Let's solve the given equations step by step.

The equations provided are:

1. $15.0 \, \text{kg} \cdot a_x = 150 \, \text{N} \cdot \cos(30^\circ) + N \cdot \cos(90^\circ) + 15.0 \, \text{kg} \cdot 19.8 \, \text{N/kg} \cdot \cos(90^\circ)$
2. $15.0 \, \text{kg} \cdot 0 = 150 \, \text{N} \cdot \sin(30^\circ) + N \cdot \sin(90^\circ) - 15.0 \, \text{kg} \cdot 19.8 \, \text{N/kg} \cdot \sin(90^\circ)$

Step 1: Simplify the equations

Equation 1: Horizontal Component

First, note that:

• $\cos(30^\circ) = \frac{\sqrt{3}}{2}$
• $\cos(90^\circ) = 0$
• $\cos(90^\circ) = 0$

Thus, equation 1 simplifies to: $15.0 \, a_x = 150 \cdot \frac{\sqrt{3}}{2}$

Calculating $150 \cdot \frac{\sqrt{3}}{2}$: $150 \cdot \frac{\sqrt{3}}{2} \approx 150 \cdot 0.866 \approx 129.9 \, \text{N}$

Therefore, we have: $15.0 \, a_x = 129.9 \implies a_x = \frac{129.9}{15.0} \approx 8.66 \, \text{m/s}^2$

Equation 2: Vertical Component

Now simplify equation 2:

• $\sin(30^\circ) = \frac{1}{2}$
• $\sin(90^\circ) = 1$

Thus, equation 2 simplifies to: $0 = 150 \cdot \frac{1}{2} + N - 15.0 \cdot 19.8$ $0 = 75 + N - 297$ $N = 297 - 75 = 222 \, \text{N}$

Summary of Results

• Acceleration in the x-direction: $a_x \approx 8.66 \, \text{m/s}^2$
• Normal Force $N$: $N = 222 \, \text{N}$

Step 2: Force Diagram

1. Weight Force ($W$): Acts downwards, $W = 15.0 \, \text{kg} \cdot 19.8 \, \text{N/kg} = 297 \, \text{N}$.
2. Normal Force ($N$): Acts upwards, $N = 222 \, \text{N}$.
3. Applied Force ($F_a$): $150 \, \text{N}$ at $30^\circ$ to the horizontal.
4. Net Force: Resultant of horizontal forces providing $a_x$.

Invented Problem

Problem Scenario: A box weighing 15.0 kg is on a flat surface. An external force of 150 N is applied at an angle of 30 degrees from the horizontal. Determine the box's acceleration and the normal force acting on it, considering the gravitational force.

This scenario uses the same physics principles represented in the equations.

Questions for Further Exploration

1. How does the angle of the applied force affect the acceleration of the box?
2. What would happen to the normal force if the box were on an incline?
3. How would the situation change if the mass of the box were increased?
4. Can you calculate the frictional force acting on the box if the coefficient of friction is known?
5. What are the implications of the net force being zero in terms of the box's motion?

Tip

When working with forces, always draw a free-body diagram to visualize all forces acting on the object; it aids in understanding the problem better. Would you like more details on any part of this solution?