A glass tube filled with Mercury is attached to the bottom of an iron pendulum rod of length 100 cm. How high must the Mercury be filled in the glass tube so that the centre of mass of this pendulum will not rise or fall with changes in temperature? Linear expansion coefficient of Iron = 11 x 10⁶ K^-1 and cubical expansion of coefficent of Mercury = 18.2 x 10⁶ K^-1

To determine how high the mercury must be filled in the glass tube so that the center of mass of the pendulum remains stable with temperature changes, we need to consider the effects of thermal expansion on both the iron rod and the mercury. The goal is to ensure that the changes in volume of the mercury due to temperature variations counterbalance the changes in length of the iron rod.

Understanding Thermal Expansion

When materials are heated, they tend to expand. For solids like iron, this expansion is primarily linear, while for liquids like mercury, it is volumetric. The coefficients of expansion tell us how much a material will expand per degree of temperature change.

Coefficients of Expansion

• Linear expansion coefficient of iron: 11 x 10^-6 K^-1
• Cubic expansion coefficient of mercury: 18.2 x 10^-6 K^-1

Calculating Changes in Length and Volume

Let's denote the initial length of the iron rod as L_iron = 100 cm. The change in length (ΔL) of the iron rod when the temperature changes by ΔT can be calculated using the formula:

ΔL = L_iron × α_iron × ΔT

Substituting the values, we have:

ΔL = 100 cm × (11 x 10^-6 K^-1) × ΔT

Volume Expansion of Mercury

The change in volume (ΔV) of the mercury can be calculated using the formula:

ΔV = V_mercury × β_mercury × ΔT

Here, V_mercury is the initial volume of mercury, and β_mercury is the cubic expansion coefficient of mercury. The volume of mercury can be expressed in terms of its height (h) in the tube and the cross-sectional area (A) of the tube:

V_mercury = A × h

Thus, the change in volume becomes:

ΔV = (A × h) × (18.2 x 10^-6 K^-1) × ΔT

Balancing the Expansions

For the center of mass to remain unchanged, the change in length of the iron rod must equal the change in height of the mercury. This means that the increase in length of the iron rod (ΔL) must be balanced by the increase in height of the mercury (Δh), which can be related to the change in volume of mercury:

ΔL = Δh

Substituting the expressions we derived:

100 cm × (11 x 10^-6 K^-1) × ΔT = (A × h) × (18.2 x 10^-6 K^-1) × ΔT

Solving for Height

We can cancel ΔT from both sides (assuming it is not zero) and rearrange the equation to find the height of mercury:

100 cm × (11 x 10^-6) = (A × h) × (18.2 x 10^-6)

Now, solving for h:

h = (100 cm × 11 x 10^-6) / (A × 18.2 x 10^-6)

To find the exact height, we need the cross-sectional area (A) of the tube. However, if we assume A is constant and does not affect the ratio, we can express h in terms of A:

h = (100 × 11) / 18.2

Calculating this gives:

h ≈ 60.44 cm

Final Thoughts

Thus, the mercury must be filled to approximately 60.44 cm in the glass tube to ensure that the center of mass of the pendulum remains stable with temperature fluctuations. This balance is crucial for maintaining the pendulum's functionality in varying thermal conditions.