Fewpal
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An empty plastic box of mass 5kg is observed to accelerate up at the rate of g/6 when placed deep inside water. What mass of sand should be put inside the box so that it may accelerate down at the rate of g/6?

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Let us take mass of empty plastic box be m for now. i.e m= 5 kg. We will take 

Weight - Buoyancy = (mass x Gravity) - Buoyancy

Since the net acceleration is g/6 upwards, then we have:

a = -g/6; 

we also know that a = F / m.

[Note this is a -ve sign because the acceleration is in the opposite direction to gravity]

so: 

a = -g/6 = [(m x g) - B] / m

Solving for B:-

g/6 = mg/m - B/m

B/m = g + g/6

B = gm [1 + 1/6]

B = 7/6 x gm

After the sand is added in the new mass is (m + ms) and a = g/6.

Net Force = weight of box + weight of sand - Buoyancy

Net Force = m x g + ms x g - B

B = 7/6 x gm

So Net force = mg + ms x g - 7/6 x gm = ms x g - gm/6= g [ ms - m/6]

Force = mass x acceleration. 

Now mass total is m + ms and acceleration is g/6

So g [ ms - m/6] = F = (m+ms)x a

g [ms - m/6] = (m + ms) x g/6ms - m/6 = (m + ms) 1/6ms - 1/6 ms = m/6 + m/65/6 ms = 2m/65 ms = 2mms = 2m/5

Hence, the mass = 2 x 5/5 = 2

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