Question

A beaker containing water is placed on the pan of a balance, which shows a reading of $M$. A lump of sugar of mass $m$ and volume $V$ is now suspended by a thread (from an independent support) in such a way that it is completely immersed in water without touching the beaker and without any overflow of water. How will the reading change as time passes on ?

• A. Reading will increase.
• B. Reading will decrease.
• C. Reading will remain the same.
• D. Insufficient information

Answer 1

The correct option is A Reading will increase.

Let $\rho$ and ${\rho }_w$ be the density of sugar and water respectively.

Initially, the balance will show the weight of water and thrust.

And we know that
$\text{Thrust}=V{\rho }_{w}g=\frac{m}{\rho }\times {\rho }_w\times g$

$[\text{as}V=\frac{m}{\rho }]$

$\therefore W_1=(M+\frac{m}{\rho }{\rho }_w)g$

$W_1=(M+\frac{m{\rho }_w}{2\rho }+\frac{m{\rho }_w}{2\rho })g........\left(1\right)$


Consider the situation when half the sugar has dissolved. In this situation the thrust reduces to $\left(\frac{m{\rho }_{w}g}{2\rho }\right)$ from $\left(\frac{m{\rho }_{w}g}{\rho }\right)$ while weight of the solution increases by $\left(\frac{mg}{2}\right)$ (due to sugar dissolved). So reading of the balance will become

$W_2=(M+\frac{m}{2}+\frac{m{\rho }_w}{2\rho })g......\left(2\right)$

Finally, when all the sugar is dissolved, thrust will become zero and the weight of the solution will increase by $mg$.
So the reading will become.

$W_3=(M+m)g$

$W_3=(M+\frac{m}{2}+\frac{m}{2})g......\left(3\right)$

Now as $\rho >1{\text{g/cm}}^3$ and ${\rho }_w=1{\text{g/cm}}^3$

$\therefore \frac{m}{2}>\frac{m}{2\rho }$

So comparing Eqs . $\left(1\right),\left(2\right)\text{and}\left(3\right)$, we find that the reading of the balance will gradually increase till all the sugar dissolves in water and finally will become constant equal to $(M+m)g$.

Hence, option $\left(\text{A}\right)$ is correct.

Alternative solution: Trick: If dissolving material have higher density as compare to water density, reading will always increase.