Calculating the Angle of a Pendulum Swing into a Peg

[StrangerDanger]

Homework Statement

With this problem I have to get the answer: cosθ = r/L * cosα - √(3)/2 * (1 - r/L)
which in other words mean I need to find angle θ with arccos[r/L * cosα - √(3)/2 * (1 - r/L)].

Here's the picture:

Lcosθ is the vertical length of the string at its lowest point.

rcosα is a fraction of that same vertical string in terms of displacement "r" (which is from the start of the string to the peg)

∠β is the angle between the peg and the horizon.

(L-r) sinβ is the height from the end of the peg and the horizontal

(L-r) cosα is the horizontal length of that same peg.

(L-r)cosα is the vertical length of the string from the ball to the peg.

So, this is not really a physics issue but more like a math issue but since this is a physics problem I've decided to put it under here.

My problem is that I am unable to continue from this point as shown on the picture of my attempt. I don't know where to continue from here on out. I am trying to find "t" for the equation but I am unsure how. Where do I continue from now?

Homework Equations

[/B]
Newtonian Position Formula:
y_f = y_i +v_iyt + .5gt²
x_f = x_i +v_ixt + .5gt²

Energy Equation:
Work of hand - force of friction * displacement = delta Kinetic Energy + delta Potential Energy

W_h - f_F*d = [.5*mv_f² - .5*mv_i²] - [mgh_f - mgh_i]

The Attempt at a Solution

Picture of Attempt:

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[BvU]

Hello Stranger,

Your position formula is only valid for uniform acceleration. You don't have that here !
Can you think of a condition you can impose on $\beta$ ?

 

[StrangerDanger]

Hello Stranger,

Your position formula is only valid for uniform acceleration. You don't have that here !
Can you think of a condition you can impose on $\beta$ ?

Thanks for the welcome :).
The only method I can think of when dealing with changing acceleration by breaking it into parts. Each part for every time the value of acceleration changes. I have not yet learned ho to derive very well but I know it exists. As for angle β I am clueless on what to impose.

 

[BvU]

Well, then perhaps you can conquer this one without dealing with changing acceleration ?
The sketch suggests a trajectory, but is it realistic ? Where must the mass run out of sped to fall on the peg ? What would happen if it ran out of speed at e.g. 85 degrees ?