Flat Plate Moving Normal to Plane in Low Pressure Gas

To determine which statements are correct, we analyze the interaction between the moving flat plate and the gas molecules using the kinetic theory of gases.

1. Pressure on the Leading and Trailing Faces

Let the plate move with velocity $v$ in the positive $x$-direction (normal to its plane). The gas has mass density $\rho = n m$, where $n$ is the number density of the gas molecules and $m$ is the mass of a single molecule. The mean thermal speed of the molecules is $u$, and $v \ll u$.

Consider a molecule with normal velocity component $v_x$ relative to the gas at rest.

• Leading Face (Front): In the reference frame of the moving plate, the relative speed of an approaching molecule is $(v_x + v)$. Upon elastic collision with the plate, the change in momentum of each molecule is: $\Delta p = 2m(v_x + v)$

  The number of molecules striking a unit area of the leading face per unit time with velocity components in the range $[v_x, v_x + dv_x]$ is given by $(v_x + v) f(v_x) dv_x$, where $f(v_x)$ is the one-dimensional velocity distribution function.

  Thus, the pressure on the leading face $P_1$ is: $P_1 = \int_{0}^{\infty} 2m(v_x + v)^2 f(v_x) dv_x$ Expanding the term inside the integral: $P_1 = 2m \int_{0}^{\infty} v_x^2 f(v_x) dv_x + 4m v \int_{0}^{\infty} v_x f(v_x) dv_x + 2m v^2 \int_{0}^{\infty} f(v_x) dv_x$

  Since $v \ll u$, we neglect the $v^2$ term: $P_1 \approx P_0 + 4m v \left(\frac{n}{2} \cdot \bar{v}_x\right)$ where $P_0$ is the static gas pressure and $\bar{v}_x \propto u$ is the average speed of molecules moving in one direction. $P_1 \approx P_0 + C \rho u v$ where $C$ is a constant of proportionality.

• Trailing Face (Back): Similarly, for the trailing face, the relative velocity of approaching molecules is $(v_x - v)$. The pressure $P_2$ on the trailing face is given by: $P_2 \approx P_0 - C \rho u v$

2. Difference in Pressure and Resistive Drag Force

The pressure difference $\Delta P$ between the leading and trailing faces is: $\Delta P = P_1 - P_2 \approx 2 C \rho u v$

• Since $\rho$ is constant, the pressure difference is directly proportional to $uv$: $\Delta P \propto uv$ Therefore, Statement [A] is correct.

The total resistive drag force $F_{\text{resistive}}$ experienced by the plate of area $A$ is: $F_{\text{resistive}} = \Delta P \cdot A = (2 C \rho A u) v$

• Since $(2 C \rho A u)$ is constant, the resistive force is directly proportional to the velocity $v$: $F_{\text{resistive}} \propto v$ Therefore, Statement [B] is correct.

3. Dynamics of the Plate

Under the influence of the constant external force $F$, the equation of motion for the plate of mass $M$ is: $M \frac{dv}{dt} = F - F_{\text{resistive}} = F - b v$ where $b = 2 C \rho A u$.

Solving this differential equation with initial condition $v(0) = 0$: $v(t) = \frac{F}{b} \left(1 - e^{-\frac{b}{M}t}\right)$

The acceleration of the plate is: $a(t) = \frac{dv}{dt} = \frac{F}{M} e^{-\frac{b}{M}t}$

• As $t \to \infty$, $a(t) \to 0$. Thus, the acceleration decreases over time and is not constant. Therefore, Statement [C] is incorrect.

• As $t \to \infty$, the velocity approaches a constant terminal velocity $v_{\text{terminal}} = \frac{F}{b}$. At this terminal state, $a = 0$, which implies: $F = F_{\text{resistive}}$ Thus, at a later time, the applied force $F$ balances the resistive drag force. Therefore, Statement [D] is correct.

Conclusion

The correct statements are [A], [B], and [D].