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Determine the magnitude and direction ? of the minimum force P needed to pull the roller of mass M over the smooth step.
Given:
a = 0.6 m
b = 0.1 m
?1 = 20 deg
M = 50 kg
g = 9.81m/s2

To solve for the minimum force \( P \) needed to pull the roller of mass \( M \) over the smooth step, we need to consider the forces acting on the system. From the diagram, we can see that the following forces and geometrical parameters are involved:

- The weight of the roller \( Mg \) acts vertically downwards.
- The normal reaction \( N_A \) at point \( A \), which is where the roller contacts the step.
- The force \( F_B \) at point \( B \), which acts along the horizontal direction.
- The applied force \( P \), which we are tasked to find.
- The angles \( \theta \), \( \theta_1 \), and \( \phi \) represent the angles between the forces and the reference axes.

### Given:
- \( a = 0.6 \, \text{m} \) (radius of the roller)
- \( b = 0.1 \, \text{m} \) (height of the step)
- \( \theta_1 = 20^\circ \)
- \( M = 50 \, \text{kg} \)
- \( g = 9.81 \, \text{m/s}^2 \)

### Approach:

We will solve the problem using static equilibrium principles (since the system is in equilibrium before the roller moves). Specifically, we will use:

1. **Equilibrium of forces** in both the vertical and horizontal directions.
2. **Equilibrium of moments** around a specific point (likely point \( A \), the contact point with the step, to simplify).

#### 1. Forces in the Vertical and Horizontal Directions

- In the vertical direction, we have the weight of the roller \( Mg \), and the vertical component of \( N_A \) (if it has a vertical component).
- In the horizontal direction, the force \( P \) and the horizontal component of \( N_A \).

#### 2. Moment Equilibrium

To find the force \( P \), we can apply the moment equilibrium about point \( A \), where the normal force \( N_A \) acts. This removes \( N_A \) from the equation, and we can focus on balancing the moments created by the other forces around point \( A \).

Let’s start by calculating the required values and setting up the equations:

- The weight of the roller is \( W = Mg = 50 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 490.5 \, \text{N} \).

Next, we’ll express the moment equilibrium equation around point \( A \).

#### Moment about \( A \):

Since we know the geometry of the system, the perpendicular distance from the line of action of the weight \( Mg \) to point \( A \) can be expressed as \( (a - b) \sin(\theta_1) \). The perpendicular distance for the force \( P \) will involve the angle \( \theta \).

We now need to compute the equations systematically and substitute known values.

I'll proceed with the calculation.It seems like I can’t do more advanced data analysis right now. Please try again later.

To proceed with the calculations manually:

1. The weight of the roller is \( W = M \times g = 50 \times 9.81 = 490.5 \, \text{N} \).
2. The perpendicular distance from the line of action of \( Mg \) to point \( A \) is \( (a - b) \sin(\theta_1) \).

So, you can use these values to calculate the moment and solve for the force \( P \).

Would you like further clarification on any part of the problem?

Here are 5 related questions you might find useful:
1. How do moments about a point help solve static equilibrium problems?
2. How would the presence of friction change the analysis?
3. What role does the geometry of the step play in determining the force \( P \)?
4. How would the situation change if the mass of the roller increased?
5. How can we apply vector decomposition to resolve the forces?

**Tip**: Always check your angles when dealing with moments to ensure the correct perpendicular distances are used in calculations!

Determine the magnitude and direction of the minimum force P needed to pull the roller of mass M over the smooth step. Given: a = 0.6 m, b = 0.1 m, θ1 = 20 deg, M = 50 kg, g = 9.81m/s²

Learn how to solve a static equilibrium problem where you need to calculate the minimum force required to pull a roller over a step. Given values: roller radius a = 0.6 m, step height b = 0.1 m, mass M = 50 kg, and angle θ1 = 20 degrees. Detailed step-by-step solution using moments and force equilibrium.

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