Question

Find the length of cycloid from one cusp to the next , where `x=a(θ + sinθ) , y=a(1-cosθ)`

Answer 1

Given curve :Cycloid `x=a(θ + sinθ) , y=a(1-cosθ)`  

The length of given curve is : 

`s=int_(θ1)^(θ2) sqrt((dx/(dθ))+(dy/(dθ))^2 dθ)` 

`dx/(dθ)=a(1+cosθ)                dy/(dθ)=a sin θ` 

∴ `(dx/(dθ))^2 +(dy/(dθ))^2=a^2[1+2 cosθ +cos^2 θ+sin^2θ ]`

                      =`2a^2[1+cosθ]`

                      = `4a^2 [cos^2  θ/2]`

∴`sqrt((dx/dθ)^2+(dy/(dθ))^2)= 2a cos  θ/2` 

∴` s= int_-pi^pi 2acos  θ/2  dθ`

= `2xxint_0^pi 2 a cos  θ/2 dθ` 

=` 4a [2 sin  θ/2]_0^pi` 

∴ `s= 8a`