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

In this website the drawing of the situation in the first exercise can be seen, for better understanding.

http://wwwprof.uniandes.edu.co/~gtel...s-semana10.pdf [Broken]

The problem asks me to determine the minimum height h to allow an amousement park car to complete a loop with radius r without falling. The inicial velocity is 0. In the beginning, i had to do it without friction, with no problem. The horribleness started when i tried to do it with friction, more specifically in the loop part.

I use v1 for the speed after it falls the height h, v2 for the time it continues by the flat, and v3 for the speed in the tallest part of the loop. (Question in the last sentencie, for the impatient readers :) )

2. Relevant equations

E2-E1=-W; W=Fk dx; Fk=µk N. E(1,2)= Ek+Ep

3. The attempt at a solution

For the beginning of the movement, the fall from the height h to the floor, assuming is a straight plane is given by:

m g h - 1/2 m v1^2 = -µk m g cosø (h/senø)<---that last is the distance from h.

v1=√(2 g h-g h µk cotø) <---- i'll go fast, in he first steps. I just simplified v1.

for the second part, when the car moved a small part on a flat, i had

1/2 m v2^2 - 1/2 mv1^2=-µk m g cos ø x <--- x=dx, distance.

v2=√(2gh - g h µk cotø -2 g x µk)

Now, in the loop.

I've tried everything, and can't come up with a reliable solution. I know i can use

1/2 m v3^2+ mg(2r) - 1/2 m v2^2=-Fk x2

well, x2 is πr because it'll just half the circle, but im having a real bad time finding the normal force for the Fk=µk N. Because, for a circular movement i got:

N= (m v^2)/r + m g cos ø

is v changing? if i integrate for ø from 0 to π it'll give 0! if i want to find the tangential acceleration i'll complicate myself even more. So, in a summary, the big question i want to ask is

How do you estimate the work made by friction on a loop with given inicial velocity, so the car won't fall? Thank you very much!

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# Friction problem with loop (work and energy)

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