Optimizing Time for a Ball on a Sloped Path: Arc or Chord?

[talisman2212]

Homework Statement

We let a ball with mass m to slide down under the influence of his weight the path defined by the chord or the path defined by the arc of the circle until the ball reach the lowest point, as it seems from the picture. In which path the ball will make the shorter time to reach the lowest point?

Homework Equations

The Attempt at a Solution

At first to compute the time at the path of the arc the problem is the same with the period T of a pendulum which is T=2[itex]\pi[/itex][itex]\sqrt{\frac{L}{g}}[/itex] and we are going to have t=[itex]\frac{T}{4}[/itex] but we must not use any known fact about pendulum. I am trying to use Newton's Law F=ma but it's still difficult, I can't find a way to express sinw with L. And even if I find it I don't think I can compute time because the acceleration is not constant.

 

[voko]

There are not too many options. Either you use the pendulum result, or derive it. Another option would be to prove that piecewise linear curves approximating the circle give the lesser time of descent the closer they approximate the circle.

 

[talisman2212]

Our teacher told us that it can be solved without using the pendulum. I have tried lot of things but can't find it.

 

[voko]

Consider the path connecting the same point that consists of two equal chords. Which path is faster: one-chord or two-chord? Then consider the path of four equal chords and so on.

 

[talisman2212]

Consider the path connecting the same point that consists of two equal chords. Which path is faster: one-chord or two-chord? Then consider the path of four equal chords and so on.

I know how to comptute the time in chord but I can't compute the time in the arc.

 

[voko]

You do not need to compute the time in the arc to answer the question in the problem.

 

[talisman2212]

You do not need to compute the time in the arc to answer the question in the problem.

I know that, but our teacher told us to compute it anyway.

 

[voko]

Your options are listed in #2.

 

[talisman2212]

Your options are listed in #2.

How I will derive it?

 

[voko]

You derive it by solving the equation of motion, which you could get from the laws of Newton or conservation of energy.

 

[talisman2212]

You derive it by solving the equation of motion, which you could get from the laws of Newton or conservation of energy.

I used the small angle approximation where sin[itex]\theta\approx\theta[/itex] and I'm here: a(t)=-[itex]\frac{L}{g}\theta(t)[/itex] where a(t) is the angular momentum in a given time t. How I will find now that T=2[itex]\pi\sqrt{\frac{L}{g}}[/itex]?

 

[voko]

Are you sure a(t) is the angular momentum? Isn't it the angular acceleration?

 

[talisman2212]

Are you sure a(t) is the angular momentum? Isn't it the angular acceleration?

Sorry, is the angular acceleration. How I find T now?

 

[voko]

You have differential equation ## \frac {d^2\theta} {d\theta^2} + \frac L g \theta = 0 ##. Solve it.