- #1
PFuser1232
- 479
- 20
Homework Statement
In a concrete mixer, cement, gravel, and water are mixed by tumbling action in a slowly rotating drum. If the drum spins too fast the ingredients stick to the drum wall instead of mixing. Assume that the drum of a 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 the time? Assume ##g = 9.8 m/s^2##.
Homework Equations
##F_r = ma_r##
The Attempt at a Solution
If a particle of mass ##m## goes around in a vertical circle on the drum wall at angular speed ##\omega##, then in polar coordinates we have:
$$-(N + mg \sin{\theta}) = -mR \omega^2$$
$$N = m(R \omega^2 - g \sin{\theta})$$
Where ##\theta## is the angle the position vector of the particle makes with the horizontal and ##N## is the magnitude of the normal reaction from the drum wall.
At the top, if we wish to find the maximum value of ##\omega## beyond which ##N## is nonzero (sticking to the wall, as described in the question), we can set ##N = 0## and solve for ##\omega_{max}##:
$$\omega_{max} = \sqrt{\frac{g}{R}}$$
Is this correct?
Also, why do we choose the ##N = 0## for ##\theta = \frac{\pi}{2}##? Is it because if ##N = 0## for any other point, the ingredients wouldn't go around in a circle, but would instead be in free fall?