Trouble with rotational kinematics

Question

I'm having a bit of trouble with the following homework problem:

My thinking is there are only three forces acting on the laundry: the force due to centripetal acceleration, the force due to gravity, and the normal force, which is orthogonal to the plane tangent to the circle where the center of the laundry's mass touches the dryer. ( The question says the dryer's drum is smooth, so I'm taking that to mean we are ignoring the force due to friction between the drum and the laundry. )

So, I drew the following free-body diagram:

Next, I applied Newton's Second Law in the i-direction:

$$ F\ =\ m\ \frac{v^2}{R} $$ $$ -N\ cos\ (\ \theta\ )\ +\ \frac{m\ v^2cos\ (\ \theta\ )}{R}\ =\ 0 $$ $$ \frac{m\ v^2cos\ (\ \theta\ )}{R}\ =\ N\ cos\ (\ \theta\ ) $$ $$ m\ v^2cos\ (\ \theta\ )\ =\ N\ cos\ (\ \theta\ )\ R $$ $$ m\ v^2\ =\ N\ R $$ $$ \frac{m\ v^2}{R}\ =\ N $$

Then, I applied Newton's Second Law in the j-direction:

$$ N\ sin(\ \theta\ )\ -\ \frac{m\ v^2\ sin\ (\ \theta\ )}{R}\ -\ mg\ =\ 0 $$ $$ \frac{m\ v^2\ sin(\ \theta\ )}{R}\ -\ \frac{m\ v^2\ sin\ (\ \theta\ )}{R}\ -\ mg\ =\ 0 $$ $$ -\ mg\ =\ 0 $$

Which--is not entirely useful ( nor is it probably true ).

I've done this question a few times now and can't seem to spot my mistake--either in approach, algebra, or both. Any help in spotting my mistake in algebra, approach, or both would be greatly appreciated.

Answer 1

There are three forces acting which then change to two.
The frictional force will not contribute to the net force towards the centre of rotation which produces the centripetal acceleration as it is tangential to the drum.
The other two forces on the washing are the normal reaction and the weight.
Draw some free body diagrams as the washing moves from the bottom.
You need the condiion when the normal reaction becomes zero.

Answer 2

I guess that your free body diagram is not correct.

I came up with the equation $m\omega^2r=N+gcos\theta$.

Substitute $N=0$ and given radius, at the $\theta=70.5$