So my cat has a favorite chair he claims as his own, it's out swivel computer chair. He likes to get a running start to go jump onto it as you're walking towards it. His running starts allows for him to get a high acceleration which transfers to centripetal acceleration onto the spinning chair. In order for him to get a really good spin he needs a high velocity since he's already a big cat with alot of mass. With such a high acceleration and velocity his landing on the chair transfers some of the landing force making the chair move. But, he hasnt perfected this yet because if he was to perfect it then the centripetal acceleration would hold him to the back of the chair so he wouldnt move. But he needs his nails to hold on so he wont fall off so he needs some work to do.
The only time his claws would be needed would be for an abrupt stop from his spinning. THe path he would take if this happened would be tangent to his path. Not only would it be tangent but it would also be a projectile path and take him a little ways away due to the force of gravity acting on him. But his nails prevent this by keeping him grounded onto the chair and absorb the force of the quick stop.