Question

A uniform spherical ball of mass $m$ and radius $r$ is dropped in liquid of coefficient of viscosity $\eta$. Density of the liquid is $\rho$, whereas density of ball is $2\rho$. The de-Broglie wavelength of the ball when it is moving with terminal speed is $\frac{2\beta \pi h\eta r}{m^{2}g}$. Find β.
($h=$ Planck's constant)

Answer 1

Terminal speed of the ball is achieved when viscous force is balanced by the difference in buoyancy and weight.
i.e $mg-\frac{mg}{2}=6\pi r\eta v$
or $v=\frac{mg}{12\pi r\eta }$

At terminal velocity $v$, de-Broglie wavelength
$\lambda =\frac{h}{p}=\frac{h}{mv}=\frac{h}{m\left(\frac{mg}{12\pi \eta r}\right)}$

$\Rightarrow \lambda =\frac{12\pi \eta hr}{m^{2}g}$
$\therefore \beta =6$