Problem $8.$

Walking on water Water striders are insects commonly found on ponds, rivers, and lakes

that appear to "walk" on water. A typical length of a water strider is about $0\mathrm{.}4$ in $.,$ and they

can cover $100$ body lengths in $1$ s $.$ It has long been recognized that it is surface tension that

keeps the water strider from sinking below the surface. What has been puzzling is how they

propel themselves at such a high speed. They can't pierce the water surface or they would

sink$.$ A team of mathematicians and engineers from the Massachusetts Institute of Tech$-$

nology $($MIT$)$ applied conventional flow visualization techniques and high$-$speed video to

examine in detail the movement of the water striders. They found that each stroke of the

insect's legs creates dimples on the surface with underwater swirling vortices sufficient to

propel it forward. It is the rearward motion of the vortices that propels the water strider

forward. To further substantiate their explanation, the MIT team built a working model of

a water strider, called Robostrider, which creates surface ripples and underwater vortices as

it moves across a water surface. Waterborne creatures, such as the water strider, provide an

interesting world dominated by surface tension.

$($a$)$ The water strider bug shown in Fig. P$1\mathrm{.}9\mathrm{.}15$ is supported on the surface of a pond by

surface tension acting along the interface between the water and the bug's legs. Determine

the minimum length of this interface needed to support the bug. Assume the bug weighs

${10}^{-4}N$ and the surface tension force acts vertically upward. $($b$)$ Repeat part $($a$)$ if surface

tension were to support a person weighing $750$ N $.$