To solve this problem, we need to analyze the dynamics of the two cylinders in a gravity-free environment. The larger cylinder has a radius of 4r and is rotating with a constant angular velocity, while the smaller cylinder, with a radius of r, is rotating inside it and remains in contact with the inner surface of the larger cylinder. Our goal is to find the angular velocity of the smaller cylinder.
Understanding the System
In this scenario, we have two cylinders: the larger one is rotating at a constant angular velocity, which we can denote as ω₁. The smaller cylinder, which is in contact with the inner surface of the larger cylinder, will experience a tangential velocity due to this rotation. The key point here is that the smaller cylinder must rotate in such a way that it maintains contact with the larger cylinder.
Angular Velocities and Relationships
Let’s denote the angular velocity of the smaller cylinder as ω₂. The tangential velocity of a point on the surface of the larger cylinder can be expressed as:
For the smaller cylinder, the tangential velocity at its surface is:
Establishing the Relationship
Since the smaller cylinder is always in contact with the larger cylinder, the tangential velocities at the point of contact must be equal. Therefore, we can set the two equations for tangential velocity equal to each other:
Solving for Angular Velocity
Now, we can simplify this equation to find ω₂:
Thus, the angular velocity of the smaller cylinder is:
Conclusion
In summary, the angular velocity of the smaller cylinder is four times that of the larger cylinder. This relationship arises from the requirement that the tangential velocities at the point of contact between the two cylinders must be equal. Therefore, if the larger cylinder rotates with an angular velocity ω₁, the smaller cylinder will rotate with an angular velocity of 4ω₁.
