Opening of glass bottle containing an ideal monoatomic gas is tightly closed with a cork fitted with a thin frictionless glass tube of cross- section area S. inner volume of the tube is negligibly small as compared to volume V of the bottle. A small glass marbles of mass m, when dropped in the tube, it starts oscillating up and down in the tube without friction. The marble exactly fits inside the tube so that there is no leakage of air. The atmospheric pressure is $P_{0}$ and consider the glass, the cork and the material of the marble as perfect insulation of heat. If the time period of oscillations of the marble is
$2 π \sqrt{\frac{K m V}{15 S (P_{0} S + m g)}}$ . The value of K is……

Question

Opening of glass bottle containing an ideal monoatomic gas is tightly closed with a cork fitted with a thin frictionless glass tube of cross- section area S. inner volume of the tube is negligibly small as compared to volume V of the bottle. A small glass marbles of mass m, when dropped in the tube, it starts oscillating up and down in the tube without friction. The marble exactly fits inside the tube so that there is no leakage of air. The atmospheric pressure is $P_{0}$ and consider the glass, the cork and the material of the marble as perfect insulation of heat. If the time period of oscillations of the marble is
$2 π \sqrt{\frac{K m V}{15 S (P_{0} S + m g)}}$ . The value of K is……

Accepted Answer

If the pressure of the gas is P when the marble is in equilibrium
$P S = P_{0} S + m g ............ (i)$
When the marble is getting at displacement y yellow the mean position In the tube,
$d P = - (\frac{γ P}{V}) (- S y) d P = (\frac{5}{3} \frac{P S}{V}) y d P = \frac{5}{3 V} (P_{0} S + m g) y .......... (i i)$

Now, $m \frac{d^{2} y}{d t^{2}} = - d P S$

$\frac{d^{2} y}{d t^{2}} = \frac{- 5 S (P_{0} S + m g) y}{3 m V}$

Time period of oscillations of the marble,
$T = 2 π \sqrt{\frac{3 m V}{5 S (p_{0} S + m g)}}$
Hence, K =9

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