# Motion of a rigid body on application of constant force perpendicular to a point which is not the center of mass?

Consider a rod which has a uniform mass distribution which is free to move in space. Now, it is known that if an impulse (perpendicular to rod) is applied which does not pass through the center of mass, it will undergo pure rotational motion (with translation).

Now consider a situation where the force is applied constantly, and it remains perpendicular to the point of application throughout the situation even as the body undergoes motion.

What would be the motion of the body in such a case? How could this motion be described mathematically?

• Now, it is known that if an impulse (perpendicular to rod) is applied which does not pass through the center of mass, it will undergo pure rotational motion. No. It will undergo rotational and translational motion unless it is attached to something preventing it from translating Commented Feb 23, 2021 at 1:06
• Yes, you are correct. I had earlier thought that since linear motion would not be accelerated, it would be appropriate to call it pure rotation, but perhaps I am wrong. I have edited the question.
– Dodo
Commented Feb 23, 2021 at 6:24

If the force is of constant magnitude, so is the torque applied about the center of mass. So the rotational motion is easy to calculate, as fixed rotational acceleration of $$\alpha = \frac{F d}{I}$$ is applied.

The orientation is thus

$$\theta(t) = \tfrac{1}{2} \alpha t^2$$

But the direction of the force changes with time and so the acceleration of the center of mass

$$\vec{a} = \frac{\vec{F}}{m}$$

As there is planar motion here, the force vector is

$$\vec{F}(t) = \pmatrix{ F \cos \theta(t) \\ -F \sin \theta(t) }$$

Unfortunately the integral $$\vec{v} = \tfrac{1}{m} \int \vec{F}(t)\,{\rm d}t$$ does not an analytical solution. But it is easily evaluated numerically, or experimentally.

• Is there any way to graph this result? Would it require to be done numerically?
– Dodo
Commented Feb 23, 2021 at 6:30
• You have to graph something of the form $\int \sin( x^2 )\,{\rm d}x$ and yes it is going to be numerical as I stated in the answer. Commented Feb 23, 2021 at 20:46