This problem involves a system with two masses connected by a pulley. One mass ($m_1 = 3.0 \, \text{kg}$) is on a horizontal frictionless surface, and the other mass ($m_2 = 1.0 \, \text{kg}$) is on an inclined plane tilted at an angle $\theta = 42^\circ$. The goal is to find the acceleration of the system when a force $F = 18.5 \, \text{N}$ is applied to the box on the incline. Here's the step-by-step approach:

Step 1: Free-body diagrams

• For the UPS box on the floor ($m_1 = 3.0 \, \text{kg}$):
  This mass is on a horizontal frictionless surface. The only force acting on it in the direction of motion is the tension $T$ in the cord connecting it to the other box.

  Newton's Second Law for this mass is: $m_1 a = T$ where $a$ is the acceleration of the system.

• For the FedEx box on the incline ($m_2 = 1.0 \, \text{kg}$):
  The forces acting on this box are:

  1. The applied force $F = 18.5 \, \text{N}$ directed up the incline.
  2. The gravitational force $m_2 g$ (which can be split into components parallel and perpendicular to the incline). The component of gravity acting down the incline is $m_2 g \sin \theta$.
  3. The tension $T$ in the cord pulling the box down the incline.

  Newton's Second Law for this mass in the direction along the incline is: $m_2 a = F - T - m_2 g \sin \theta$

Step 2: Set up the system of equations

We now have two equations:

1. $m_1 a = T$
2. $m_2 a = F - T - m_2 g \sin \theta$

We can substitute $T$ from the first equation into the second:

$m_2 a = F - m_1 a - m_2 g \sin \theta$

Step 3: Solve for $a$

Rearranging the equation for $a$:

$a(m_1 + m_2) = F - m_2 g \sin \theta$

$a = \frac{F - m_2 g \sin \theta}{m_1 + m_2}$

Step 4: Plug in the values

Given:

• $m_1 = 3.0 \, \text{kg}$
• $m_2 = 1.0 \, \text{kg}$
• $\theta = 42^\circ$
• $F = 18.5 \, \text{N}$
• $g = 9.8 \, \text{m/s}^2$

We calculate the acceleration $a$:

$a = \frac{18.5 - (1.0)(9.8) \sin 42^\circ}{3.0 + 1.0}$

Let's calculate the final answer.The acceleration of the system is approximately $2.99 \, \text{m/s}^2$.

Would you like further details on how this was derived, or any clarification? Here are five related questions you can explore:

1. How would the problem change if there was friction on the incline?
2. What if the angle of the incline was different, say 30° or 60°?
3. How does the tension in the cord compare to the force applied?
4. What would happen if the applied force was less than the force needed to overcome gravity on the incline?
5. How would the system behave if both masses were placed on the horizontal surface?

Tip: In problems involving inclined planes, always break forces into parallel and perpendicular components to simplify the analysis.