$1.$ A single cylinder diesel engine generating power in every revolution of the crank shaft has a cylinder bore of $150$ mm and a stroke length of $200$ mm $.$ The engine fires at the rate of $500$ cycles per minute and the mean effective pressure is $800$ kPa $.$ If the net load on brake drum is $50$ kg and its diameter is $100$ cm $.$ Calculate the indicated power, brake power and mechanical efficiency of the engine.