In order to obtain the equations of the motion of a rigid body, I need the applied torque about the center of mass $T_\mathrm{c}$. However, I have a torque that is applied off center of mass at the point $r_\mathrm{s}$ of the rigid body and I want to replace it with a torque about the center of mass $T_c$ and a force.
Suppose the rigid body is composed of $p$ particles, then
$$
T_\mathrm{s} = \sum_{i=1}^p (r_i – r_\mathrm{s}) \times F_i = \sum_{i=1}^p (r_i – r_\mathrm{c}) \times F_i + (r_\mathrm{c} – r_\mathrm{s}) \times \sum_{i=1}^p F_i
$$
where $T_\mathrm{c}$ is the torque about the center of mass, $F_\mathrm{r}$ is the resultant of forces and $r_\mathrm{s} = r_\mathrm{c} + d_\mathrm{s}$. How can I determine $T_\mathrm{c}$ and $F_\mathrm{r}$? There are multiple solutions, I think, are all equivalent? For instance one can take $F_\mathrm{r} = 0$ which will make the c.o.m to stand still, while under $T_\mathrm{s}$ it is surely moving.
Best Answer
A pure torque does not have a point of application. It is shared among the entire rigid body. Only torque as as result of a force at a distance needs specification of the point of measurement.
For the equations of motion you need the net torque about the center of mass $\vec{T}_C$. If this torque is a result of an applied pure torque $\vec{\tau}$ and a force $\vec{F}$ located at a point A specified by $\vec{r}_{A/C}$ relative to the center of mass then $$\boxed{\vec{T}_C = \vec{\tau} + \vec{r}_{A/C} \times \vec{F}}$$
To see the equations of motion expressed on a point different from the center of mass read this answer about Derivation of Newton-Euler equations of motion not at the center of mass.
In this answer $\vec{T}_A$ is the torque at a point not at the center of mass.