Homer Simpson spinning inside an hourglass

[ajejebrazorf]

Homework Statement

Homer Simpson (mass 123 kg) wants to complete a full lap on a giant hourglass with his motorcycle (mass 158 kg). The cross-section of the hourglass in the zy-plane is given by z = k|y| with k = 0.53, at a height h = 3.5 meters from its center. Knowing that the coefficient of static friction between the motorcycle tires and the surface of the hourglass is μs = 0.6, determine:
(a) the minimum rotational speed ωmin;
(b) the maximum velocity vmax;
(c) the maximum normal force Nmax at the point of contact if the velocity is maximum.

Relevant Equations

N*sin(α) - N*μs*cos(α) = m*an
N* cos(α) + N*μs*sin(α) -m*g = 0

Hi. I've spent close to 3 hours trying to answer to the 1st question, but everytime I get the same mathematically impossible result ωmin = sqrt(negative number).
Do you see any mistakes? Do you think there's a mistake in the solutions?

 

[Lnewqban]

Welcome, @ajejebrazorf !
It is better to show your equations in the form of LaTeX, which guide you can find over the left bottom corner of this window.

The response for a) indicates that the motorcycle will be able to remain parked on the surface without the help of any centrifugal effect.

Try again your free body diagram, not including any horizontal acceleration.

 

[ajejebrazorf]

Welcome, @ajejebrazorf !
It is better to show your equations in the form of LaTeX, which guide you can find over the left bottom corner of this window.

The response for a) indicates that the motorcycle will be able to remain parked on the surface without the help of any centrifugal effect.

Try again your free body diagram, not including any horizontal acceleration.

Thank you for the reply!
But I don't understand how it would be able to remain parked while being inclined inside a cone?
And supposing that the centripetal acceleration is 0 implies that $tan(α) = μs$. Which doesn't make sense as it's equal to k. Thank you in advance

 

[Lnewqban]

 

[Lnewqban]

And supposing that the centripetal acceleration is 0 implies that $tan(α) = μs$. Which doesn't make sense as it's equal to k.

$tan(α) = μs$ only applies to the angle at which sliding becomes imminent.
Actually, that is the practical way to determine the value of the coefficient of static friction between two materials.

For smaller angles, the magnitude of static friction, which is nothing else than the Newton's third law reaction to the force trying to induce a slide, is only what it needs to be to keep the static balance.