# Will a disk (or any rigid body) without fixed axis on a frictionless surface rotate?

If there is a disk (or any non-particle object) placed rest on a frictionless table, and a force is applied on its rim. Will it rotate and have translational motion, or will it only have translational motion?

One of my teacher told me that without friction, the object will not rotate but another teacher told me that it will have both motion.

• Apr 22 at 14:00

If the axis of the force does not pass through the center of gravity, then it will exert a torque and cause a rotational movement.

Consider disc of uniform mass density for simplicity.

Factors on which rotation will depend on frictionless surface:

• Torque about Centre of Mass [COM]
• Direction of Force

There can be $$3$$ possible cases to apply the Force: Tangentially, Normally & Along a chord of disc

Observe the situations shown in the image carefully.

Case I: Tangentially

Considering there the rim has enough friction, when a force is applied to the rim, there is a torque acting about the COM. This torque gives the disc an angular acceleration to rotate.

Case II: Normally

When force is applied normally, which means the line of force passes through the COM, the torque about COM is zero. Therefore, we conclude that no angular acceleration hence, no rotation.

Case III: Along a chord (other than a diameter) of disc

There will be rotation in this case because the line of force do not pass through the COM producing a non-zero torque about the COM. Consequently, producing angular acceleration.

In all cases translation will happen because: $$F = m.a_{com}$$ where,

$$a_{com}$$ - linear acceleration

• What will be the translational acceleration in the 3 cases. IN case 2, it will be F/m. Would it be the same in case of 1 and 3 ? Especially what will be the translational acceleration in case 1 ? will it be 0 ? Will it just rotate in place ? Apr 22 at 18:00
• As F is the only force in horizontal direction then it should have F/m as translational acceleration. Apr 23 at 1:04

One of my teacher told me that without friction, the object will not rotate but another teacher told me that it will have both motion.

Maybe the first teacher was talking about the friction between the rim and whatever is applying the tangential force to its rim. If there is no friction on its rim , then the tangential force will just "slide" along the rim and not rotate it