A small block is pressed against the inner wall of a vertical cylindrical drum of radius 4 m. The drum is initially rotating at a high speed about its vertical axis but is slowing down with a constant angular deceleration of 6 rad/s ^2 . The coefficient of static friction between the block and the wall is 0.26 . The block will begin to slip relative to the wall when the angular speed of the drum drops to what value?

Step-by-step solution

The normal force N provides the centripetal force for the block: N = m ^2 R Because the drum is decelerating, the block experiences a tangential acceleration a_t = R . Thus, the static friction must provide two perpendicular forces along the surface of the wall: 1. A vertical force to balance gravity: f_y = mg 2. A horizontal tangential force to provide the deceleration: f_t = m a_t = m R The total required static friction f is the vector sum of these two perpendicular components: f = f_y^2 + f_t^2 = (mg)^2 + (m R)