IPhO Training: Circular Motion Homework

[jaumzaum]

Homework Statement

i'm being trained to the IphO (Internataional Physics Olimpyad), and I've come across the following exercise:

A F Force is applied to a body initially stopped in a rollercoaster. The F force makes the body describe a circular trajectory at the rollercoaster (loop). As the body is getting speed, its centripetal force raises too,and if we have a friction force, and forgetting the gravity, (making the F force, the centripetal force and its reaction and the friction forc the only forces applied to the body), we have that the reaction of the centripetal force is the "Normal" for the friction force, and as the time passes, the speed raises, the centripetal force raises, and the friction force raises too. The question is:

a) Calculate the function Velocity in function of time
b) Will the velocity converge or diverge? To how?
c) Will the body stay in MU any time? If so, calculate when.




If you could help me, I thank you

[]'s
John

Homework Equations

m=mass
F1=F Force
R=Radius of the RollerCoaster
Fc=Centripetal Force = mv²/R
Fa=Friction Force = Fc.u, u = friction coefficient
Ar = Resulting Acceleration = (F1-Fa)/m

The Attempt at a Solution

Ar = F1/m - v²u/R


Assuming:

F1/m = a
u/R = b


But v is already function of Ar, and I don't know how to continue, I really stopped here.
I've tried some infinitesimal calculus, but nothing so revelant:

Calling V_n as the velocity in the time n.dt, where dt is a infinitesimal part of time, like 1/Infinity

T=0.dt
V₀=0
A₀=a

T=1.dt
V₁=V₀+A₀. dt = a.dt
A₁=a-V₁²b = a-a.b.dt²

T=n.dt
V_n=V_n-1+A_n-1.dt = V_n-1 +(a-V_n-1²b) dt



I've tried to calculate V₀ to V₅, and being x= a.b.dt²



V₀/a.dt = 0

V₁/a.dt = 1

V₂/a.dt = 2-x

V₃/a.dt = 3 - x (5 + (-4 + x) x) =

3 - 5 x + 4 x² - x³

V₄/a.dt = (2 - x) (2 + x (-6 + x (14 + x (-18 + x (14 + (-6 + x) x))))) =

4 - 14 x + 34 x² - 50 x³ + 46 x⁴ - 26 x⁵ + 8 x⁶ - x⁷

V₅/a.dt = 5 - x (30 + x (-146 + x (518 + x (-1398 + x (2950 + x (-4904 + x (6421 + x (-6600 + x (5288 + x (-3260 + x (1512 + x (-508 + x (116 + (-16 + x) x))))))))))))) =

5 - 30 x + 146 x² - 518 x³ + 1398 x⁴ - 2950 x⁵ + 4904 x⁶ - 6421 x⁷ + 6600 x⁸ - 5288 x⁹ + 3260 x¹⁰ - 1512 x¹¹ + 508 x¹² - 16 x¹³ + 16 x¹⁴ - x¹⁵



I didn't see any relation between these numbers, I've tried to solve it integrating the Force, getting the Job, but then I had the velocity that was the derivate of distance, I don't know, it seems easy problem at first, but I really can't solve it!

 

[ideasrule]

Ar = F1/m - v²u/R


Assuming:

F1/m = a
u/R = b

Ar is dv/dt, so you have dv/dt=a-bv^2. Rearranging, we have dv/(a-bv^2)=dt. Do you know how to integrate this equation?

Good luck in IPhO! Which country are you representing?