Question #203995

Find the curvature and torsion of the curve x=a(u-sinu),y=(1-cosu),z=bu


1
Expert's answer
2021-06-07T18:44:47-0400

let

r=[a(usinu),1cosu,bu]r=[a(u-sinu),1-cosu,bu]

so

rr^{'} (single deriverative of r) =[a(1cosu),sinu,0]=[a(1-cosu),sinu,0]

r=[asinu,cosu,0]r^{''}=[asinu,cosu,0]

r=acosu,sinu,0r^{'''}=acosu,-sinu,0


Now :


Curvature =r×rr3=\dfrac{|r^{'} \times r^{''}|}{|r^{'}|^3}


Now: r×r=r^{'} \times r^{''}= i[bcosu]j[absinu]+k[a(cosucos2u)asin2u]i[-bcosu]-j[-absinu]+k[a(cosu-cos^2u)-asin^2u]

=(bcosu)i+(absinu)j+(acosua)k=(-bcosu)i +(absinu)j+(acosu-a)k


r×r=b2(cos2u+a2sin2u)+a2+a2(cos2u2cosu)|r^{'} \times r^{''}|= \sqrt{b^2(cos^2u+a^2sin^2u)+a^2+a^2(cos^2u-2cosu)}


r=a2(1+cos2u)2acosu+sin2u+b|r^{'}|= \sqrt{a^2(1+cos^2u)-2acosu+sin^2u+b}


Curvature =r×rr3=b2(cos2u+a2sin2u)+a2+a2(cos2u2cosu)(a2(1+cos2u)2acosu+sin2u+b)3/2= \dfrac{|r^{'} \times r^{''}|}{|r^{'}|^3} = \dfrac{\sqrt{b^2(cos^2u+a^2sin^2u)+a^2+a^2(cos^2u-2cosu)}}{(a^2(1+cos^2u)-2acosu+sin^2u+b)^{3/2}}


Torsion =[r,r,r]r×r2= \dfrac{[r^{'},r^{''},r^{'''}]}{|r^{'} \times r^{''}|^2}


[r,r,r]=[a(1cosu)(0)][sinu(0)]+b(asin2uacos2u)=ab[r^{'},r^{''},r^{'''}]=[a(1-cosu)(0)] -[sinu(0)] +b(-asin^2u-acos^2u)= -ab


so Torsion =abb2(cos2u+a2sin2u)+a2+a2(cos2u2cosu)= \dfrac{-ab}{b^2(cos^2u+a^2sin^2u)+a^2+a^2(cos^2u-2cosu)}



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS