# Can single force acting on a rigid body lead to rotational motion?

Can a single force acting on a rigid body be the cause of pure rotational motion of the body? Suppose a single force acts on a disc in space at rest at a place other than its COM would the disc show both translational and rotational motion?

It is always true that:

$$F = \frac{dp}{dt} \tag{1}$$

and:

$$\tau = \frac{dL}{dt} \tag{2}$$

where $$F$$ and $$\tau$$ are the (net) force and the torque, and $$p$$ and $$L$$ are the linear and angular momenta. So if you apply a force it must change the linear momentum in accordance with equation (1) i.e. you cannot have pure rotational motion.

Can a single force acting on a rigid body be the cause of pure rotational motion of the body?

Yes. Assume that the body starts with some linear momentum $$\vec p$$ but no rotational motion. And suppose that we apply a force $$\vec F =-\vec p/t$$ for a time $$t$$. Then the body will end with no translational motion and if $$\vec F$$ is applied off center then it will have some rotational motion.

Can a single force acting on a rigid body be the cause of pure rotational motion of the body?

You can only produce pure rotation with a couple. A couple consists of two equal and opposite parallel forces.

A single force whose line of action is not through the center of mass will produce rotation plus translation.

A single force whose line of action is through the center of mass will produce pure translation.

See the figures below.

Hope this helps. 