Properties Of Matter Ques 5
- A flat plane is moving normal to its plane through a gas under the action of a constant force $F$. The gas is kept at a very low pressure. The speed of the plate $v$ is much less than the average speed $u$ of the gas molecules. Which of the following options is/are true?
(2017 Adv.)
(a) At a later time the external force $F$ balances the resistive force
(b) The plate will continue to move with constant non-zero acceleration, at all times
(c) The resistive force experienced by the plate is proportioal to $v$
(d) The pressure difference between the leading and trailing faces of the plate is proportional to $u v$
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Answer:
Correct Answer: 5.( a, c, d )
Solution:
$ \begin{aligned} v_1 & =u+2 v \\ \Delta v_1 & =(2 u+2 v) \\ F_1 & =\frac{d p_1}{d t} \\ & =\rho A(u+v)(2 u+2 v) \\ & =2 \rho A(u+v)^2 \\ v_2 & =(u-2 v) \\ \Delta v_2 & =(2 u-2 v) \\ F_2 & =\frac{d p_2}{d t} \\ & =\rho A(u-v)(2 u-2 v)=2 \rho A(u-v)^2 \end{aligned} $
[ $\triangle F$ is the net force due to the air molecules on the plate]
$ \begin{gathered} \Delta F=2 \rho A(4 u v)=8 \rho A u v \\ P=\frac{\Delta F}{A}=8 \rho(u v) \\ F_{\mathrm{Det}}=(F-\Delta F)=m a \\ F-(8 \rho A u) v=m a \quad[m \text { is mass of the plate }] \end{gathered} $
$F-(8 \rho A u) v=m a$
BETA
