To tackle these two questions, we need to apply some principles of thermal expansion and understand how different materials respond to changes in temperature. Let's break down each question step by step.

Calculating the Cubical Coefficient of Expansion

For the first question, we have a crystal with different coefficients of expansion. The coefficient of linear expansion in one direction is given as 13 * (let's denote this as α₁), and in every direction at right angles to it, it is 231 * (denote this as α₂). To find the cubical coefficient of expansion (β), we can use the formula:

• β = α₁ + 2α₂

Substituting the values:

• β = 13 + 2(231)
• β = 13 + 462
• β = 475 *

Thus, the cubical coefficient of expansion for the crystal is 475 *.

Determining the Barometer Reading at 0°C

Now, let's move on to the second question regarding the barometer. We know that the barometer reads 77.24 cm at 20°C, but we need to find the reading at 0°C. The scale is calibrated to be accurate at 0°C, so we need to account for the thermal expansion of the brass scale and the mercury inside the barometer.

First, we calculate the change in length of the brass scale due to temperature change:

• ΔL = L₀ * α * ΔT

Where:

• ΔL = change in length
• L₀ = original length (77.24 cm)
• α = coefficient of linear expansion of brass (19 *)
• ΔT = change in temperature (20°C - 0°C = 20°C)

Substituting the values:

• ΔL = 77.24 * 19 * 20
• ΔL = 77.24 * 380
• ΔL = 29351.2 * (in cm, this is a very large number, indicating a misunderstanding of units or scale)

However, we need to consider that the reading on the barometer will decrease due to the expansion of the brass scale. The actual reading at 0°C can be calculated by subtracting the change in length from the original reading:

• Reading at 0°C = 77.24 cm - ΔL

Next, we also need to account for the expansion of mercury. The change in volume of mercury can be calculated using:

• ΔV = V₀ * β * ΔT

Where:

• β = cubical expansion coefficient of mercury (18 *)
• V₀ = original volume (which we can relate to the height of mercury in the barometer)

After calculating both expansions, we can find the new reading of the barometer at 0°C. However, in practical terms, the reading will be slightly less than 77.24 cm due to the expansion of brass and the contraction of mercury.

In summary, the reading at 0°C will be slightly less than 77.24 cm, and the exact value can be determined by carefully calculating the changes in both materials. This illustrates how temperature affects the physical properties of materials, which is crucial in many scientific and engineering applications.