Question

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 r=0.5m 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 g=9.8m/s2.

Answer 1

The moment when the ingredients do not stick to the wall, this means that there is no force on the ingredients by the wall. Therefore, the only force acting on the ingredients is simply due to the weight of the ingredients due to gravity g:

F = mg                         ---> 1

where m = mass of ingredients             

This force also provides centripetal force. We know that the formula for centripetal force is given as:

F = mv^2/r                    ---> 2

where v = speed of rotation of the drum and r is the redius

Therefore equating 1 and 2:
mv^2/r = mg 
Dividing by m 
v^2/r = g 
v^2 = gr 
v = sqrt(g r) 

v = sqrt(9.8 * 0.5)

v = 2.21 m/s

If the problem ask for the angular velocity then we simply have to divide by the radius:

w = angular velocity

w = v/r

w = 2.21/0.5

w = 4.43 rad/s