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If we have a general object and we have multiple forces acting on it at several points. enter image description here

Assuming we know the center of mass, mass and moment of inertia of the object:

  • Will the object rotate only around the center of mass or do i need to consider the torques around other points?
  • can the object rotate and translate at the same time? I mean can we just sum the forces at the center and the torques around it; and then calculate the linear and angular movement separately.
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The instantenous center of rotation of a planar body is generally not the center of mass, unless the net forces applied all cancel out. In 2D, the general motion is a rotation about a specific point. If the point is at infinity then the body is said to purely translate.

The rules of motion lead us the following equivalent statments that are valid for both 2D and 3D bodies:

  1. A pure force thorugh the center of gravity (with no net torque) will purely translate a rigid body (any point on the body).
  2. A pure torque any point on the body (with no net force) will purely rotate a rigid body about its center of gravity

The equations of motion in 3D are described using the following

  1. The net force vector acting on a rigid body equals the derivative of linear momentum, or more commonly mass times the acceleration vector of the center of mass. $$ \sum_i (\vec{F}_i) = \frac{{\rm d}(m \vec{v}_{cm})}{{\rm d}t}= m \,\vec{a}_{cm}$$
  2. The net torque vector about the center of mass acting on a rigid body equals the derivative of angular momentum $$\sum_i (\vec{\tau}_i+\vec{r}_i\times \vec{F}_i) = \frac{{\rm d}(I_{cm} \vec{\omega})}{{\rm d}t} = I_{cm} \vec{\alpha} + \vec{\omega} \times I_{cm} \vec{\omega}$$

($\times$ is the vector cross product)

  • So yes, torques need to be considered including the torque arms of the forces (the $\vec{r}_i\times \vec{F}_i$ parts)
  • So yes, the general motion in a plane is translation and rotation about a point, and in 3D rotation about an axis with a parallel translation along the axis (like a flying football or a bullet with a screw type motion).
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  • $\begingroup$ Could you please help me understand the 2nd: "A pure torque any point on the body (with no net force) will purely rotate a rigid body about its center of gravity". If I hit tangentially sphere, why does it not only rotates (I assume pure torque), but moves from me as well? How to identify net force here which forces it to move away from me? $\endgroup$
    – Damir Tenishev
    Dec 15, 2022 at 17:42
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    $\begingroup$ @DamirTenishev - a single tangential force results in both net force and net torque not being zero. The proof is $F_{\rm net} = m a_C$ so when net force is zero (pure torque) the center of mass does not accelerate. So the motion can be described as a rotation about the center of mass, since any co-moving reference frame would also be an inertial frame without loss of generality. For a mathematical proof look at any book that derives the equations of motion from a system of particles the moves together and Newton's 2nd law. $\endgroup$
    – John Alexiou
    Dec 16, 2022 at 14:07
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    $\begingroup$ Thank you, it helps. I will read these chapters again. It seems that you are experienced in the subject, could you please consider my question Rotation of a system of bodies on an axis and the root cause for it Rotation of the systems of two bodies connected by a motor? Maybe you can help me to figure out what is a missing part in my understanding of rotation. $\endgroup$
    – Damir Tenishev
    Dec 16, 2022 at 14:33
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    $\begingroup$ @DamirTenishev - this NASA paper might answer some questions for you. $\endgroup$
    – John Alexiou
    Dec 16, 2022 at 21:44
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    $\begingroup$ John, thank you so much both for the reference to the paper and detailed explanation for my question mentioned by me above. The paper is exactly for my case, I will read it cover to cover. Sorry for the delay with the response on your answer on the topics of the sticks, I will need days to understand it. So, I will avoid hasty replies there. One thing makes me puzzled. Namely, I just solved the task with a couple of formulas and my simulation looks realistic and stable; still puzzled, why. Maybe I miss some special cases, looking into it in different scenarios. $\endgroup$
    – Damir Tenishev
    Dec 17, 2022 at 22:08
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When dealing with torques, we need to specify the origin. If we choose the origin to be at the center of mass, then if the torques don't entirely cancel, there will be an angular acceleration of the object. If we choose the origin to be some other point, then the rotational motion is some combination of rotation about the center of mass and rotation of the center of mass itself about the origin. If the net force on the object is not zero, the center of mass will accelerate, and if the net torque about the center of mass in not zero, it will angularly accelerate.

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° To start with take the centre of mass as the reference point, because it would be important as well as simple to calculate the net torque about it. Now, you can take any arbitrary point on the mass, pivot or pin joint(rotating) it, and calculate the net torque about it. ° The body can, if having unbalanced torque rotate only if hinged/pivoted, hence it will only translate otherwise. :)

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  • $\begingroup$ If I get what you're saying, I don't think that's correct. :( If you are saying that a body can only rotate if its pivoted, then that isn't actually the case. A free body in space can be made to rotate if a net torque is applied to it. Sorry if I misunderstood your answer. $\endgroup$
    – Involute
    Apr 22, 2015 at 12:43

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