$($b$)$ A single cylinder reciprocating engine has speed $240$ rpm $,$ stroke $300$ mm $,$ mass of the reciprocating parts $50$ kg and mass of the revolving parts $40$ kg at $150$ mm radius. If two$-$third of the reciprocating parts and all the revolving parts are to be balanced, find $($a$)$ the balance mass required at a radius of $400$ mm $,$ and $($b$)$ the residual unbalanced force when the crank has rotated $60$ n from top dead centre.