A balloon is made of a material of surface te | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The pressure inside the balloon due to surface tension is $P = \frac{2S}{R}$.Using Bernoulli's principle,the velocity of the gas escaping is $v =

Vedclass · Accepted Answer

$a=-\frac{1}{2}, \alpha=-\frac{1}{2}, \beta=-1, \gamma=\frac{1}{2}, \delta=\frac{7}{2}$

Vedclass · Answer

(C) The pressure inside the balloon due to surface tension is $P = \frac{2S}{R}$.Using Bernoulli's principle,the velocity of the gas escaping is $v = \sqrt{\frac{2P}{\rho}} = \sqrt{\frac{4S}{\rho R}} = 2S^{1/2} \rho^{-1/2} R^{-1/2}$.Given $\psi(r) \propto r^\alpha$,comparing with $v \propto R^{-1/2}$,we get $\alpha = -1/2$.The rate of change of volume is $\frac{dV}{dt} = -A \cdot v$.Since $V = \frac{4}{3}\pi R^3$,$\frac{dV}{dt} = 4\pi R^2 \frac{dR}{dt}$.So,$4\pi R^2 \frac{dR}{dt} = -A \cdot k \cdot S^{1/2} \rho^{-1/2} R^{-1/2}$,where $k$ is a constant.$R^{5/2} dR = -C \cdot S^{1/2} \rho^{-1/2} A dt$.Integrating from $R$ to $0$ and $0$ to $T$: $\int_0^R R^{5/2} dR = \int_0^T C' S^{1/2} \rho^{-1/2} A dt$.$\frac{2}{7} R^{7/2} = C' S^{1/2} \rho^{-1/2} A T$.Thus,$T \propto S^{-1/2} A^{-1} \rho^{1/2} R^{7/2}$.Comparing with $T \propto S^a A^\beta \rho^\gamma R^\delta$,we get $a = -1/2, \alpha = -1/2, \beta = -1, \gamma = 1/2, \delta = 7/2$.

Similar Questions

An air bubble of volume $V_0$ is released by a fish at a depth $h$ in a lake. The bubble rises to the surface. Assume constant temperature and standard atmospheric pressure $P$ above the lake. The volume of the bubble just before touching the surface will be (density of water is $\rho$):

$A$ small air bubble is at the inner surface of the bottom of a beaker filled with cold water. Now,the water in the beaker is heated. The size of the bubble increases. The reason for this may be:

$A$ liquid flows through a horizontal tube. The velocities of the liquid in the two sections,which have areas of cross-section $A_1$ and $A_2$,are $v_1$ and $v_2$ respectively. The difference in the levels of the liquid in the two vertical tubes is $h$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module