An empty plastic box of mass m is found to accelerate up at the rate of g/6 when placed deep inside water. How much sand should be put inside the box so that it may accelerate down at the rate of g/6?

$\frac{2m}{5}$

The upward acceleration is due to force of buoyancy applied by water on the plastic box. Let force of buoyancy is $F.$

Now F - mg = m $\frac{g}{6}$

⇒ F = $\frac{7mg}{6}$

Let M be the mass of sand to be put to give acceleration $g/6$ down$.$ Then from equation of motion$,$

$(m+M)g\; -\; \$\backslash frac\{7mg\}\{6\}\; =\; \backslash frac\{(m+M)g\}\{6\}\$$

mg +Mg - 7mg/6 = mg/6 + Mg/6

$Mg\times 5/6=mg2/6$

$M\; =\; \$\backslash frac\{2m\}\{5\}\$$