I think it's rotational speed will decrease because it's some of it's kinetic energy get converted into heat energy

Explanation:

When a solid metal sphere is rotating about its diameter with a constant angular speed (ω), its temperature can increase appreciably due to various factors such as heat transfer, friction, or external heating. This increase in temperature can affect the rotational speed of the sphere.

Effect of Temperature on Rotational Speed:

When the temperature of a solid metal sphere increases, it causes thermal expansion, which leads to a change in the sphere's dimensions. This change in dimensions can affect the rotational speed of the sphere due to the conservation of angular momentum.

Conservation of Angular Momentum:

Angular momentum (L) is the product of the moment of inertia (I) and the angular velocity (ω) of the rotating object. According to the law of conservation of angular momentum, the angular momentum of a closed system remains constant unless acted upon by an external torque.

Effect of Thermal Expansion:

When the temperature of the metal sphere increases, it undergoes thermal expansion, causing an increase in its dimensions. As a result, the moment of inertia of the sphere changes since it depends on the mass distribution and the shape of the object.

Increase in Moment of Inertia:

When the temperature of the sphere increases, its moment of inertia also increases due to the change in dimensions. This increase in moment of inertia affects the rotational speed of the sphere.

Change in Rotational Speed:

As the moment of inertia increases, the angular velocity of the sphere decreases to conserve angular momentum. This means that the rotational speed of the sphere decreases when its temperature increases appreciably.

Summary:

In summary, when a solid metal sphere is rotating about its diameter with a constant angular speed and its temperature increases appreciably, the rotational speed of the sphere will decrease. This is due to the increase in the moment of inertia of the sphere caused by thermal expansion, which leads to a decrease in angular velocity to conserve angular momentum.