A glass test tube with a plane base has a diameter of 4 cm and mass 30 g. Centre of gravity of the empty test tube is at a height of 10 cm from the base. Find the amount of water to be taken in the test tube so that when it floats vertially, its centre of gravity shifts to the midpoint of the immersed portionof the tube.

A glass test tube with a plane base has a diameter of 4 cm and mass 30

1.	A glass test tube with a plane base has a diameter of 4 cm and mass 30 g. Centre of gravity of the empty test tube is at a height of 10 cm from the base. Find the amount of water to be taken in the test tube so that when it floats vertially, its centre of gravity shifts to the midpoint of the immersed portionof the tube.
Answer» Solution :The centre of gravity P of the empty tube is 10 CM above the point O. Let the HEIGHT of water level taken in the tube be h cm. Midpoint of the test tube-water system be at R. Hence, part of the tube immersed in water, AC = `2xxOR`. Volume of water in the tube = `PI(2)^(2)h=4pihcm^(3)` Mass of the tube + water in it = `(30+4pih)g` From the condition of floatation, `pi(2^(2))xxAC=30+4pih` `THEREFORE" "AC=2xxOR=30/(4pi)+h=d` (say). TAKING moment about the point R, `4pihxxRQ=30xxPR` or, `4pih[OR-OQ]=30xx[OP-OR]` or, `4pih[d/2-h/2]=30[10-d/2]` or, `4pih[1/2(30/(4pi)+h)-h/2]=30[10-1/2(30/(4pi)+h)]` or, h = 8.806 cm Hence the required mass of water = `4pixx8.806=110.66g`.