Consider the following system consisting of a box sliding down a plane. The coefficient of friction between the plane and the box is $\mu$. A pendulum is attached to the top of the box as shown.
The acceleration of the box+pendulum is $a=g(\sin\theta-\mu\cos\theta)$ I believe.
In a non-inertial frame attached to the box, the free-body diagram for the pendulum is
My goal is to find the angle $\phi$ from the equilibrium of these 3 forces. I have to pick x and y axes to decompose these forces. If I pick the x axis along the fictitious force and the y perpendicular to it I get (I think)
$$m_P a+T\sin\phi=m_P g\sin\theta$$
which I can then solve for $\phi$ :
Is this right?