Chapter 16 Curvature-I
16.1 Exercise 9
16.1.1 Question 1
Find the radius of curvature at any point \((x,y)\) for the curve
\(y^2 = 4ax\) (TU 2057)
\(x^{2/3} + y^{2/3} = a^{2/3}\)
\(y=c \cos \text{h} \left(\frac{x}{c}\right)\)
\(xy=c^2\)
\(x=a \cos \theta, y = a \sin \theta\)
\(x = a \cos \phi, y = b \sin \phi\)
16.1.2 Question 2
In the cycloid \(x=a(\theta + \sin \theta), y = a(1-\cos \theta)\) at \(\theta =0\), prove that \(\rho = 4a\).
16.1.3 Question 3
Show that the radius of curvature at a point \((r, \theta)\) for the curve \(r=ae^{\theta \cot \alpha}\) is \(\rho = r \csc \alpha\).
16.1.5 Question 5
Show that the radius of curvature for the curve \(r^m = a^m \cos m \theta\) is \(\frac{a^m}{(m+1)r^{m-1}}\).
16.1.6 Question 6
Find the radius of curvature at the point \((p,r)\) of the curve \(r^{m+1} = a^m p\).