Question

When a metal ball is immersed in a liquid, the apparent weight of the ball at$0^\circ {\text{C}}$ and $50^\circ {\text{C}}$ are found to be ${W_1}$ and ${W_2}$ respectively. If ${\gamma _s} < {\gamma _l}$
A. ${W_1} > {W_2}$
B. ${W_1} = {W_2}$
C. ${W_2} > {W_1}$
D. Data insufficient

Answer 1

We need to find out the relation between the coefficient of expansion and the temperature to find the relation between apparent weight and temperature. Here, we can use the concept of buoyancy to find the relation.

Complete step by step answer:
We know that when a metal is immersed in any liquid it experiences an upthrust or upward force which is the buoyant force and thus the weight of the metal changes which is known as the apparent weight.
Now apparent weight = The weight of the metal - upthrust

Since the weight of the metal is constant for the same metal in any situation therefore the apparent weight is inversely proportional to the upthrust.
Now, upthrust is defined as
${F_{up}} = {\rho _l} \times {V_l} \times g$
Where ${F_{up}}$ is the upward force, ${\rho _l}$ is the density of the liquid, ${V_l}$ is the volume of the displaced liquid by the metal which is equal to the volume of the metal and g is the acceleration due to gravity.

Now with an increase in temperature, the weight of the displaced liquid will decrease.
It is given that ${\gamma _s} < {\gamma _l}$ that is the coefficient of expansion of solid is less than the coefficient of expansion of the liquid. Therefore the upthrust will decrease as the temperature will increase and hence the apparent weight of the metal will increase with an increase in temperature. Thus ${W_2} > {W_1}$.

Hence, option C is correct.

Note: The buoyant or the upward force is the same force we experience when we go into a water body and if the upward force/ upthrust is more than our weight we would be able to float in water like in a dead sea. The coefficient of expansion $\gamma$ is the change in the volume caused by the change in temperature.