Question 423: Properties of Matter

Question

A train with cross-sectional area S
t
is moving with speed vt
inside a long tunnel of cross-sectional
area S
0
(S
0
= 4S
t
). Assume that almost all the air (density $$\rho $$) in front of the train flows back between
its sides and the walls of the tunnel. Also, the air flow with respect to the train is steady and laminar.
Take the ambient pressure and that inside the train to be p
0
. If the pressure in the region between the
sides of the train and the tunnel walls is p, then
p
0
- p = $${7 \over {2N}}\rho v_t^2$$. The value of 𝑁 is ________.

Accepted Answer

Applying Bernoulli's equation,
$${p_0} + {1 \over 2}ho v_1^2 = p + {1 \over 2}ho {v^2}$$
$${p_0} - p = {1 \over 2}ho ({v^2} - v_1^2)$$ .... (i)
From equation of continuity,
$$4{S_t}{v_t} = v 	imes 3{S_t}$$
$$ \Rightarrow v = {4 \over 3}{v_t}$$ ..... (ii)
From Eqs. (i) and (ii), we get
$${p_0} - p = {1 \over 2}ho \left( {{{16} \over 9}v_t^2 - v_t^2} ight) = {1 \over 2}ho {{7v_t^2} \over 9}$$
$$	herefore$$ N = 9

Properties of Matter

Integer / Numeric Answer Type