You have been hired to build a large sand mound in an indoor playground and must be careful about the stress that the sand will put on the floor. Consulting research literature, you are surprised to find that the greatest stress occurs, not directly beneath the apex (top) of the mound, but at points that are a distance $r^{\max }$ from that central point (Fig. $13-52 a$ ). This outward displacement of the maximum stress is presumably due to the sand grains forming arches within the mound. For a mound of height FIGURE 13-52 = $H=3.00 \mathrm{~m}$ and angle $\theta=33^{\circ}$, and $\quad$ Problem $48 .$
with sand of density $\rho=$ $1800 \mathrm{~kg} / \mathrm{m}^{3}$, Fig. $13-52 b$ gives the stress $\sigma$ as a function of radius $r$ from the central point of the mound's base. In that figure, $\sigma^{\text {center }}=40000 \mathrm{~N} / \mathrm{m}^{2}, \sigma^{\max }=40024$
$\mathrm{N} / \mathrm{m}^{2}$, and $r^{\max }=1.82 \mathrm{~m}$
(a) What is the volume of sand contained in the mound for $r \leq$ $r^{\max } / 2 ?$ (Hint: The volume is that of a vertical cylinder plus a cone on top of the cylinder. The volume of the cone is $\pi R^{2} h / 3$, where $R$ is the cone's radius and $h$ is the cone's height.) (b) What is the weight $W$ of that volume of sand? (c) Use Fig. $13-52 b$ to write an expression for the stress $\sigma$ on the floor as a function of radius $r$, for $r \leq r^{\max }$. (d) $\mathrm{On}$ the floor, what is the area $d A$ of a thin ring of radius $r$ centered on the mound's central axis and with radial width $d r ?$ (e) What then is the magnitude $d F$ of the downward force on the ring due to the sand? (f) What is the magnitude $F$ of the net downward force on the floor due to all the sand contained in the mound for $r \leq r^{\max } / 2 ?$ [ Hint: Integrate the expression of (e) from $r=0$ to $r=r^{\max / 2 .]}$ Now note the surprise: This force magnitude $F$ on the floor is less than the weight $W$ of the sand above the floor, as found in (b). (g) By what