A tube is fixed in a vertical plane as shown in figure .Form point A a sphere of mass 0.314 kg is released .During its motionin the tubeitfacesa constant resistiveforce R . At B itsvelocitybecomes zero . Calculate(i) theconstant resistiveforce R and (if) work done byresistive force (average radisu of semicircularpath is 1 m ) .

A tube is fixed in a vertical plane as shown in figure .Form point A a

1.	A tube is fixed in a vertical plane as shown in figure .Form point A a sphere of mass 0.314 kg is released .During its motionin the tubeitfacesa constant resistiveforce R . At B itsvelocitybecomes zero . Calculate(i) theconstant resistiveforce R and (if) work done byresistive force (average radisu of semicircularpath is 1 m ) .
Answer» Solution :Let D be the lowest point on its path . Let us assume that the potential energy at D to zero , so potential energy at A is , `v_(A) = mgr ""….(1)` `v_(B) = (mgr)/2 ""……(2)` ` [ :. "In right TRIANGLE " Delta OB. B , angle OBB = 30^(@) and OB = r ."Theoppositeside of " 30^(@) " in right triangle is the half value of diagonal "] ` Work DONE by constant resistive force R , `W_(R) = " Force " xx " displacement " ` ` = R xx arc AB ` ` = R xx theta xx r ` ` = R xx (pi/2 +pi/3) r ` ` :. W_(R) = R xx (5pi)/6 r "".....(3)` Now `V_(A)=V_(B) +W_(R)` ` mgr = (mgr)/2 +R (5pi)/6 r ` ` [ :. " From equation (1) ,(2) and (3) "] ` ` :. (mgr)/2= R xx (5pi)/5 r ` ` :.R = (3mg)/(5pi)` ` = (3xx0.314 xx10)/(5xx3.14)` ` :. R = 0.6 N ` Work doneby resistiveforce R, `W_(R) = R xx` displacement ` = R xx arcAB ` ` = R xx(5pir)/6` ` = (0.6 xx 5xx 3.14 xx1)/6` ` :. W_(R) = 1.57 J`