The shown 2kg collar fits loosely on the inclined rod which

The shown 2-kg collar fits loosely on the inclined rod which the coefficient of static friction is mu_s = 0.2 The rod is rotating around the vertical (z) axis with a constant angular speed. Determine the maximum constant speed the rod can have so that the collar does not slip up the rod.

Solution

Resolving vertical mg force into normal and tangential components,

N = mg*cos theta

Force along the incline = mg*sin theta

Centrifugal force acting horizontally = mrw^2

Resolving it along and normal to incline,

Centrifugal force along the incline = mrw^2*cos theta

Centrifugal force normal to incline = mrw^2*sin theta

Total normal force = mg*cos theta + mrw^2 *sin theta

Friction force Ff = 0.2*Normal = 0.2*(mg*cos theta + mrw^2 *sin theta)

When at the verge of upward movement:

mrw^2 *cos theta = mg*sin theta + 0.2*(mg*cos theta + mrw^2 *sin theta)

rw^2 *cos theta = g*sin theta + 0.2*(g*cos theta + rw^2 *sin theta)

Cos theta = 4/5, Sin theta = 3/5

r = 0.25*cos theta = 0.25*4/5 = 0.2

0.2*w^2 *4/5 = 9.81*3/5 + 0.2*(9.81*4/5 + 0.2*w^2 *3/5)

w = 7.4 rad/s