A flat plate 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

Question

A flat plate 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]

Smart Achievers · Accepted Answer

(1, 2, 4)

$1 = \frac{v_{1} - v}{v + u}$ $1 = \frac{v + v_{2}}{u - v}$

$∴$ $v_{1} = u + 2 v$ $∴$ $v_{2} = u - 2 v$

$∴$ $Δ v_{1} = 2 u + 2 v$ and $Δ v_{2} = 2 u - 2 v$

$Now F_{1} = \frac{d p_{1}}{d t} = ρ A (u + v) (2 u + 2 v)$

$and F_{2} = \frac{d p_{2}}{d t} = ρ A (u - v) (2 u - 2 v)$

$∴$ $F_{1} = 2 ρ A {(u + v)}^{2} and F_{2} = 2 ρ A {(u - v)}^{2}$

$Δ F is the net force due to the air molecules on the plate.$

$Δ F = F_{1} - F_{2} = 8 ρ A u v$ $∴$ $P = \frac{Δ F}{A} = 8 ρ u v$

$The net force F_{net} = F - Δ F = m a$

$∴$ $F - 8 ρ A u v = m a$

Due to viscosity, the plate will eventually reach terminal velocity. So now the plate will move with constant velocity.

Solution

Q.
A flat plate 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]

1 The pressure difference between the leading and trailing faces of the plate is proportional to uv

2 The resistive force experienced by the plate is proportional to v

3 The plate will continue to move with constant non-zero acceleration, at all times

4 At a later time the external force F balances the resistive force

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