Locating the Locus of a Falling Rod on a Smooth Plane

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The discussion explores the motion of a vertical rod of length 'l' as it falls on a smooth, frictionless plane after a slight disturbance. The locus of the center of the rod moves in a straight line perpendicular to the plane, originating from the initial contact point. A point halfway between the center and the bottom of the rod also follows a straight line perpendicular to the plane, positioned between the initial contact and the bottom. In contrast, the top of the rod traces a parabolic curve, with the initial contact point as the focus and the bottom of the rod as the directrix. This analysis highlights the distinct trajectories of various points on the rod during its fall.
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?
 
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Silverbackman said:
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?
 


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.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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