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Question
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?
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Solution
Let U be the upward force of water acting on the plastic box.
Let m be the initial mass of the plastic box.
When the empty plastic box is accelerating upward,
\[U - mg = \frac{mg}{6}\]
\[ \Rightarrow U = \frac{7 mg}{6}\]
\[\Rightarrow m = \frac{6U}{7g} . . . . \left( i \right)\]
Let M be the final mass of the box after putting some sand in it.
\[Mg - U = \frac{Mg}{6}\]
\[ \Rightarrow Mg - \frac{Mg}{6} = U\]
\[ \Rightarrow M = \frac{6U}{5g} . . . . \left( ii \right)\]
Mass added
\[= \frac{6U}{5g} - \frac{6U}{7g}\]
\[= \frac{6U\left( 7 - 5 \right)}{35 g}\]
\[ = \frac{6U \cdot 2}{35 g}\]
From equation (i),
\[m = \frac{6U}{7g}\]
∴ Mass added
\[= \frac{2}{5}m\]
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