In butt welding operation on plates, the heat input necessary is given by

$$Q = 8\,K\,{T_c}\,t\,\,\,\left[ {0.2 + {{Vb} \over {4\alpha }}} \right]$$
Where $$K=$$ thermal conductivity,
$${T_c} = $$ temp increases from room temp up to $$MP,$$
$$t=$$ thickness of plate,
$$V=$$ welding speed,
$$b=$$ width of weld,
$$\alpha = thermal diffusivity

Two alloy steel plates as shown in fig are to be welded using a power source rated at $$5$$ $$kVA$$ having a duty cycle of $$75\% .$$ Using the given data, determine the maximum welding speed for the job given in $$mm/s$$

Data : $$K = 45\,W/{m^0}C,\,\,{T_c} = {1450^ \circ }C,$$ and
$$\alpha = 1.2 \times {10^{ - 5}}\,\,{m^2}/s.$$

What is the approximate $$\% $$ change in the life, $$t,$$ of the tool with zero rake angle used in orthogonal cutting when its clearance angle, $$\alpha ,$$ is changed from $$10$$ to $$7$$ deg?

(Hint: flank wear rate is proportional to $$\cot \,\alpha $$)

In an orthogonal cutting experiment with a tool of rake angle $$7$$$$deg$$, the chip thickness was found to be $$2.5mm$$ when the uncut chip thickness was set to $$1mm,$$ find the shear angle $$\phi $$ and find the friction angle $$\beta $$ assuming that Merchants's formula holds good.

The lives of two tools $$A$$ & $$B$$, governed by the equation $$V{T^{0.125}} = 2.5$$ and $$V{T^{0.25}} = 7$$ respectively in certain machining operation where $$V$$ is the cutting velocity in $$m/s$$ and $$T$$ is the tool life in sec. Find out the speed $$V$$ at which both the tools will have the same life.

Also calculate the corresponding tool life. If you have to machine at a cutting speed of $$1m/s$$, then which one of these tools will you choose in order to have less frequent tool changes?