# Homework Help: Pendulum and Inclined plane.

1. Oct 3, 2012

### talisman2212

1. The problem statement, all variables and given/known data
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?

2. Relevant equations

3. 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$\pi$$\sqrt{\frac{L}{g}}$ and we are going to have t=$\frac{T}{4}$ 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.

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2. Oct 3, 2012

### 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.

3. Oct 3, 2012

### 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.

4. Oct 3, 2012

### 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.

5. Oct 4, 2012

### talisman2212

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

6. Oct 4, 2012

### voko

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

7. Oct 4, 2012

### talisman2212

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

8. Oct 4, 2012

### voko

Your options are listed in #2.

9. Oct 5, 2012

### talisman2212

How I will derive it?

10. Oct 5, 2012

### voko

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

11. Oct 6, 2012

### talisman2212

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

12. Oct 6, 2012

### voko

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

13. Oct 7, 2012

### talisman2212

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

14. Oct 7, 2012

### voko

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