Two identical solid spheres, each of mass M and radius R, are connected by a massless rigid rod. Initially, the centers of the spheres are separated by a distance of 4R. The system rotates with an angular speed ω₀ about an axis perpendicular to the rod and passing through its center. While rotating, internal forces pull the spheres closer until their centers are separated by a distance of 2R (i.e., they are just touching). Assuming no external torques act on the system during this process, what is the ratio of the new angular speed (ωf) to the original angular speed (ω₀)? (The moment of inertia of a solid sphere about its diameter is (2/5)MR²).
Correct: C
This problem involves the conservation of angular momentum because no external torques act on the system. The angular momentum (L) is given by L = Iω, where I is the moment of inertia and ω is the angular speed.
First, calculate the initial moment of inertia (I₀):
Each sphere has a mass M and radius R. Its moment of inertia about its own center of mass (diameter) is I_CM = (2/5)MR². The axis of rotation for the system is perpendicular to the rod and passes through its center. Initially, the centers of the spheres are 4R apart, so each sphere's center is 2R from the central axis of rotation.
Using the parallel-axis theorem, I = I_CM + Md², where d is the distance from the CM to the axis of rotation.
For one sphere: I_single_sphere = (2/5)MR² + M(2R)² = (2/5)MR² + 4MR² = (2/5 + 20/5)MR² = (22/5)MR².
Since there are two identical spheres, the total initial moment of inertia is: I₀ = 2 * (22/5)MR² = (44/5)MR².
Next, calculate the final moment of inertia (If):
In the final configuration, the centers of the spheres are 2R apart. This means each sphere's center is R from the central axis of rotation.
For one sphere: I_single_sphere = (2/5)MR² + M(R)² = (2/5)MR² + MR² = (2/5 + 5/5)MR² = (7/5)MR².
The total final moment of inertia is: If = 2 * (7/5)MR² = (14/5)MR².
Now, apply the conservation of angular momentum:
L₀ = Lf
I₀ω₀ = Ifωf
(44/5)MR² * ω₀ = (14/5)MR² * ωf
To find the ratio ωf / ω₀:
ωf / ω₀ = I₀ / If
ωf / ω₀ = [ (44/5)MR² ] / [ (14/5)MR² ]
ωf / ω₀ = 44 / 14
ωf / ω₀ = 22 / 7
The correct choice is C.