Question

Two friends Prakash and Naveen are trying to build castles on a seashore in the form of a conical pile of sand. For this they are provided with the following data:
$1)$ Volume of sand $=225\,cm^{3}$
$2)$ Co-efficient of friction between sand grains, $\mu _{S}= \, \sqrt{1.6}$
With the recently acquired knowledge about the friction, they realize that for a given amount of sand, they are able to build the pile only till certain maximum height $h$ . Find the approximate maximum height $h$ (in $cm$ ).

Answer 1

For sand net to fall $tan \theta =\mu _{s}$
$\frac{h}{r}=\mu _{s}$
$V=\frac{1}{3}\pi r^{2}h$
$225=\frac{1}{3}\times \pi \times \frac{h^{2}}{\mu _{s}^{2}}\times h$
$h^{3}=\frac{225 \times 3 \times \mu _{s}^{2}}{\pi }$
$h^{3}=\frac{225 \times 3 \times 1.6}{22}\times 7$
$h^{3}\approx343$
$h=7\,cm$