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Falling Rod on Smooth Floor

We have a uniform rod of length 2l balanced vertically on the floor, and then allowed to fall. I need the angular velocity and angular acceleration when the rod makes an angle (theta) WITH THE VERTICAL. In this situation, the floor is smooth and slipping occurs. I need to solve this using both the conservation law AND equations of motion.

I have attached my attempt at a solution using the energy method, which I understand is NOT correct, since the base of the ladder can slip. However, I do not understand why it is wrong. I apparently need an additional term for horizontal motion at the point of contact (the entire horizontal coordinate would be something like lsin(theta) + x, where x is not constant). If you have no argument with my energy solution, I will pay full credits for the solution via only the equations of motion. By equations of motion, I mean specifically using the translational (ma_x and ma_y) and rotational (I*alpha) equations.

Assume the initial angle between the vertical and the rod is zero, not small. The floor is completely frictionless.

This problem was previously posted and not solved according to the above specifications. If you can solve this problem as I've requested, let me know. I do not want to lose more credits to incorrect assumptions or incomplete solutions. The solution must be solved using the equations of motion. I will pay more for both solutions.


Solution Preview

This problem need to be solved by energy method. BY equation of motion, somewhere you need energy law equations. (see solution)

As surface is smooth (frictionless), there will be no force in horizontal direction. Hence, Centre of mass (C.M.) of the the rod will not have any horizontal acceleration, velocity and displacement. Due to gravity, it will fall vertically downward.

As, one of the ends of the rod is in-contact with the horizontal surface, that end of the rod will slide on the surface, & because of that slowly rod will have increasing angle with the vertical & finally will hit the ground in horizontal position.

Note -- C.M. will have only vertical ...

Solution Summary

A falling rod on smooth floors are examined.