To calculate the minimum force $F_{\text{min}}$ that the spring-loaded arm must apply to hold the package on the ramp, we need to consider the forces acting on the package and the effect of static friction.

Step-by-Step Breakdown:

1. Forces Acting on the Package:

• Gravitational Force ($F_g$): This acts vertically downward and is given by: $F_g = m \cdot g = 7.75 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 76.0275 \, \text{N}$ where $m = 7.75 \, \text{kg}$ is the mass of the package and $g = 9.81 \, \text{m/s}^2$ is the acceleration due to gravity.

• Normal Force ($F_N$): This is the force perpendicular to the surface of the ramp, opposing the component of gravity normal to the ramp. The component of the gravitational force perpendicular to the ramp is: $F_N = F_g \cdot \cos(\theta) = 76.0275 \cdot \cos(41.5^\circ)$

• Frictional Force ($F_f$): The frictional force resists the motion of the package and is proportional to the normal force. The maximum static friction force is: $F_f = \mu_s \cdot F_N = 0.445 \cdot F_N$ where $\mu_s = 0.445$ is the coefficient of static friction.

• Parallel Component of Gravitational Force ($F_{\parallel}$): This is the component of the gravitational force acting down the ramp, trying to pull the package downwards: $F_{\parallel} = F_g \cdot \sin(\theta) = 76.0275 \cdot \sin(41.5^\circ)$

2. Force Balance Along the Ramp:

To hold the package stationary, the force applied by the arm ($F_{\text{min}}$) must counteract the net force along the ramp. The frictional force $F_f$ helps in resisting motion, so the force applied by the arm must counteract the remaining part of the parallel gravitational force.

The balance of forces along the ramp direction is: $F_{\text{min}} + F_f = F_{\parallel}$ Thus, the minimum force the spring-loaded arm must apply is: $F_{\text{min}} = F_{\parallel} - F_f$

3. Substitute the Values:

Now, let's calculate each term.

1. Normal force $F_N$: $F_N = 76.0275 \cdot \cos(41.5^\circ)$

2. Frictional force $F_f$: $F_f = 0.445 \cdot F_N$

3. Parallel component of gravitational force $F_{\parallel}$: $F_{\parallel} = 76.0275 \cdot \sin(41.5^\circ)$

I'll compute these now.The minimum force $F_{\text{min}}$ that the spring-loaded arm must apply to hold the package on the ramp is approximately 25.04 N.

Would you like any further details or have any questions about this calculation?

Here are five related questions to explore further:

1. How would the minimum force change if the angle of the ramp was increased?
2. What would happen if the coefficient of static friction was reduced?
3. How does the weight of the package affect the required force?
4. How can we determine the maximum force the arm can apply before moving the package up the ramp?
5. How does kinetic friction differ from static friction in this scenario?

Tip: Always remember to consider both friction and the gravitational components when analyzing forces on an inclined plane.