Question

A plate of mass $M$ is moved with constant velocity $v$ against dust particles moving with velocity $u$ in opposite direction as shown. The density of the dust is $\rho$ and plate area is $A$. Find the force $F$ required to keep the plate moving with uniform velocity.

• A. $\rho A(u+v)$
• B. $\rho (u+v{)}^2$
• C. $\rho A{u}^2$
• D. $\rho A(u+v{)}^2$

Answer 1

The correct option is D $\rho A(u+v{)}^2$


Velocity of plate relative to dust = $(v+u)$
In time ${}^{\prime }d{t}^{\prime }$, the plate comes into contact with volume of dust particles
$dV=A\cdot dx=A(v+u)dt$

So, mass of dust particles striking the plate in time $dt$ is
$dm=\rho dV=\rho A(v+u)dt$ ...(1)

If we consider (plate + $dm$ mass of dust) as a system and apply impulse momentum equation over time $dt$,


Change in momentum $△P=P_f-P_i$
$△P=(M+dm)v-[Mv-dm.u]$
$=Mv+dm.v-Mv+dm.u$
$=dm.(v+u)$
$\Delta P=F.t$
$\Rightarrow F=\frac{dm}{dt}(u+v)$

From eq. (1)
$F=\frac{\rho A(v+u)dt.(v+u)}{dt}$
$F=\rho A(u+v{)}^2$ is the force acting on the plate due to the dust.
Hence, the same force $F$ is required to keep the plate moving with uniform velocity.