To determine if the force $P$ is sufficient to keep the mass in equilibrium on the inclined plane, we need to analyze the forces acting on the mass and see if they balance out.

Given Data:

• Mass: $m = 5 \, \text{lbm}$
• Incline angle: $\theta = 15^\circ$
• Force $P = 2.3 \, \text{lbf}$ applied at an angle of $10^\circ$ from the incline.
• Acceleration due to gravity $g = 32.2 \, \text{ft/s}^2$ (standard in imperial units).

Note: In imperial units, mass $m$ in $\text{lbm}$ needs to be multiplied by $g$ to get the weight $W$ in pounds-force (lbf). So:

$W = m \times g = 5 \, \text{lbm} \times 32.2 \, \text{ft/s}^2 = 5 \, \text{lbf}$

This weight acts directly downward.

Forces acting on the mass:

1. Weight $W$: The weight of the object is split into two components:

   • Component parallel to the incline: $W_{\parallel} = W \sin(\theta)$
   • Component perpendicular to the incline: $W_{\perp} = W \cos(\theta)$

2. Applied Force $P$: This force is also split into components:

   • Component parallel to the incline: $P_{\parallel} = P \cos(10^\circ)$
   • Component perpendicular to the incline: $P_{\perp} = P \sin(10^\circ)$

Equilibrium Condition:

To achieve equilibrium, the sum of the forces parallel to the incline must be zero.

$P_{\parallel} = W_{\parallel}$

If $P_{\parallel}$ is less than $W_{\parallel}$, the mass will slide down. If it's greater, the mass will slide up.

Let’s calculate the values:

• $W_{\parallel} = W \sin(15^\circ)$
• $P_{\parallel} = P \cos(10^\circ)$

Let me compute these values.The parallel components of the forces are:

• The weight component parallel to the incline: $W_{\parallel} = 1.29 \, \text{lbf}$
• The applied force component parallel to the incline: $P_{\parallel} = 2.27 \, \text{lbf}$

Conclusion:

Since the applied force $P_{\parallel}$ (2.27 lbf) is greater than the weight component $W_{\parallel}$ (1.29 lbf), the applied force will push the mass up the incline, meaning the mass will not remain in equilibrium.

Do you need more details or have any questions?

Here are 5 related questions:

1. How would adding friction to the incline change the result?
2. What would happen if the angle of the incline were steeper?
3. How do we calculate the normal force acting on the object?
4. How does increasing the applied force angle affect the outcome?
5. What effect does the mass of the object have on the equilibrium?

Tip: When solving force equilibrium problems, always break forces into components along the incline to simplify calculations.