Problem $1(5$ marks$).$ The figure consists of an apparatus that is one$-$fourth of a torus of rectangular

cross section constructed of poly$($methyl methacrylate$).$ The torus is attached to a lever arm and rotates

about a pivot point $($reasonably$).$ Assume the fluid is water at standard ambient conditions.

The torus details are as follows: the torus has inside and outside radii $R_i$ and $R_O,$ respectively.

The center of the radii is at the pivot point of the lever arm.

The torus is submerged in a liquid, and there will exist an unbalanced force that is exerted on the

plane of dimensions $hw.$

To bring back the torus and lever arm back to their balanced position, a weight $W$ is added to the weight

hanger. Determine what must the weight be $([=]N)$ to keep the balance position. Take the following

dimensions: $L=33\mathrm{.}6cm,h=10\mathrm{.}2cm,w=7\mathrm{.}6cm,d_2=8\mathrm{.}9cm($tank bottom to bottom of torus$),$ pivot to

tank bottom $d_l=31\mathrm{.}8cm,z=6\mathrm{.}05cm.$

$3-$D view

Figure $1$: Schematic of a toroidal gate.