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 answer this question, we need to understand the concept of conservation of angular momentum.
Angular momentum is the rotational equivalent of linear momentum, and according to the conservation of angular momentum, the total angular momentum of a system remains constant if no net external torque acts on it. In other words, the initial angular momentum of the system must be equal to its final angular momentum.
Now let's analyze the situation step by step:
(a) Does the angular speed increase or decrease as the wheel heats up? Why?
From the given information, we know that the bicycle wheel is rotating freely with an initial angular speed of 18.00 rad/s. When the temperature changes, it affects the dimensions of the wheel, including its radius and moment of inertia.
As the temperature increases, the wheel expands due to thermal expansion. This leads to an increase in the radius of the wheel, which in turn changes its moment of inertia. Consequently, the moment of inertia of the wheel increases.
Since angular momentum is the product of moment of inertia and angular speed (L = Iω), the conservation of angular momentum implies that if the moment of inertia increases, the angular speed must decrease to keep the total angular momentum constant.
Therefore, as the wheel heats up, its angular speed decreases.
(b) What is the angular speed at the higher temperature?
To find the angular speed at the higher temperature, we can use the conservation of angular momentum principle mentioned above. Since the total angular momentum of the wheel remains constant, we can set the initial angular momentum equal to the final angular momentum.
Initial Angular Momentum = Final Angular Momentum
The initial angular momentum is given by L_initial = I_initial * ω_initial, where ω_initial is the initial angular speed of 18.00 rad/s.
The final angular momentum is defined as L_final = I_final * ω_final, where ω_final is the angular speed at the higher temperature.
Since no net external torque acts on the wheel, we can assume that the initial and final moments of inertia are related by the equation:
I_final = k * I_initial,
where k is a constant.
Substituting these values into the conservation equation, we get:
I_initial * ω_initial = k * I_initial * ω_final.
The moment of inertia cancels out, and we have:
ω_final = ω_initial / k.
To find the value of k, we need additional information that relates the change in temperature to the change in the radius or moment of inertia. Unfortunately, this information is not provided in the given question.
Therefore, we cannot determine the exact value of ω_final without knowing the specific relationship between temperature change and the change in the moment of inertia for the given wheel.
In summary, without more information, we cannot determine the exact angular speed at the higher temperature, but we can conclude that it will be less than the initial angular speed of 18.00 rad/s due to the increase in moment of inertia.