An iron ball has a diameter of 6 cm and is 0.010 mm too...

An iron ball has a diameter of $6 \mathrm{~cm}$ and is $0.010 \mathrm{~mm}$ too large to pass through a hole in a brass plate when the ball and plate are at a temperature of $30^{\circ} \mathrm{C}$. At what temperature (the same for ball and plate) will the ball just pass through the hole? $\alpha=1.2 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}$ and $1.9 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}$ for iron and brass, respectively.

Key Concepts

Thermal Expansion

Thermal expansion refers to the tendency of materials to change in size—typically to expand—when they are heated and contract when cooled. This phenomenon is fundamental in understanding how the dimensions of objects vary with temperature changes and is crucial in many engineering applications, particularly in scenarios where precise fits between different parts are required.

Coefficient of Thermal Expansion

The coefficient of thermal expansion is a material-specific constant that quantifies the degree to which a material expands per unit temperature increase. It is fundamental for predicting how much an object’s dimensions will change with a temperature variation and is an essential parameter when comparing the thermal behavior of different materials.

Linear Expansion Formula

The linear expansion formula, often expressed as ?L = L???T, relates the change in a material's length ?L to its original length L?, the coefficient of thermal expansion ?, and the change in temperature ?T. This equation is widely used to calculate the dimensional changes in materials due to temperature fluctuations, which is key in designing mechanical systems and ensuring compatibility between interfacing components.

Differential Expansion

Differential expansion occurs when two materials with different coefficients of thermal expansion are subjected to the same temperature change, resulting in different amounts of expansion or contraction. This concept is important in engineering applications where materials in contact or in assembly must accommodate these differences to avoid interference or mechanical failure.

Transcript

Please give Ace some feedback