A wheel$-$tractor unit, operating in its fourth$-$gear range and at its full rated rpm$,$ is observed to maintain a steady speed of $6\mathrm{.}5mph$ when operating under the following conditions. The ambient air temperature is $52F$ and the altitude is at $250$ feet above sea level. The tractor is climbing a $6\%$ grade with a rolling resistance of $100lbs$ per ton, and it is towing a pneumatic$-$tired trailer loaded with sand and gravel. The single$-$axle tractor has an empty operating weight of $75,000lbs.$ The loaded trailer has a weight of $40,000lbs.$ The weight distribution for the combined tractor$-$trailer unit is $51\%$ to the drive axle and $49\%$ to the rear axle.

a$.$ The manufacturer, for the conditions described above, rates the tractive effort of the tractor at $340$ rimpull horsepower. What percentage of this rated rimpull hp does the tractor actually develop? Assume "the $20-lb$ of rimpull required per ton of weight per $\%$ of slope" approximation will be

$2$

acceptable. Assume traction is not a limiting factor. Use the following equation to compute the required $hp$ :

Required $hp=\frac{\text{R}\text{e}\text{q}\text{u}\text{i}\text{r}\text{e}\text{d}\text{R}\text{i}\text{m}\text{p}\text{u}\text{l}\text{l}(lbs)}{33000ft-lb\frac{s}{m}in}(\frac{\text{S}\text{p}\text{e}\text{e}\text{d}(mph)}{60\frac{\text{m}\text{i}\text{n}}{h}r})\frac{5280ft}{\text{m}\text{i}\text{l}\text{e}}$

b$.$ What is the value of the coefficient of traction if the drive wheels of the tractor are at the point of incipient slippage for the conditions stated above?