Introduction:
In this problem, we have two uniform spheres of mass M with radii R and 2R. These spheres are rotating about a fixed axis through a diameter. We are given that the rotational kinetic energies of the spheres are identical, and we need to find the ratio of the magnitude of their angular momenta.

Angular momentum:
Angular momentum is a vector quantity that measures the rotational motion of an object. It depends on the mass, radius, and rotational speed of the object. The formula for the magnitude of angular momentum is given by L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.

Moment of inertia:
The moment of inertia (I) measures the resistance of an object to changes in its rotational motion. It depends on the mass distribution and the axis of rotation. The moment of inertia for a solid sphere rotating about an axis through its diameter is given by I = (2/5)MR², where M is the mass of the sphere and R is the radius.

Rotational kinetic energy:
Rotational kinetic energy is the energy possessed by an object due to its rotational motion. It depends on the moment of inertia and the square of the angular velocity. The formula for rotational kinetic energy is given by K = (1/2)Iω², where K is the rotational kinetic energy.

Equal rotational kinetic energies:
Given that the rotational kinetic energies of the spheres are identical, we can equate their expressions using the formulas mentioned above. Let's denote the rotational kinetic energy of the sphere with radius R as K₁ and the rotational kinetic energy of the sphere with radius 2R as K₂.

K₁ = (1/2)I₁ω₁²
K₂ = (1/2)I₂ω₂²

We are given that K₁ = K₂. Substituting the expressions for moment of inertia, we get:

(1/2)(2/5)MR²ω₁² = (1/2)(2/5)M(2R)²ω₂²

Simplifying the equation, we get:

(1/5)MR²ω₁² = (1/5)M(4R²)ω₂²

Ratio of angular momenta:
The ratio of the magnitude of the angular momenta can be found by dividing the angular momentum of the sphere with radius R by the angular momentum of the sphere with radius 2R. Let's denote the angular momentum of the sphere with radius R as L₁ and the angular momentum of the sphere with radius 2R as L₂.

L₁ = I₁ω₁
L₂ = I₂ω₂

To find the ratio L₁/L₂, we can substitute the expressions for moment of inertia from above:

L₁ = (2/5)MR²ω₁
L₂ = (2/5)M(2R)²ω₂

Taking the ratio, we get:

L₁/L₂ = [(2/5)MR²ω₁] / [(2/5)M(2R)²ω₂]

Simplifying the expression, we get:

L₁/L₂ = (R/4R) * (ω₁/ω₂)

Cancelling out the common terms, we

Ans is 1/4.