Question

A thin walled cylinder of mass m height h and cross section area S is filled with gas and floats on the surface of water. As a result to leakage from lower part of cylinder the depth of its submergence has increased by a. Determine the initial pressure x in cylinder if initial atm pressure is p atm.

Answer 1

A thin walled cylinder of mass m height h and cross section area S is filled with gas and floats on the surface of water. As a result to leakage from lower part of cylinder the depth of its submergence has increased by a. Determine the initial pressure x in cylinder if initial atm pressure is p atm.

Solution:

Connection between the pressures p1 and p2 ($p_1 = p_{\text{initial}}$ - pressure before entering water, $p_2$ - pressure after entering water).

Mendeleev-Clapeyron equation for gas in the first and in the second case:

$$p_1 V = vRT; V = h * S$$ $$p_2 V = vRT; V = (h - a) * S$$ $$(1) = (2): p_1 h = p_2 (h - a)$$ $$p_2 = p_1 \left(\frac{h}{h - a}\right)$$

Newton's second law for the cylinder:

$$F_{atm} + mg - F_{gas} = 0$$ $$F_{atm} = p * S$$ $$F_{gas} = p_2 S$$

(2') and (5) and (4) in (3): $pS + mg - p_1 \left(\frac{h}{h - a}\right) S = 0$

$$p_1 \left(\frac{h}{h - a}\right) = \frac{mg}{S} + p$$ $$p_1 = \frac{\frac{mg}{S} + p}{\frac{h}{h - a}} = \left(\frac{mg}{S} + p\right) \left(1 - \frac{a}{h}\right)$$

Answer: initial pressure $p_1 = x = \left(\frac{mg}{S} + p\right) \left(1 - \frac{a}{h}\right)$.