature at any point on the curve Ans.

6. Radius of curvature of polar curves Q. Find the radius of curvature at any point of the curve . Ans :

7. Application: When engineers design train tracks, they need to ensure the curvature of the track will be safe and provide a comfortable ride for the given speed of the trains.

8. THANK YOU Reference : Engineering Mathematics by R.K. Jain and S.R.K Iyenger