Solving the Friction Dilemma Homework Statement

[Solaxys]

Homework Statement

Q: A worker wishes to pile a cone of sand onto a circular area in his yard. The radious of the circle is R, and no sand is to spill into the surrounding area. If is the static coefficient of friction between each layer of sand along the slope and the sand beneath it (along which it might slip), show that the greatest volume of sand that can be stored in this manner is
(pi**R3)/3.
(The volme of a cone is Ah/3, where A is the base area and h is the cone's height).


sorry, should've stated:
I've sat here thinking about this question for about 3 hrs now to no avail.

the only thing I can scratch up is the fact that since V =Ah/3 and V(cone)= (pi**R3)/3.
that means h = R.

But I don't know how that works...

 

[kaeltemittel]

Hi! You´ve written V =(pi**R3)/3. Does your R3 correspond to my R^3 (three-times R) ?
Then everything is o.k.
You´ve already written V =Ah/3 and so V = (pi*R^2*h)/3.
What you need to know now: How can I express “h” by ?
Look at the rectangular triangle formed by R and L (it’s the “half” of the cone).

I
III
IIIII
IIIIII L
IIIIIII
IIIIIIII
IIIIIIIII
IIIIIIIIII
R



Two things: 1)What is the tangent of the angle between R and L? Call it phi.
2)Remember that tan phi = ? This is the largest angle at which sth. can lie on the plane without slipping (you can derive this relation from the “inclined plane with friction” if you want to…)
Now substitute…

Greetings from Germany...

 

[Solaxys]

TYVM!
Made so much sense =).
TYVM.