How Do You Calculate Force for a Marble to Complete a Loop in a Rollercoaster?

[Breanne]

Homework Statement

I am trying to calculate the speed and force needed to make a marble complete a loop in the rollercoaster that i have been instructed to make. I do not know how to calculate f though. I can not use v=d/t because I can't find time until the rollercoaster is built which I cannot build until I find the force needed to get the marble to go through the loop. How do i find f?
Radius (arbitrary) = 11 cm
= 0.11 m
mass = approx 3.56g

Homework Equations

T = 1/f
v = 2πr/T
FR =mv^2 / r

The Attempt at a Solution

v = 2πr/T
= 2πr/(1/f)
= 2π0.11/(1/f)
=0.6911503838/(1/f)
FR = mv^2/r
= 3.56v^2 / 0.11
= 32.36363636v^2
= 32.36363636(0.6911503838/(1/f))

π is supposed to be pi, the alt code I found did not work sorry

 

[PhanthomJay]

You need to determine the height of the top of the takeoff ramp such that its speed at the top of the circle is just enough to prevent the marble from falling when it reaches the top of the loop. Once youi calculate the theoretical value of the marble's speeed, there will be friction and drag to consider. Use conservation of energy and the centripetalacceleration equation, and assume no contact force at the top. You'll probably have to experiment to determine the proper height, or make the ramp high enough or the radius larger to exceed the minimum value for the speed.

 

[Breanne]

Thank you for the help, the links where especially useful :)

 

[sean-820]

I would use F=(mv^2) /r

m=mass
v=velocity
r=radius

You can find the minimum force needed by F=mg then sub into the first equation. This is assuming the radius is constant. Real rollercoasters are clothoid loops (oval) , not circles if you wanted to do an actual roller coaster. These loops make the forces more balanced and so you can do a lower velocity and experience the same force. i doubt it would matter if you just need to find the minimum velocity to do a loop, but its worth a shot. Like said already you will need a bit more velocity to overcome friction

 

[Breanne]

Thank you both for your help. after following the above friction link I found the equation for work done by friction: W= ∫ μK N ds
I did not know what two of the variables represent and went to my teacher for help. She told me not to use this equation because I haven't taken calculus. I still need to calculate how much of the energy in my rollercoaster has be converted or lost due to friction. I would greatly appreciate it if the following variables could be explained to me: ∫ and s .
Thanks :)