IPhO Training: Circular Motion Homework

In summary, the conversation discusses a physics exercise involving a force being applied to a body in a rollercoaster, causing it to follow a circular trajectory. The question asks to calculate the velocity as a function of time, determine if it will converge or diverge, and calculate when the body will stay in motion. The conversation also mentions attempting to solve the problem using infinitesimal calculus.
  • #1
jaumzaum
434
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!
 
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  • #2
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?
 

FAQ: IPhO Training: Circular Motion Homework

1. What is the purpose of Mechanic -> IPhO Training?

The purpose of this training is to prepare students for the International Physics Olympiad (IPhO), an annual physics competition for high school students from around the world. The training focuses on developing problem-solving skills and deepening understanding of physics concepts.

2. Who is eligible to participate in Mechanic -> IPhO Training?

Students who are interested in physics and have a strong foundation in the subject are eligible to participate in this training. Typically, these are high school students between the ages of 15-19 years.

3. What topics are covered in Mechanic -> IPhO Training?

The training covers a wide range of topics in mechanics, which is a branch of physics that deals with the motion and forces of objects. Some of the topics that are covered include kinematics, Newton's laws of motion, work and energy, and rotational motion.

4. How long is the Mechanic -> IPhO Training?

The duration of the training can vary, but it typically lasts for a few months to a year. It depends on the intensity of the training and the schedule set by the organizing committee. Some programs may also have a preliminary round of training before selecting students for the final round.

5. How can I participate in Mechanic -> IPhO Training?

In order to participate in the training, you need to first qualify for your country's national physics Olympiad and be selected to represent your country at the IPhO. You can also reach out to your national physics society or organization for more information on how to join the training program.

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