Calculate the acceleration of a system with a 10 kg mass on a 30-degree incline and a 15 kg hanging mass, assuming a frictionless system and pulley.

Mass on the inclined plane, $m_1 = 10 \text{ kg}$.
Angle of inclination, $\theta = 30^\circ$.
Hanging mass, $m_2 = 15 \text{ kg}$.
Acceleration due to gravity, $g = 9.8 \text{ m/s}^2$.
The plane is smooth, so there is no friction. The pulley is also smooth.

Assume the system accelerates such that $m_2$ moves downwards and $m_1$ moves up the incline. Let $T$ be the tension in the string and $a$ be the acceleration of the system.

For mass $m_1$ (on the inclined plane):
The forces acting along the incline are the tension $T$ (up the incline) and the component of gravity $m_1 g \sin\theta$ (down the incline).

For mass $m_2$ (hanging vertically):
The forces acting are gravity $m_2 g$ (downwards) and tension $T$ (upwards).

From equation (2), we can express $T$:

Substitute this expression for $T$ into equation (1):

Rearrange the terms to solve for $a$:

We know $\sin(30^\circ) = 0.5$.

The acceleration of the system is $3.92 \text{ m/s}^2$.