# Very hard mechanics question

## Homework Statement

a pendulum of mass m and length L is pulled back an angle 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 that cosθ=r/Lcosα-√3/2(1-r/L)

## The Attempt at a Solution

I tried to find the forces on the pendulum at the initial and final position
Lcosθ-mg=0
Lsinθ=ma
rcosθ-mg=0
rsinθ=ma
but it doesn't seem to work at all. what I get is that r=L which is impossible.

You need to set up equations for each of the following:
1. The projectile motion problem that starts when the string is no longer taught.
2. The position (angle) at which the ball enters projectile motion (when the gravitational force inward surpasses the necessary centripetal force).
3. The ball's velocity (using conservation of energy) when it enters projectile motion.

Then you will need to combine all of those so that the angle you found in (2) cancels out, the time component you introduced in (1) also cancels out, and the velocity you solved for in (3) should cancel out. Once you solve this for cos(θ) you should get the answer written.