homework and exercises - Force applied off center on an object

Assume there is a rigid body in deep space with mass $m$ and moment of inertia $I$. A force that varies with time, $F(t)$, is applied to the body off-center at a distance $r$ from its center of mass. How do I calculate the instantaneous acceleration, rotational acceleration, and trajectory of this object, assuming it starts from rest?

Answer

If the position of the center of mass is $\vec{r}_C$ and the location of the force application $\vec{r}_A$ then the Euler-Newton equations of motion for rigid body are:

$$\vec{F} = m\,\vec{a}_C \\ (\vec{r}_A-\vec{r}_C)\times \vec{F} = I_C \vec{\alpha} + \vec{\omega}\times I_C \vec{\omega}$$

with c.g. velocity $\vec{v}_C = \dot{\vec{r}_C}$, c.m. acceleration $\vec{a}_C = \ddot{\vec{r}_C}$, $I_C$ the moment of inertia tensor about the c.m.

In 2D when $(x,y)$ is the location of the c.m. Point C this becomes

$$\begin{vmatrix} F_x \\ F_y \\ 0 \end{vmatrix} = m \begin{vmatrix} \ddot{x} \\ \ddot{y} \\ 0 \end{vmatrix} \\ \begin{vmatrix} c_x \cos\theta \\ c_y \sin\theta \\ 0 \end{vmatrix} \times \begin{vmatrix} F_x \\ F_y \\ 0 \end{vmatrix} = \begin{vmatrix} I_x & &\\& I_y & \\ & & I_z \end{vmatrix} \begin{vmatrix} 0 \\ 0 \\ \ddot{\theta} \end{vmatrix} + \begin{vmatrix} 0 \\ 0 \\ 0 \end{vmatrix}$$

where $(c_x,c_y)$ is the position of point A from the c.m. when the body orientation is $\theta=0$ (initially).

By component then the equations are

$$\ddot{x} = F_x/m \\ \ddot{y} = F_y/m \\ \ddot{\theta} = \frac{-c_y \sin\theta F_x + c_x \cos\theta F_y}{I}$$

If the force is rotating with the body, and initially located at $(cx,0)$ pointing in the +y direction then

$$\ddot{\theta} = \frac{c_x F_y}{I}$$