A wheeled inclined plane with an object: Newton's equation of motion

Question:
Consider an inclined cart with motorized wheels as shown. Assume that the coefficients of static and kinetic frictions are equal on the surface of the cart.
(1) After an object is put on the cart gently, it starts falling with acceleration of $a$. The cart is fixed, so it does not move at all. Suppose the inclined angle is the angle where the object just starts falling. Find the coefficient of static friction.
(2) The cart moves ahead and begins accelerated. The, the object becomes stopped falling. Find the acceleration of the cart.

Answer:
(1) From the diagram, we can set up two equations of motion for the moving axis and the axis perpendicular to it.
\begin{eqnarray}
mg\sin\theta -\mu N &=& ma  \\
N - mg\cos\theta &=& 0
\end{eqnarray}
where $\mu$, $N$, and $\mu N$ are the coefficient of static friction, the normal force and the frictional force, $f$. From (2), we obtain the normal force, $N = mg\cos\theta$. Then, equation (1) becomes
\begin{equation}
mg\sin\theta - \mu mg\cos\theta = ma
\end{equation}
All $m$'s are cancelled out. Solve for $\mu$.
\begin{eqnarray}
& & g\sin\theta - \mu g\cos\theta = a  \\
& & \rightarrow \mu g\cos\theta = g\sin\theta - a  \\
& & \rightarrow \mu = \frac{g\sin\theta - a}{g\cos\theta}
\end{eqnarray}

(2) As shown in the figure, the acceleration of the cart works against the gravitational falling force. The equation of motion in the moving direction becomes
\begin{equation}
mg\sin\theta - \mu N -ma\cos\theta = 0
\end{equation}
The left hand side is the net forces in that direction. The right hand side is zero because there is no motion. Now, set up the other equation of motion for the axis perpendicular to the moving direction.
\begin{equation}
N - ma\sin\theta - mg\cos\theta = 0
\end{equation}
From (8), solve for $N$.
\begin{equation}
N  = ma\sin\theta + mg\cos\theta
\end{equation}
Plug this into (7).
\begin{eqnarray}
& &mg\sin\theta - \mu (ma\sin\theta + mg\cos\theta) -ma\cos\theta = 0  \\
& & \rightarrow g(\sin\theta - \mu\cos\theta) - a(\mu\sin\theta + \cos\theta) = 0  \\
& & \rightarrow a = \frac{\sin\theta - \mu\cos\theta}{\mu\sin\theta + \cos\theta}g
\end{eqnarray}

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