A steel bicycle wheel (without the rubber tire) is rotating freely with an angular speed of 18.00 rad/s. The temperature of the wheel changes from -100.0 to +300.0 degrees Celcius. No net external torque acts on the wheel, and the mass of the spokes is negligible. (a) Does the angular speed increase or decrease as the wheel heats up? Why? (b) What is the angular speed at the higher temperature?
To determine whether the angular speed of the wheel increases or decreases as it heats up, we need to understand the relationship between angular speed and moment of inertia.
(a) Angular speed, represented as ω, is related to the moment of inertia, represented as I, by the equation:
I = I0 + αt,
where I0 is the initial moment of inertia, α is the angular acceleration, and t is the time. If there is no net external torque acting on the wheel, the angular acceleration α will be zero, and thus the moment of inertia will remain constant.
Now, the moment of inertia of a solid cylindrical wheel can be calculated using the formula:
I = 0.5 * m * r^2,
where m is the mass of the wheel and r is its radius. Since the steel wheel does not have a rubber tire, we can assume the mass is distributed uniformly, so the moment of inertia remains constant as the temperature changes.
Therefore, as the wheel heats up, the moment of inertia remains the same, and hence the angular speed also remains constant. The angular speed does not increase or decrease.
(b) Thus, the angular speed at the higher temperature will be 18.00 rad/s, the same as the initial angular speed.