Elastic Collision and Acceleration Dynamics in a Rotating System

Understanding the dynamics of a rotating system and the resulting collisions with attached objects is crucial in various engineering and physics applications. In this article, we examine a specific scenario where an elastic collison is involved, exploring the complex interplay between a rotating rim and a block attached by a flexible string.

The Scenario

The problem in question involves a 20 kg rim with an arc speed of 10 m/s. Attached at the top of the rim’s surface via a string, a 5 kg block initially rests on dry ice. The goal is to determine the final speed of the 5 kg block after the system reaches a stable state. However, as noted by Torsten Hehl, this scenario is open to various interpretations depending on the specifics of the setup.

Assumptions and Calculations

To simplify the problem, we make the following assumptions:

The string is flexible and can stretch to accommodate the transfer of momentum without breaking. The system operates on a frictionless plane, allowing the block to move freely after being released. For the first stage, we assume the block will move in a straight line while momentum is transferred from the rim to the block.

Using the principles of physics, specifically the conservation of angular and linear momentum, we can derive the final speeds:

Calculating Final Speeds

We use the equations derived from the conservation of momentum:

tp_{initial} p_{final}
v_{block} frac{2 m_{rim}}{m_{rim} m_{block}} u
v_{rim} frac{m_{rim} - m_{block}}{m_{rim} m_{block}} u

Given the initial speed of the rim u is 10 m/s:

v_{block} frac{2 cdot 20}{20 5} cdot 10
v_{block} frac{400}{25}
v_{block} 16 , text{m/s}.

Subsequent Stages

Once the block reaches the rim, it will experience both rotational and linear motion dynamics. We consider two potential outcomes:

Full inelastic collision: The block sticks to the rim, resulting in a combined tangential speed. Continued string dynamics: The block remains attached, which involves de-spinning and acceleration due to gravity.

Inelastic Collision Outcome

If the block collides inelastically with the rim:

v_{final} frac{m_{rim} cdot v_{rim} m_{block} cdot v_{block}}{m_{rim} m_{block}}
v_{final} frac{20 cdot 6 5 cdot 16}{25}
v_{final} frac{120 80}{25}
v_{final} frac{200}{25}
v_{final} 8 , text{m/s}.

Continued String Dynamics

In the scenario where the block remains attached to the string, the system involves a complex mix of rotational and linear motion:

The block will initially move faster than the rim due to its greater speed. As it approaches the rim, the string will stretch, transferring momentum and de-spinning the system. The final speed of the block and the rim will be determined by the balance of these forces, which is difficult to predict without numerical simulation.

Conclusion

Given the setup and initial conditions, the block can accelerate to a speed of 16 m/s. In a fully inelastic collision, the final speed of the combined system will be 8 m/s. In the case where the block remains attached to the string, the exact final speed will depend on the specifics of the de-spinning and acceleration processes, which may require numerical methods for precise prediction.

Keywords

elastic collision, rim speed, string dynamics, rotational motion