A horizontal single$-$cylinder engine has a cylinder diameter of $100$ mm $,$ a stroke of $120$ mm and a connecting rod length of $250$ mm $.$ The mass of the piston is $0\mathrm{.}7$ kg and that of the connecting rod is $2$ kg $.$ The centre of mass of the connecting rod is $100$ mm from the big end bearing centre and the effect of the connecting rod mass may be allowed for by apportioning it between the two ends. When the crank is $20$ from the top dead centre position on the working stroke, the gas pressure is $700k\frac{N}{m^2}.$

$($a$)$ Determine, for this position, the crankshaft torque when the speed is $2000\frac{r}{m}in.$

$($b$)$ At what speed would the crankshaft torque be zero for this crank position, $($the gas pressure remaining unchanged$)?$

$($Answer: $26\mathrm{.}7$ Nm; $2227\frac{r}{m}in)$