Force not acting on the center of mass

Question

There are a few questions here which are almost the same. But I still had a doubt I read that a force applied anywhere on a rigid body produces the same acceleration on the center of mass. How is that possible?

If I apply a force on the center of mass it just accelerates but if I apply a force off the center of mass it accelerates linearly and rotates. Since it rotates some of the force goes into rotating the body so the linear acceleration cannot be the same? Isn't it?

And is there a mathematical proof that a force applied any where on the rigid body produces the same acceleration? There was a proof in one the answers which used the action of force on discrete particles of the body. Somebody pls give a better proof

Answer 1

This is a result of Newton's 2nd law. Force is the time derivative of linear momentum. And momentum of a collection of particles is defined as $${\bf p} = \sum_i m_i {\bf v}_i = \left( \sum_i m_i \right) {\bf v}_C $$ where $m_i$ is the individual mass, ${\bf v}_i$ the individual velocity and ${\bf v}_C$ the velocity of the center of mass. By taking the derivative of the above you get the relationship between force ${\bf F}$ and center of mass acceleration $\dot{{\bf v}}_C$

$${\bf F} = \left( \sum_i m_i \right) \dot{{\bf v}}_C $$

The center of mass location ${\bf r}_C$ is defined by

$$ \sum_i m_i {\bf r}_i = \left( \sum_i m_i \right) {\bf r}_C $$

and by direct differentiation of the above, you get

$$ \sum_i m_i {\bf v}_i = \left( \sum_i m_i \right) {\bf v}_C $$

_{where ${\bf v}_i = \dot{{\bf r}}_i$ and ${\bf v}_C = \dot{{\bf r}}_C$}

So what happens when a force is applied away from the center of mass?

The point where the force is applied will accelerate at least as much as the center of mass. In general, it will accelerate more due to the rotation. The force will feel a reduced mass given by the relationship

$$ m_{eff} = \left( \frac{1}{m} + \frac{c^2}{I} \right)^{-1} $$ where $m$ is the mass, $I$ is the mass moment of inertia about the axis of rotation and $c$ is the moment arm of the force as seem by the center of mass.