# Mechanic -> IPhO Training

1. Jun 27, 2011

### jaumzaum

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

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

2. Relevant equations

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

3. 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 Vn 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
Vn=Vn-1+An-1.dt = Vn-1 +(a-Vn-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!

Last edited by a moderator: Jun 27, 2011
2. Jun 28, 2011

### ideasrule

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?