MHB Friction on a half pipe in the middle location

[Dustinsfl]

Consider a half pipe of height L. The middle section, non sloping part, has a friction coefficient of \(\mu_k = 0.1\) and frictionless every where else. The length of this section is L. How many times can the skateboarder go back and forth before he stops?

In the friction section,
\[
\sum F_x = v_x - F_f = v_x - .1N
\]
since \(F_f = \mu_k N\).

Not sure how to determine how many times the skateboarder can go back and forth.

 

[gleem]

Assume the skateboarder starts at the top L meter above the bottom with Lmg potential energy. When he crosses the bottom he always losses 0.1 mgL of energy no matter how high he had reached before. So having Lmg energy to start he can traverse the bottom N(0.1mgL) times.

Lmg = N(0.1mgL)

So N =10