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(A) The pressure inside a soap bubble of radius $r$ and surface tension $T$ is given by $P_{in} = P_{ext} + \frac{4T}{r}$.Initially, the pressure insi

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$8 P_1 + \frac{24 T}{r}$

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(A) The pressure inside a soap bubble of radius $r$ and surface tension $T$ is given by $P_{in} = P_{ext} + \frac{4T}{r}$.Initially, the pressure inside the bubble is $P_{in,1} = P_1 + \frac{4T}{r}$.Assuming the temperature remains constant, the air inside the bubble follows Boyle's Law, $P_{in,1} V_1 = P_{in,2} V_2$.The volume of the bubble is $V = \frac{4}{3} \pi r^3$.So, $(P_1 + \frac{4T}{r}) \cdot \frac{4}{3} \pi r^3 = (P_2 + \frac{4T}{r/2}) \cdot \frac{4}{3} \pi (r/2)^3$.$(P_1 + \frac{4T}{r}) r^3 = (P_2 + \frac{8T}{r}) \frac{r^3}{8}$.$8(P_1 + \frac{4T}{r}) = P_2 + \frac{8T}{r}$.$8P_1 + \frac{32T}{r} = P_2 + \frac{8T}{r}$.$P_2 = 8P_1 + \frac{24T}{r}$.

In a cylinder provided with a piston, air is under pressure $P_1$ at a constant temperature $t$. $A$ soap bubble with radius $r$ and surface tension $T$ is lying inside the cylinder. To reduce the radius of the soap bubble to half, the required air pressure inside the cylinder is

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