To find the cart's acceleration when rolling friction is neglected, we will only consider the forces acting on the cart due to gravity along the incline.

Step-by-step solution:

1. Identify forces:

   • The gravitational force acting on the cart is $F_g = m \cdot g$, where:
     • $m = 50 \, \text{kg}$ is the mass of the cart,
     • $g = 9.81 \, \text{m/s}^2$ is the acceleration due to gravity.

2. Component of gravitational force along the incline: The force acting along the incline is the component of the gravitational force parallel to the surface: $F_{\text{parallel}} = F_g \cdot \sin(\theta)$ where $\theta = 15^\circ$ is the angle of the incline.

   Therefore: $F_{\text{parallel}} = m \cdot g \cdot \sin(15^\circ)$

3. Newton's second law: Since the only force acting on the cart along the incline is $F_{\text{parallel}}$, the net force $F_{\text{net}}$ is equal to $F_{\text{parallel}}$.

   Using Newton’s second law, $F_{\text{net}} = m \cdot a$, where $a$ is the acceleration of the cart. Thus: $m \cdot a = m \cdot g \cdot \sin(15^\circ)$ Simplifying by canceling the mass $m$ from both sides: $a = g \cdot \sin(15^\circ)$

4. Calculate the acceleration: $a = 9.81 \, \text{m/s}^2 \cdot \sin(15^\circ)$ Using $\sin(15^\circ) \approx 0.2588$, we get: $a \approx 9.81 \, \text{m/s}^2 \cdot 0.2588$ $a \approx 2.54 \, \text{m/s}^2$

Final answer:

The acceleration of the cart, neglecting rolling friction, is approximately $2.54 \, \text{m/s}^2$.

Would you like more details on how rolling friction would affect the result or how to include it? Here are some related questions to explore:

1. How does rolling friction impact acceleration?
2. How can we calculate acceleration including rolling friction?
3. What forces contribute to the motion of a cart on an inclined plane?
4. How is the normal force calculated for an inclined plane?
5. What is the difference between sliding friction and rolling friction?

Tip: When dealing with inclined planes, always break down forces into components parallel and perpendicular to the surface to simplify your calculations.