- #1
Mr Davis 97
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Homework Statement
We have a concrete mixer, and if the drum spins too fast, the ingredients will stick to the wall of the drum rather than mix. Assume that the drum of the mixer has radius R = 0.5 m and that it is mounted with its axle horizontal. What is the fastest the drum can rotate without the ingredients sticking to the wall all of the time?
Homework Equations
Newton's 2nd law
The Attempt at a Solution
So I believe that I have solved this problem. We define that the ingredients stick to the wall when, at the very top of the motion, there is some normal force pushing the ingredients toward the center. Thus to find the maximum angular velocity, we just need to find the angular velocity at which there is only gravity acting at the top and no normal force.
Also, we are working in polar coordinates, so we define our reference frame such that towards the center of rotation is positive, and outward is negative.
Thus, (imagine that our analysis is on a single particle, at the very top of the rotation)
##F_g = m a_{radial}##
##mg = m (\ddot{r} - r \dot{\theta^2}_{max})##
##g = -r \dot{\theta^2}_{max}##
##\displaystyle \dot{\theta}_{max} = \sqrt{-\frac{g}{r}}##
This is the correct equation (I think), except there is a negative. Why is there a negative? Did I define my coordinate system wrong?