So here is a simple problem and here is a diagram that I find found online. Why is the frictional force pointing downwards. I mean I get the correct answer if I follow this diagram and math checks out but intuitively. I thought the frictional force should be acting against the person falling off the bed due to downward gravity so I thought, frictional force should be pointing upwards NOT downwards. Can someone explain why? The questions asks for the smallest angle at which the person will begin to slip off. correct answer is provided in the bottom for your reference
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Suppose there is no rotational motion. Then the frictional force must point upwards along the plane to keep the person from falling.
Now consider the situation when there is no friction and angular speed of rotation is very high. In the rotating frame, the centrifugal force will point outwards and its component along the plane will be higher (when angular speed is high enough) than the component of gravity downward along the plane. (You could do the whole calculation from inertial frame without invoking pseudo forces, but the situation be harder to visualize). If you turn on friction at this moment, it will point downwards in order to resist the upward motion of the man.
So, for low angular velocities, friction will point upward, and gradually to zero at a certain angular velocity (you can show that this angular speed is given by $\omega^2 = g \tan \theta /R$, where $R$ is the radius of the motion of the center of mass), and for higher angular velocities friction will point downward.