Analytical Mechanics- constraints/lagrange

[Elvex]

Homework Statement

Consider a point mass m moving under the influence of the gravitational force F= -mg e_y . The mass is constrained to slide along a given curve y= f(x) in the x-y plane. You may set z=0 from the start and consider two dimensional motion.

c) A Skier descends a slope with profile y= -ax^n with a>0 and n>0. She starts at the top at (x,y) = (0,0) with zero velocity, and slides straight down without friction under the influence of gravity. If the slope steepens sufficiently, the skis will leave the ground at some point. Formulate a condition for when this happens. For what values of the parameter n, and at which point, do the skis leave the ground?


OK, so I already formulated the constraint and solved for the constraint force as a function of x. It's pretty messy.

My question is more a conceptual one. How do I define a condition for when the skier falls off the curve?

Does this have to do with relating the constraint force to the gravitational force? The tangent line of the constraint force? I'm not really sure how to start this.

 

[NeoDevin]

You can look at it two ways:

When the normal force from the hill is greater than the gravitational force perpendicular to it

or,

When the hill is dropping faster than the skier.

Hope that helps

 

[Elvex]

I got it, I had to set the constraint force to zero, or my lagrange multiplier really cause the gradient of my constraint is trivial in setting F = 0.

I then dropped the E term cause E = 0 in this case, U defined as being negative... and then got a term with a's and n's = 1, which can only be satisfied for n > 2. Good problem.

 

[AlephZero]

... and then got a term with a's and n's = 1, which can only be satisfied for n > 2. Good problem.

And that answer is obviously correct, because if the ground was not there the skier would be falling in a parabola. i .e. with n = 2.