Properties of Matter 4 Question 15
17. If the density of air is $\rho _a$ and that of the liquid $\rho _l$, then for a given piston speed the rate (volume per unit time) at which the liquid is sprayed will be proportional to
(a) $\sqrt{\frac{\rho _a}{\rho _l}}$
(b) $\sqrt{\rho _a \rho _l}$
(c) $\sqrt{\frac{\rho _l}{\rho _a}}$
(d) $\rho _l$
Passage 2
A small spherical monoatomic ideal gas bubble $(\gamma=5 / 3)$ is trapped inside a liquid of density $\rho _l$ (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains $n$ moles of gas. The temperature of the gas when the bubble is at the bottom is $T _0$, the height of the liquid is $H$ and the atmospheric pressure is $p _0$ (Neglect surface tension)
$(2008,4$ M)
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Answer:
Correct Answer: 17. $\frac{4 T}{\rho v^{2}}$
Solution:
- Free body diagram of the wire is as shown in figure.
Considering the equilibrium of wire in vertical direction, we have
$$ 2 T l \cos \theta=\lambda l g $$
$$ \text { For } \quad y«a, \cos \theta \approx \frac{y}{a} $$
Substituting the values in Eq. (i), we get
$$ T=\frac{\lambda a}{2 y} $$
