Question

A bucket of water is suspended from a spring balance. What happens to reading of balance (a) when a piece of stone suspended from a string is immersed in the water without touching the bucket (b) when a piece of lead or cork is put in the water in the bucket.

Answer 1

Hint:Use the concept of floatation law to determine the readings of the spring balance for different conditions. Also use the formula for upward buoyant force acting on the object completely or partly immersed in a liquid. Hence, determine the new reading of the spring balance using the concept of floatation law when stone is suspended in water and when cork or lead is put in the water.

Formula used:
The buoyant force \[{F_B}\] acting on an object immersed in a fluid in the upward direction is given by
\[{F_B} = \rho Vg\] …… (1)
Here, \[\rho \] is density of the fluid, \[V\] is volume of the object immersed in the fluid and \[g\] is acceleration due to gravity.

Complete step by step answer:
(a) We have asked the effect of the reading of the spring balance if a stone is suspended in the water with the help of a sting without touching the water.As the stone is immersed and suspending in the water in bucket, according to the law of floatation, the water exerts an upward buoyant force on the stone.Let \[{W_0}\] be the initial reading of the spring balance.Let \[m\] be mass of the stone, \[{\rho _s}\] be density of stone and \[{\rho _w}\] be density of water. Rewrite equation (1) for buoyant force acting on the stone.
\[{F_B} = {\rho _w}Vg\]
\[ \Rightarrow {F_B} = {\rho _w}\dfrac{m}{{{\rho _s}}}g\]
\[ \Rightarrow {F_B} = \dfrac{{{\rho _w}}}{{{\rho _s}}}mg\]

This expression gives the weight of the water displaced by the stone. As the density of stone is more than the density of water, the buoyant force will be less than actual weight of the stone.Thus, the new reading of the spring balance will be
\[W = {W_0} + {F_B}\]

Hence, the reading of the spring balance increases when the stone is suspended in the bucket.

(b) In the second case, a piece of cork or lead is placed in the bucket.As the density of cork is less than the density of water, the cork will float on the surface of water and as the density of lead is more than the density of water, the piece of lead will submerge in water and reach at the bottom of the bucket. Let \[w\] be the weight of the cork and lead piece. Thus, according to floatation law, the new reading of the spring balance will be
\[W = {W_0} + w\]

Hence, the reading of the spring balance increases when a piece of cork or lead is put in the bucket.

Note:The students should keep in mind that the upward force acting on the stone represents the weight of the liquid displaced by the stone. So, the upward force adds in the new reading of spring balance. Also, since the cork floats on the surface of water, the upward thrust is equal to weight of cork and weight of lead acts at the bottom of the bucket. So, these two values of weights of cork and lead should be added in initial reading of the spring balance.