Question

A balloon is filled with helium at the atmospheric pressure $P_0$. The volume of spherical balloon is $v_0$. After being released, the balloon burst at an altitude where the atmospheric pressure is $\frac{P_o}{2}$. Before bursting the balloon has volume $\frac{5V_0}{4}$, temperature remaining the same. If the mass of material of balloon is m and density at the time of bursting is ${\rho }_o$, then:

Answer 1

Let us denote the final volume and pressure by,

$V^{\prime }=\frac{5}{4}{V}_0\&P^{\prime }$

Helium as an ideol gas,

$P_0{V}_0=P^{\prime }{V}^{\prime }$ (isothermol)

$P^{\prime }=\frac{P_0{V}_0}{V^{\prime }}=0.8P_0$

Since external pressure $=\frac{P_0}{2}=0.5P_0$

Net outword pressure on the surface of balloon $=0.8P_0-0.5P_0=0.3P_0$

Net pressure force $=PA=0.3P_0\pi R^2$

Aslo $\frac{5}{4}{V}_0=\frac{4}{3}\pi R^3$

But this must just balance the elastic force $\sigma \Delta A$ where $\Delta A=2\pi Rt$ (t is thickness of balloon material obtained from density

${\rho }_0=\frac{m}{4\pi R^{2}t}$

$\text{So},[\sigma =\frac{9P_0{V}_0{\rho }_0}{16m}]$
Hence, the correct options are, $\text{(}b,d)$