Thermal Expansion

To determine how the angular speed of a uniform metal solid sphere changes when its temperature is raised, we need to consider the effect of thermal expansion on the sphere's moment of inertia. The angular speed of a rotating object is inversely proportional to its moment of inertia, so any change in the moment of inertia will affect the angular speed.

Step 1: Understand the relationship between angular speed and moment of inertia The angular momentum of a rotating object is given by: where:
- is the moment of inertia,
- is the angular speed. If no external torque acts on the system, angular momentum is conserved ( ). Thus: where:
- is the initial moment of inertia,
- is the initial angular speed,
- is the moment of inertia after the temperature change,
- is the angular speed after the temperature change. Rearranging for :

Step 2: Determine the change in moment of inertia due to thermal expansion The moment of inertia of a solid sphere is given by: where:
- is the mass of the sphere (constant),
- is the radius of the sphere. When the temperature is raised, the radius of the sphere increases due to thermal expansion. The new radius is: where:
- is the coefficient of linear expansion of the metal,
- is the change in temperature. Given:
- ,
- . Substitute these values: The new moment of inertia is:

Step 3: Calculate the new angular speed Using the relationship , substitute : Final Answer: The angular speed after the temperature increase is: