Question 5.C-A.68: The length of a uniform test tube of mass 50 g is 10 cm. It ...
The length of a uniform test tube of mass 50 g is 10 cm. It can float vertically in water with half of its length in the water. Find the maximum and minimum density of the liquids that can be measured, with help of this test tube. (Assume that the test tube can be balanced when minimum of 1 cm of tube is inside any liquid.)
Case 1: The test tube immerses completely in a liquid of minimum density.
The length of the test tube = 10 cm
The weight of the test tube = 50 g
Let the area of cross section of the test tube = a Let the density of test tube = D_{TT}
If half of the length of the test tube floats inside the water
Weight of the floating body = weight of the displaced liquid
50 g_{ wt }=\frac{v}{2} d_{ w } g\Rightarrow V . D_{ TT } \cdot g=\frac{v}{2}(1) g \Rightarrow D_{ TT }
=\frac{1}{2}=0.5 g cm ^{-3}
The depth of immersion of the test tube inside this liiquide 10 cm
\Rightarrow V . D_{ TT } \cdot g =V \cdot D_{\ell} \cdot gTherefore, the minimum density =0.5 g cm ^{-3}
The density of this liquid = density of test tube =0.5 g cm ^{-3}
Case 2: In a liquid with maximum density the test tube floats with a minimum height inside the liquid.
The minimum height at which the test tube can be balanced is 1 cm = h/10.
∴ Volume of the test tube = a.h
\Rightarrow V . D_{ TT } \cdot g=\frac{V}{10} \cdot d_{2} \cdot gd_{L}=10 \times D_{ TT }=10 \times 0.5 g cm ^{-3}
=5 g cm ^{-3}
The maximum density of the liquid that can be measured using this test tube is 5 g cm ^{-3}