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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?

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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.

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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.

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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.

enter image description here

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