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 motion in the tube it faces a constant resistive force R . At B its velocity becomes zero . Calculate (i) the constant resistive force R and (if) work done by resistive force (average radisu of semicircular path is 1 m ) .

Explanation

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 ." The opposite side 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 done by resistive force R,
`W_(R) = R xx` displacement
` = R xx arc AB `
` = R xx(5pir)/6`
` = (0.6 xx 5xx 3.14 xx1)/6`
` :. W_(R) = 1.57 J`