- #1

frozenguy

- 192

- 0

## Homework Statement

A uniform stick of mass m and length L, initially

upright on a frictionless horizontal surface, starts falling. The circle at the center of the

stick marks the center of mass. Derive an expression for the speed of the center of mass

as a function of y and θ if the stick falls as shown (with the center of mass moving

straight downward).

## Homework Equations

[tex]v=\frac{dy}{dt}[/tex]; [tex]\omega=\frac{d\theta}{dt}[/tex]

[tex]v_{cm}[/tex][tex]=r\omega[/tex]; [tex]I=\frac{1}{12}[/tex][tex]mL^{2}[/tex]

[tex]K_{rot}[/tex]=[tex]\frac{1}{2}[/tex][tex]I\omega^2[/tex]

[tex]K=\frac{1}{2}mv^2[/tex]

## The Attempt at a Solution

There are no non-conservative forces so [tex]E_{mech}[/tex] is conserved.

Therefore I figure: [tex]U_{i}+K_{i}=U_{f}+K_{f}[/tex]

So: [tex]mg\frac{1}{2}L=\frac{1}{2}mv^{2}_{cm}+\frac{1}{2}I\omega^2[/tex]

Then subed in [tex]v=\frac{dy}{dt}[/tex] and [tex]\omega=\frac{d\theta}{dt}[/tex] and

[tex]I=\frac{1}{12}[/tex]mL[tex]^{2}[/tex], canceled out the (1/2) and m and attempted to integrate the equation.

mg and L are all constants right? So I got [tex]0=2m\frac{dy}{dt}y+\frac{1}{12}2mL^2\frac{d\theta}{dt}\theta[/tex] which I don't think is right..