glass and a card

Question

Take a cylindrical glass of height $L$ and radius $R$. Fill the glass to a height $h$. When you place a card that completely covers the opening of the glass, you can turn the glass over without the water running out. This is because the tension force keeps the water in the glass over the small distance between the card and the glass.

1. Calculate the pressure differential on the card by the increase in mass above it.
2. This pressure differential is compensated by an isothermal expansion of the air above the water. Calculate the expansion of the air. How much does the card drop relative to the edge of the glass (a height $\delta k$)?
3. Assume that the tension force can bridge a maximum distance $\epsilon$. What are the conditions on $h$ and $L$ such that $\delta k$ does not exceed $\epsilon$?

My attempt:

1. I tried to use the hydrostatic equilibrium.

$F_{fluid}=-\rho_{fluid} g V_{fluid}$. Because the fluid is in a cilinder and the volume of the fluid doesn't chang by turning the glass: $V_{fluid}=\pi R^{2} h$.

$F_{air in glass}=-P_{air in glass} A_{fluid}=-P_{air in glass} \pi R^{2}$

$F_{air on card}=P_{air on card} \pi R^{2}$

$=> \triangle P= \rho_{fluid}gh$

2. Now for the second question I'm thinking I maybe solved the first question wrong. I don't really know how to calculated this one.

Answer 1

For your first question, you could also have used Bernouilli's principle.

For your second question, you can use the ideal gas law assuming that the temperature stays constant, we get that $$ P_f \, V_f = P_i \, V_i \iff V_f = \frac{P_i}{P_f} V_i. $$ Where the $i$ index stands for the situation where the glass is filled with water and the $f$ index stands for the glass turned.

Now, you can fill in everything that you know:

• $P_i = p_0 $, atmospheric pressure;
• $V_i = \pi R^2 (L-h) $;
• $P_f = \Delta p - p_0 $;
• $V_f = V_i + \Delta V = \pi R^2 (L-h) + \pi R^2 \delta k $.

Therefore $\delta k = \frac{\Delta p (L-h)}{p_0 - \Delta p} $.

To answer your third question, you can just work with the inequality: $\delta k < \epsilon$.