Creating a Plinko Launcher for My Board: Ensuring the Puck Makes it Up

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Narzu6425
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I am trying to create a plinko launcher for my plinko board (as you would see in the price is right). I created a pinball type spring launcher and used a bell crank in order to change the horizontal force into a vertical force. I am using a 3.5in spring with a spring constant of 1.09lbs/in. I plan on having about 2 in of compression which results in a spring force of 9.697Newtons.

I want to ensure that the plinko will make it up the board and into its appropriate slots. I am having trouble coming up with right calculation in order to prove that the plinko puck (4oz or .2lb) will make it up the board. I also was wondering if anyone had any input of how to come up with the appropriate angle of the curve in order to ensure the plinko will make a smooth turn. Thank you all so much for your help.
 
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I would get the work done by the spring, and then that will give you the KE of the mass.

∫f(x)dx gives you the work done by the spring

I converted to metric and here is what I got

f(x) = 190.88x
so the integral is

∫0 - .0508 95.44x2
gave me ≈.246J

If it is launched vertically you should subtract gravity .89N(gravity on puck) * .0508m = .045J

.246 - .045 = .201J Which should be the KE of the puck when it leaves the spring.
If I did my math right that will only give you .23m(8.9in) of height. So depends on the height of your board, and if it contacts anything on the way up(slides up a slanted surface or anything) You should also subtract the work done by friction.
 
Thank you for the quick response! I am going to give this a shot and see what I can come up with.

I realized that I will need much more height than that so I think I am going to try a spring with a different K value. Unfortunately the launcher has too be place within tight constraints so I can't increase the x value for compression.
 
One more quick question...Once i find my Kinetic energy how do I determine my maximum height?

Do I find the initial velocity first and then use gravity as a negative acceleration?
 
Just find the potential energy in the spring before it is released,
[tex]PE = \frac{1}{2}kx^2[/tex]
Where [itex]k[/itex] is the spring constant and [itex]x[/itex] is the distance it is compressed. This value is then equal to,
[tex]mg(h - \cos (\theta) \mu d)[/tex]
With [itex]m[/itex] being the mass of the puck, [itex]g[/itex] being the gravitational force, [itex]h[/itex] being the height that it moves up, [itex]\theta[/itex] being the angle of any surface the puck moves against measured from the ground, [itex]\mu[/itex] being the constant of kinetic friction for the surface and the puck and [itex]d[/itex] being the distance it moves along this surface.
 
Awesome, thank you! I was able to figure out my calculations from all of your help :)