A businessman uses a faulty balance of unequal arms. He buys some old papers from a person and for this he uses a (1)/(2) kg counterpoising weight. He then readily agrees to weigh the papers alternately by changing the pans of the balance during successive weighings. Show that he gains in every 1 kg of purchase. [ Upthrust due to air is neglected.]

A businessman uses a faulty balance of unequal arms. He buys some old

1.	A businessman uses a faulty balance of unequal arms. He buys some old papers from a person and for this he uses a (1)/(2) kg counterpoising weight. He then readily agrees to weigh the papers alternately by changing the pans of the balance during successive weighings. Show that he gains in every 1 kg of purchase. [ Upthrust due to air is neglected.]
Answer» SOLUTION :Let the length of the left and the right arms of the faulty balance be x and y ( `x gt y)` Fig. Suppose `W_(1)` kg of paper on the right pan balances the `(1)/(2)` kg counterpoising weight on the left pan. `:.""(1)/(2).x=W_(1).y" ""or",W_(1)=(1)/(2).(x)/(y)` Suppose `W_(2)` kg of paper is required on the left pan to balance `(1)/(2)` kg counterpoising weight on the right pan. `:. ""W_(2)x=(1)/(2)x" ""or", W_(2)=(1)/(2).(y)/(x)` Paper received by the businessman, `W_(1)+W_(2)=(1)/(2)((x)/(y)+(y)/(x))=(1)/(2).((x^(2)+y^(2))/(xy))` `=(1)/(2).[2+(x-y)^(2)/(xy)]=1+(1)/(2)((x-y)^(2))/(xy)` As `XGT0, ygt0 " ""and" " "(x-y)^(2)gt0, W_(1)+W_(2)GT1`kg Hence the businessman gains in EVERY kilogram of purchase.