Terragravitic Induction

[eccentrically1]

I guess the interesting thing about it is that a point on a circle traces the curve so it's easy to draw or trace it on something and then build it.
It would be interesting to know what approaches the 5 guys took to solve the problem back in the day.
Now if there were only a fast path up the slope :/

I suppose there is if you rotate the slope.

Regards

The fast path is there but is it a cycloid?

[agor95]

The fast path is there but is it a cycloid?

To be honest it is Saturday night an I am in front of a nice fire with a old molt whisky.

The idea was to spin the ball up the parabola slope to the top then spiral the ball down a center helix to recover potential and K.E.

Regards

[johannesbender]

Here is a Brachistochrone curve in dxf format , scaled it to just over 1m width , feel free to import it in to your sims.

ETA: here's a mirrored one too.
ETA: here's a closed looped one too.

[Fletcher]

Thanx for the help jb .. just imported the zip files and extracted them - then imported into WM - nice.

[agor95]

With the Brachistochrone curve and shortest distance straight ramp to study.

We appreciate the Brachistochrone curve is longer distance than the straight ramp.
And we see the ball arrives first following the curve.

So the ball is moving with a greater speed on the curve.
Does that mean it has more K.E. ?

We expect both tracks to convert the equal P.E. to equal K.E. at the end of their
respective paths.

The speed of each ball are different and their tracks.
The rate of change in their acceleration is also different.
However this change is the same on average over the length of the ramps.
If we look down at the ramps then could they be placed on a curve like a circle.
Or a spiral away or towards this circular base line?

If we treat circular variants affect as Ersatz gravity then the ball can run down this radial slope. The more the speed or radius the more E-gravity effects.

Two study points 1. The ball being rubber and the circular variants will push with different amount on the leading and trailing surface. 2. The inverted Brachistochrone curve should be analyzed.

That means the ball accelerates slowly as the spiral out develops.
The rate of acceleration increases along the ramp.

[johannesbender]

Same KE at the end , the difference you refer to is not a difference at the final end , the difference is during the travel along the curve when the ratio of fall vs lateral movement changes first to higher rate of fall which build momentum etc , then a gradual change to a higher rate of lateral movement where it expends again in its own movement , there is no extra energy as can be seen at the end .

if one is less effective than the other at going down and sideward it does not mean extra energy is available or shows a break , for instance falling straight down would beat the momentum or KE of going down the curve at some time periods before they reach the end positions , some of the GPE lost is converted in a motion towards the side with the curve while the other is more efficiently just falling down at that time .

[Leafy]

Gravity, motion, and collision.

Gravity can facilitate motion
Slopes can facilitate collision

Both is essential

[Leafy]

Asymmetric?

[Leafy]

Let me put the brischtochromeometer thing in a different perspective.

Suppose we have two balls, one plastic and one metal, drop at the same height and speed.

Both ball hits the surface at the same time, but the metal rebounded and on its way up while the plastic takes it time to rebound.

We can say that the energy rate, or power of the metal one is higher while they both have the same energy.

The slopes is comparison to elasticity I supposed.

[Fletcher]

The brachistochrone (shortest time) problem is one of optimization ..

It minimizes the time taken by a ball released from a standing start on a track and accelerated in a gravity field to a lower destination height.

It does this by finding the track shape of most efficient compromise between the path that best accelerates the ball and the shortest path to destination.

The track is steep at the beginning of the path so that the ball can quickly get up to speed.

Once it has initially gained a lot of speed the path can level out and start heading toward the destination.

The track path can even go below the level of its destination and then climb back up to destination. This longer distance route gives the ball a little bit of extra initial speed that will save it just a little bit of extra time.

** At ANY vertical height from release the KE of the ball on different track shapes is the same ( mgh = m1/2v^2 .. or .. GPE = KE ) as it is upon arrival at destination. All that changes with track shape is the distance traveled and the time taken.

[WaltzCee]

Thanx for the help jb .. just imported the zip files and extracted them - then imported into WM - nice.

Hello Fletcher,

I have a 6 body SIM I was wondering if I described it, would you SIM it for me?

There's one ridgid joint & 8 pivoting pin joints.

If you'll attempted it, I'll start a thread.

[Fletcher]

For you Walt .. I'll give it a go :7)

Usual caveat's (t's&c's) apply ..

[agor95]

Thank you Fletcher & JB

Your descriptions are sound. I am just thinking on the apparent weight of a ball and it's speed
during various slope paths.

Regard

[Fletcher]

FWIW .. In my experience Agor it helps to build a simple mind picture and context when attempting to analyze an answer to a question of a mechanical nature such as the Brachistochrone.

In this case I extended the context to a bob-weighted pendulum or a rim-weight on a circle wheel acting under gravity. n.b. both pend rod and circle background having zero mass.

In both examples above the radius doesn't change as they swing downwards and then upwards .. and their KE's at any given height loss or regain are identical .. just like the Brachistochrone.

[Senax]

FWIW .. In my experience Agor it helps to build a simple mind picture and context when attempting to analyze an answer to a question of a mechanical nature such as the Brachistochrone.

In this case I extended the context to a bob-weighted pendulum or a rim-weight on a circle wheel acting under gravity. n.b. both pend rod and circle background having zero mass.

In both examples above the radius doesn't change as they swing downwards and then upwards .. and their KE's at any given height loss or regain are identical .. just like the Brachistochrone.

But for a shallow pendulum swing from points A and B at the same height
and a deep pendulum swing from points the same points A to B the
time taken is not the same.
The shallow swing takes longer than the deep swing.