Which Bead Reaches the Bottom First?

[tacutamon]

Homework Statement

Two thin rods are fastened to the inside of a circular ring as shown in the figure below. One rod of length D is vertical, and the other of length L makes an angle θ with the horizontal. The two rods and the ring lie in a vertical plane. Two small beads are free to slide without friction along the rods.

If the two beads are released from rest simultaneously from the positions shown, use your intuition and guess which bead reaches the bottom first.

Homework Equations

Equations for Red Ball:
(1) D = .5*g*t^{2}

Equations for Blue Ball:
(2) a_{b} = g*sin(θ)

(3) L = .5*a_{b}*t^{2}

(4) L = D*sin(θ)

The Attempt at a Solution

The correct answer is that they will hit the ground at the same time. I am still curious in how I would explain this mathematically, rather than intuitively as the question states. I believe I have all the necessary equations to do so, but I am not sure how to compare them to show this. Would I show that subbing in equation 4 and 2 into equation 3 would give me equation 1?

Thanks so much for your help!

 

[TSny]

Hello, tacutamon and welcome to PF!

Homework Equations

Equations for Red Ball:
(1) D = .5*g*t^{2}

Equations for Blue Ball:
(2) a_{b} = g*sin(θ)

(3) L = .5*a_{b}*t^{2}

(4) L = D*sin(θ)

Would I show that subbing in equation 4 and 2 into equation 3 would give me equation 1?

Yes, that would work. Or, you could solve (1) for the red ball's time and solve (3) for the blue ball's time and try to show the times are equal.

 

[tacutamon]

Thanks so much! That really clears up a lot.