To solve this problem, let's analyze it step by step. The goal is to determine the magnitude and direction of the minimum force $P$ needed to pull a roller of mass $M$ over a smooth step.

Given Information:

• $a = 0.6 \, \text{m}$ (distance from the center of the roller to the point of force application)
• $b = 0.1 \, \text{m}$ (height of the step)
• $\theta_1 = 20^\circ$ (angle of force $Mg$ relative to the vertical)
• $M = 50 \, \text{kg}$ (mass of the roller)
• $g = 9.81 \, \text{m/s}^2$ (acceleration due to gravity)

We are tasked with finding:

1. The angle $\theta$ (direction of the force $P$)
2. The magnitude of the force $P$

Given answer:

• $\theta = 33.6^\circ$
• $P = 395 \, \text{N}$

Solution Outline:

1. Determine the weight $Mg$ of the roller: $Mg = 50 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 490.5 \, \text{N}$

2. Analyze the torque about point $B$: To ensure the roller just begins to lift over the step, the moment (or torque) due to the force $P$ around point $B$ should balance the moment due to the weight $Mg$ around the same point.

   Let $\phi$ be the angle between $P$ and the line $(a - b)$. From the diagram, $a \sin \phi$ component is perpendicular to $P$, creating the lever arm for torque calculation.

3. Set up the equations for static equilibrium: Use the sum of moments around point $B$ to solve for $P$.

Let me know if you would like a detailed step-by-step calculation for each of these stages to solve for $P$ and $\theta$, or if you have any questions about specific parts.

Follow-up Questions:

1. What is the relationship between torque and force in static equilibrium problems?
2. How do changes in the angle $\theta$ affect the required pulling force $P$?
3. Why is taking moments around point $B$ effective in simplifying the solution?
4. How would the solution change if the step height $b$ increased?
5. What role does the smooth surface assumption play in this problem?

Tip:

In static equilibrium problems, always check if summing moments around specific points can simplify the equations by eliminating unknown forces.