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Thermal expansion is the process where materials expand as they increase in temperature. This occurs because as atoms vibrate faster with heat, they tend to take up more space. For metals, this is defined by a constant known as the coefficient of linear expansion, denoted by \( \alpha \). In our exercise, steel is the material in question with \( \alpha \approx 11 \times 10^{-6} / ^{\circ}C \).

As the temperature of the bicycle wheel moves from \(-100.0^{\circ}C\) to \(+300.0^{\circ}C\), we notice a direct temperature change, \( \Delta T = 400^{\circ}C \).

• In thermal expansion, any length of the object changes in line with the formula: \( L_f = L_i(1 + \alpha \Delta T) \)
• For circular objects like the bicycle wheel, this affected dimension is the radius, which in turn changes the moment of inertia.

These changes, though tiny on a per-degree scale, are compounded in large temperature differences, affecting other physical properties, such as rotational motion.

Angular speed is a measure of how fast an object rotates or spins. It is expressed as radians per second (rad/s) and is crucial in determining how fast a rotation or cycle is completed.

In this exercise, the wheel initially rotates at 18.00 rad/s. When the wheel undergoes thermal expansion, it changes some dynamics. Without external interference, this is accompanied by a change in angular speed. Why? Because angular momentum is conserved.

• Without external torque, the conservation law implies \( I_i \omega_i = I_f \omega_f \)
• Since \( I_f = kI_i \), we rearrange to get the new angular speed: \( \omega_f = \omega_i / k \)

Upon heating, the wheel has a slightly slower angular speed: about 17.86 rad/s, indicating the system's balance between increasing inertia and diminishing speed.

Moment of inertia helps us understand how mass distribution affects a rotating object. It is crucial for calculations involving rotational motion. For a wheel, the moment of inertia relies heavily on the radius and mass distribution along that radius.

In our scenario, as the wheel expands, the radius increases and thereby the moment of inertia increases too. We denote the initial and final moments of inertia as \( I_i \) and \( I_f \) respectively. With thermal expansion, the relationship modifies as:

• \( I \propto r^2 \), means a small change in radius induces a larger change in inertia.
• Linear expansion adds the formula: \( I_f = I_i(1 + 2\alpha \Delta T) \)

By this relationship, we calculate an increase factor \( k \sim 1.008 \), confirming that when \( r \) changes even slightly, inertia adopts a consequential change.

This understanding is crucial as it informs why the wheel slows down, keeping angular momentum constant despite thermal changes.