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A man with a telescope can just observe the point A on the circumference of the base of an empty cylindrical vessel. When the vessel is filled completely with a liquid of refractive index 1.5, the man can just observe the middle point B of the base of the vessel without moving either the vessel or the telescope. If the diameter of the base of the vessel is 10 cm, what is the height of the vessel? |
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Answer» Solution :When the vessel is empty, a light ray from the point A enters the telescope T following the straight path AO [Fig. 2.55]. When the vessel is filled with the liquid, aray of light from the point B moves along BO and after REFRACTION in AIR enters the telescope. Let h be the height of the vessel. `"Here" angleBOC = i and angleAOC = r` According to the FIGURE, `(SINI)/(SINR) = (1)/(mu) or, mu = (sinr)/(sini)` `or, "" 1.5 = ((AC)/(AO))/((BC)/(BO)) = (AC)/(AO) xx (BO)/(BC) = (AC)/(BC) xx (BO)/(AO)` `or, "" 1.5 = (10)/(5) xx (sqrt(BC^(2) + CO^(2)))/(sqrt(AC^(2) + CO^(2))) = 2 xx (sqrt(25 + h^(2)))/(sqrt(100 + h^(2)))` `or, " " 2.25 = (4(25 + h^(2)))/(100 + h^(2))` `or, "" h = 8.45 cm` 
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