Question 421: Properties of Matter

Question

When water is filled carefully in a glass, one can fill it to a height h above the rim of the glass due to
the surface tension of water. To calculate h just before water starts flowing, model the shape of the
water above the rim as a disc of thickness h having semicircular edges, as shown schematically in the
figure. When the pressure of water at the bottom of this disc exceeds what can be withstood due to
the surface tension, the water surface breaks near the rim and water starts flowing from there. If the
density of water, its surface tension and the acceleration
due to gravity are 10
3
kg m
−3
, 0.07 Nm
−1
and 10 ms
−2
, respectively, the value of h (in mm) is _________.

Accepted Answer

Pressure at the bottom of disc = Pressure due to surface tension $$ ho gh = T\left( {{1 \over {{R_1}}} + {1 \over {{R_2}}}} ight)$$ $${R_1} > > > {R_2}$$ So, $${1 \over {{R_1}}} < < < {1 \over {{R_2}}}$$ and $${R_2} = h/2$$ $$ herefore $$ $$ ho gh = T\left( {{1 \over {{R_1}}} + {1 \over {{R_2}}}} ight) = T\left( {0 + {1 \over {h/2}}} ight)$$ $${h^2} = {{2T} \over { ho g}} \Rightarrow h = \sqrt {{{2T} \over { ho g}}} $$ $$ \Rightarrow \, = \sqrt {{{2 imes 0.07} \over {{{10}^3} imes 10}}} = \sqrt {{{14 imes 100} \over {{{10}^4} imes 100}}} $$ $$h = \sqrt {14} $$ mm = 3.741

Properties of Matter

Integer / Numeric Answer Type