Problem $2.$

You use a handheld drill with the drill bit shown. This bit has three sections that all vary in length and diameters $($all dimensions provided$).$ The shank is held in the drill's chuck, so to simplify this problem, let's assume that only an external point torque, $\backslash ($ T$\_\{$l$\}\backslash ),$ is applied by the drill at $15$ mm from the leftmost end as shown.

Let's also assume that the flutes $($the cutting ridges on the right end of the drill$)$ are within a hole that you drilled into a piece of wood causing a distributed load to act along the $30$ mm length of the flutes. No torque is applied along the $4$ mm long "step" between the shaft and the flutes $($section $\backslash ($ B C $\backslash )).$

This bit is made of steel, with a shear yield stress of $80$ MPa $.$

$($a$)$ You drill into a piece of wood and apply a torque of $\backslash ($ T$\_\{$l$\}=0\mathrm{.}3\backslash )$ N$.$m$.$ What is the resisting torque per unit length $\backslash ($ t $\backslash )$ applied by the wood along the length of the flutes in units of $($N$.$m$)/$m$?$

$($b$)$ Next, determine the internal torque, shear stress, and twist as a function of $\backslash ($ x $\backslash )$ along the entire length of the bit. Clearly delineate each section by the letters $\backslash ($ A$,$ B$,$ C$,$ D $\backslash )$ in your response. Assume a nominal diameter $($which is provided as $4$ mm $)$ and minimal circular cross$-$section along the flutes.

$($c$)$ For the functions determined in part b$,$ plot the internal torque, shear stress, and twist as a function of x along the length of the bit. Clearly delineate each section by the letters $\backslash ($ A$,$ B $\backslash ),\backslash ($ C$,$ D $\backslash )$ in your response.

$($d$)$ Determine if failure $($yielding$)$ occurred in this bit based on maximum distortion energy theory and maximum shear stress theory. If failure has not occurred, what is the factor of safety for this drill bit?

$($e$)$ Last, you can see a fillet in the image provided at point $\backslash ($ B $\backslash ).$ So$,$ let's add one to this design $($fillets are often used to reduce stress concentrations when the diameter of a shaft is changed$).$ Assume that the drill bit is made from a brittle steel. What is the minimum fillet radius you can use to ensure that this drill bit will not fail under the torques applied in this problem? $($Note $-$ use the stress concentration tables in chapter $5$ for torsion!!!$)$

Can you please just do C