Do rigid bodies experience work greater than or equal to that of point particles?

Question

Consider the following two scenarios in outer space:

1. A yo-yo whose string is pulled by a constant force.
2. The same yo-yo as before although this time, the string is fully unwound and attached to the center of the spool-holder (so that the string does not torque the yo-yo as it is pulled---i.e., 0 moment arm). The same force is then applied to the string.

Take the tension $T$ to point along the $x$ direction. Then, in each scenario, \begin{equation*} dW_x = \left(\sum F_x\right)\,dx = T\,dx = dK_\text{cm} \end{equation*} So the final velocity of the center of mass should be identical. However, in the first case the tension should also induce rotation and hence give the spool more energy the further from the center of mass the string is attached.

My question is akin to this one and while knzhou's response is convincing, I still don't understand what's going on. At face-value this still feels like a violation of energy conservation and I'm not sure how to explain why it isn't... The same force can give an object more or less energy depending on where it is applied? Is that true?

Answer 1

In case 1 the $dx$ is larger than the $dx$ in case 2. That is the difference between the cases that leads to the additional energy in case 1. The force is the same and the acceleration of the center of mass is the same, but the distance that the force is applied is larger in the first case. The linear KE is the same in the two cases and the additional work goes into the rotational KE.