One end of a uniform rod of mass m and length l is clamped. The rod lies on a smooth horizontal surface and rotates on it about the clamped end at a uniform angular velocity ω. The force exerted by the clamp on the rod has a horizontal component.

Given Information

• A uniform rod of mass m and length l is clamped at one end.
• The rod is rotating about the clamped end with an angular velocity ω.
• The rod lies on a smooth horizontal surface, implying there is no friction between the rod and the surface.

Horizontal Force on the Rod:

As the rod rotates, each small element of the rod experiences a centripetal force directed towards the axis of rotation. The center of mass of the rod, which is located at a distance l/2 from the clamped end, will also experience a centripetal force. This force will be directed horizontally, since the rod is rotating in a horizontal plane.

Horizontal Force Exerted by the Clamp:

The clamp needs to exert a force to keep the rod rotating and prevent it from flying off due to the centripetal force. The centripetal force acting on the center of mass of the rod is given by:

F_cm = m * (l/2) * ω²

This force is directed horizontally, and the clamp must exert an equal and opposite force to balance it.

So, the horizontal force exerted by the clamp is:

F_horizontal = 1/2 * m * ω² * l

Final Answer

1/2 mω²l