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Radius of Curvature

Find the radi...

Question

Find the radius of curvature $R$ at each point of trajectory as a function of $x$ .

R = (\frac{a}{b}) {[1 + {(\frac{x b}{a})}^{2}]}^{\frac{5}{2}}

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R = (\frac{a}{b}) {[1 + {(\frac{x b}{a})}^{2}]}^{\frac{2}{3}}

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R = (\frac{a}{b}) {[1 + {(\frac{x b}{a})}^{2}]}^{\frac{3}{2}}

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R = (\frac{a}{b}) {[1 + {(\frac{x b}{a})}^{2}]}^{\frac{3}{5}}

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Solution

The correct option is B $R = (\frac{a}{b}) {[1 + {(\frac{x b}{a})}^{2}]}^{\frac{3}{2}}$
$v_{x} = a s o, x = a t v_{y} = b x = a b t s o, y = \frac{a b t^{2}}{2} = \frac{b x^{2}}{2 a} . . . . . b y i n t e g r a t i o n a n g l e, tan θ = \frac{b x}{a} a_{x} = \frac{d v_{x}}{d t} = 0 & a_{y} = \frac{d v_{y}}{d t} = a b A c c e l e r a t i o n, a_{n e t} = a b a l o n g y - a x i s N o r m a l a c c e l e r a t i o n = a_{n} = a_{n e t} cos θ = a b \frac{a}{\sqrt{a^{2} + {(b x)}^{2}}} R a d i u s, R = \frac{v^{2}}{a_{n}} = \frac{{(a^{2} + {(b x)}^{2})}^{3 / 2}}{a^{2} b} = \frac{a}{b} {[[1 + {(\frac{x b}{a})}^{2}]]}^{3 / 2}$

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Find the radius of curvature R at each point of trajectory as a function of x.

The correct option is B R=(ab)[1+(xba)2]32vx=aso,x=atvy=bx=abtso,y=abt22=bx22a.....byintegrationangle,tanθ=bxaax=dvxdt=0&ay=dvydt=abAcceleration,anet=abalongy−axisNormalacceleration=an=anetcosθ=aba√a2+(bx)2Radius, R=v2an=(a2+(bx)2)3/2a2b=ab[[1+(xba)2]]3/2

Find the radius of curvature $R$ at each point of trajectory as a function of $x$ .