Locating the Locus of a Falling Rod on a Smooth Plane

[Silverbackman]

A vertical rod of length 'l' rests on a smooth plane and starts falling over under the influence of a slight disturbance. What is the locus of 1) center of the rod 2) A point halfway between the center and the bottom 3) The top of the rod?

 

[OlderDan]

A vertical rod of length 'l' rests on a smooth plane and starts falling over under the influence of a slight disturbance. What is the locus of 1) center of the rod 2) A point halfway between the center and the bottom 3) The top of the rod?

"Smooth" means frictionless, so there can be no horizontal force applied to the rod. The motion of the center of mass of a rigid body is determined by the sum of forces acting on the body. How must the center of mass move? Where is the center of mass? What happens to the point in contact with the plane? What happens to the other points in question?

 

[thepatientmental]

1) The locus of the center of the rod can be described as a straight line perpendicular to the smooth plane, passing through the initial point of contact between the rod and the plane.

2) The locus of a point halfway between the center and the bottom of the rod can also be described as a straight line perpendicular to the smooth plane, passing through the midpoint of the initial point of contact and the bottom of the rod.

3) The locus of the top of the rod can be described as a parabolic curve, with the initial point of contact between the rod and the plane as the focus and the bottom of the rod as the directrix. As the rod falls, the top will trace out this parabolic curve.