IPhO Training: Circular Motion Homework

  • Thread starter Thread starter jaumzaum
  • Start date Start date
  • Tags Tags
    Ipho Mechanic
Click For Summary
SUMMARY

The discussion revolves around a physics problem related to circular motion, specifically in the context of rollercoaster dynamics. The problem involves calculating the velocity of a body subjected to a force, centripetal force, and friction, while neglecting gravity. Key equations include the centripetal force equation (Fc = mv²/R) and the friction force equation (Fa = Fc.u). The user seeks assistance in deriving a function for velocity over time and determining its convergence or divergence.

PREREQUISITES
  • Understanding of circular motion dynamics
  • Familiarity with centripetal force and friction force equations
  • Basic knowledge of calculus, particularly integration
  • Concept of acceleration as a function of force and mass
NEXT STEPS
  • Learn how to integrate differential equations in physics contexts
  • Study the relationship between acceleration, velocity, and time in circular motion
  • Explore the implications of friction in dynamic systems
  • Review advanced topics in mechanics, such as energy conservation in circular motion
USEFUL FOR

Students preparing for physics competitions, educators teaching mechanics, and anyone interested in advanced applications of circular motion principles.

jaumzaum
Messages
433
Reaction score
33

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 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:
Physics news on Phys.org
jaumzaum said:
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?