I am working on the following problem:
Note that 0.5*N is the force due to friction pictured in the FBD.
When I try to solve this, my intuition is to draw the friction force upwards rather than downwards. If the "cushion" wasn't spinning, then the guy would slide downwards, and the friction force would be upwards. Depending on how quickly he is spinning, the friction force COULD be upwards, but how do we know with this type of problem?
I tried solving the problem my way, and I do get a different answer.
Best Answer
You can decide on the direction of the frictional force in the following way.
When $\theta = 90^\circ$ the frictional force must be upwards to balance the weight of the man.
As you decrease the angle $\theta$ the upward frictional force must decrease because eventually there will be an angle of inclination where no frictional force is needed to keep the man rotating. This is sometimes called the "ideal" banking angle given by $\tan \theta_i = \dfrac {v^2}{rg}$.
Decreasing the angle further must mean that the frictional force now point down the slope.
Eventually that frictional force is just sufficient to keep the man from sliding up the slope. That is then the minimum angle of the slope required by the questioner.