To understand why angular momentum is not affected when the radius of a rotating solid sphere is increased while keeping the mass constant, let's consider the definition of angular momentum and how it relates to the variables involved.


Angular momentum (L) is a vector quantity defined as the cross product of the moment of inertia (I) and the angular velocity (ω) of an object. Mathematically, it can be expressed as:
L = I × ω


Angular velocity is the rate of change of angular displacement with respect to time and is given by the formula:
ω = Δθ / Δt


Moment of inertia is a scalar quantity that represents an object's resistance to changes in its rotational motion. For a solid sphere, the moment of inertia can be calculated using the formula:
I = (2/5) * m * r^2
where m is the mass of the sphere and r is its radius.


Rotational kinetic energy (K) is the energy possessed by an object due to its rotational motion and is given by the formula:
K = (1/2) * I * ω^2


When the radius of the sphere is increased while keeping the mass constant, the following changes occur:

- Angular velocity (ω): The angular velocity remains unaffected because it depends on the rate of change of angular displacement, which is independent of the size or radius of the object.

- Angular momentum (L): As mentioned earlier, angular momentum is the cross product of moment of inertia and angular velocity. Since angular velocity remains constant while the moment of inertia increases with the square of the radius, the angular momentum also remains constant. Therefore, increasing the radius does not affect the angular momentum.

- Moment of inertia (I): The moment of inertia increases with the square of the radius due to its dependence on the distribution of mass. Therefore, when the radius is increased while keeping the mass constant, the moment of inertia also increases.

- Rotational kinetic energy (K): The rotational kinetic energy depends on both the moment of inertia and the square of the angular velocity. As the moment of inertia increases while the angular velocity remains constant, the rotational kinetic energy also increases.


In conclusion, when the radius of a rotating solid sphere is increased while keeping the mass constant, the angular velocity and angular momentum remain unchanged. However, the moment of inertia and rotational kinetic energy both increase.