The plane is flying along a circular path at a constant speed v on the vertical plane. The radius of the circular path of the plane is $2000$ m$,$ and the pilot weighs $72$ kg$.($a$)$ If the plane exerts no force on the crew when it is at the apex of the circular path, the pilot will feel 'weightless' at this point. Draw the pilot's free$-$body diagram, and use Newton's second law to find the speed v required to create this condition. $($b$)$ Try to ensure that the force exerted by the plane on the pilot is not more than four times the weight of the pilot. If the pilot is flying at v$=200$ m$/$s$,$ what is the minimum radius of the allowable circular path?