To determine how the fraction of the volume of the metal submerged in mercury changes when both the metal and mercury are heated, we need to consider the concepts of thermal expansion and buoyancy. Let's break this down step by step.

Understanding Thermal Expansion

When materials are heated, they expand. The extent to which they expand is described by their coefficient of volume expansion. For our scenario, the coefficients are:

• Metal: 1
• Mercury (Hg): 2

This means that for every degree of temperature increase, the volume of the metal increases by a factor of 1, while the volume of mercury increases by a factor of 2.

Initial Conditions

Let's denote the initial volume of the metal as V_m and the initial volume of mercury displaced by the metal as V_d. The fraction of the volume of the metal submerged in mercury can be expressed as:

Fraction submerged = V_d / V_m

Effects of Heating

When we increase the temperature by an amount T, the new volumes after expansion will be:

• New volume of metal: V_m' = V_m (1 + 1T)
• New volume of mercury: V_d' = V_d (1 + 2T)

Buoyancy and Volume Displacement

The principle of buoyancy states that the weight of the fluid displaced by the submerged part of the object equals the weight of the object itself. As the temperature increases, both the metal and the mercury expand, but at different rates due to their coefficients of volume expansion.

Calculating the New Fraction Submerged

After heating, the new fraction of the volume of the metal submerged in mercury becomes:

New Fraction submerged = V_d' / V_m'

Substituting the expanded volumes:

New Fraction submerged = (V_d (1 + 2T)) / (V_m (1 + 1T))

Analyzing the Change

To find out how the fraction changes, we can compare the new fraction to the original fraction:

Change in Fraction = (V_d (1 + 2T)) / (V_m (1 + 1T)) / (V_d / V_m)

This simplifies to:

Change in Fraction = (1 + 2T) / (1 + 1T)

Final Thoughts

As you can see, the factor by which the fraction of the volume of the metal submerged in mercury changes is given by the expression:

Factor of change = (1 + 2T) / (1 + 1T)

This result shows that as the temperature increases, the fraction of the metal submerged in mercury will change based on the relative expansion rates of the two materials. If you plug in specific values for T, you can calculate the exact change in the fraction submerged. This illustrates how thermal expansion affects buoyancy and the behavior of objects in fluids.