A light rod connects two masses M_1 and M_2 , and the system is...

Let $a$ be the common acceleration of the system and $T$ be the force exerted by the rod.

The equations of motion for the two masses along the incline are:

$M_1 g \sin\theta - \mu M_1 g \cos\theta + T = M_1 a$

$M_2 g \sin\theta - \mu M_2 g \cos\theta - T = M_2 a$

Adding the two equations, we get:

$(M_1 + M_2) g \sin\theta - \mu (M_1 + M_2) g \cos\theta = (M_1 + M_2) a$

$a = g(\sin\theta - \mu\cos\theta)$

Substituting the value of $a$ in the first equation:

$M_1 g \sin\theta - \mu M_1 g \cos\theta + T = M_1 g(\sin\theta - \mu\cos\theta)$

$T = 0$

Thus, the acceleration of the system is $g(\sin\theta - \mu\cos\theta)$ and the force exerted by the rod is zero.