How Is Time Calculated for a Car Sliding Along a Cycloid?

[Liquidxlax]

Homework Statement

consider a sigle loop of the cycloid with a fixed value of a. A car is released at a point P₀ from rest anywhere on the track between the origin and the lowest point P, that is P₀ has a parameter 0<theta₀ < pi. show that the time for the cart to roll from P₀ to P is given by the integral

time( P0]/sub] -> P) = $\sqrt{\frac{a}{g}}\int \sqrt{\frac{1 - cos\vartheta}{cos\vartheta_{0} - cos\vartheta}}d\vartheta$

integral is from theta naught to pi

and prove the integral equals $\pi\sqrt{\frac{a}{g}}$ the integral may be tricky and you can use theta = pi -2(alpha)

Homework Equations

$\frac{df}{dx}=\frac{d}{dy}\frac{df/dx'}$

The Attempt at a Solution

1/2 mv² = mg(y-y₁

v = $\sqrt{2g(y-y₁}$

dt = ds/v

T = $\int\frac{\sqrt{1+(x')²}}{\sqrt{2g(y-y₁}}$ dy

test the Euler formula to get

y = a(1 - cos(theta)) y' = asin(theta)

x = a (theta - sin(theta)) x' = a - acos(theta)

This is now the part I am having a problem with

Now i substitute my y and x' in

T = $\int\frac{\sqrt{1 + (a( 1 - cos(\vartheta))²}}{\sqrt{2ga(cos(\vartheta) - cos(\vartheta₀)}}(asin(\vartheta)$

and somehow that equals the top equation. Then i need to integrate it, which i get no where near close to what it wants no matter what the substitution and trig identities

not sure why all the itex isn't working

 

[Spinnor]

Do this help?

 

[Liquidxlax]

Do this help?

it sure did sometimes I'm such a retard, x and y do not depend on each other they depend on theta...

because i was doing sqrt ( 1 y'^2) dx

thank you very much

 

[Spinnor]

it sure did sometimes I'm such a retard, x and y do not depend on each other they depend on theta...

because i was doing sqrt ( 1 y'^2) dx

thank you very much

x and y do depend on each other but the answer had to be in terms of theta so that was a more useful parameter?