Pendulum swinging and hitting a peg

[angrymasteryoda]

Homework Statement

a pendulum of mass m and length L is pulled back an angle of θ and released. After the pendulum swings through its lowest point it encounters a peg α degrees out and r meters from the top of the string. The mass swings up about the peg until the string becomes slack with the mass falling inward and hitting the peg. Show for this condition cosθ=(r/L)cosα - (sqrt(3)/2)(1-(r/L))
http://michael-tech.hostzi.com/ec.png

Homework Equations

Energy from start to slack
mgh = mgh₂ + (mv²)/2
forces on slacking pendulum
∑F = mg = (mv²)/L

The Attempt at a Solution

so i have gotten the energy from start to where it slacks

with forces when it slacks i get

solving for v^2

plugging and chugging with energy i get

if image is broken http://michael-tech.hostzi.com/e1.gif

plugging back into forces i get

now it is just a projectile and this is where i am stuck. Here is what i know at this point.

= 0

= 0

= don't think i need, think its 0 though v_y = don't think i need, think its 0 though
a = 0 a = -g
t=t

I am getting stuck at the Vnot should it be vcosβ. Also does the above make sense so far

 

[haruspex]

The initial height is not L cos(θ), unless you are measuring height downwards.
You do need to care about both velocity components at the point where the string goes slack. What you not care about is final velocities, so use the vertical SUVAT equation that does not involve final velocity. Time only matters to the extent that x and y must coincide with the peg at the same point in time.