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Una barra de 3,60 m de longitud que pesa 80 kgf por metro tiene en su extremo izquierdo una carga de 100 kgf. Calcular a) la fuerza p necesaria para poder equilibrarla en el punto A b) la fuerza de apoyo en O

To solve this problem involving a uniform beam with a load at its end, we will approach it by using the principles of static equilibrium. Let's break down the problem:

### Given:
- The total length of the beam is \( 3.60 \, \text{m} \).
- The weight of the beam per unit length is \( 80 \, \text{kgf/m} \).
- The beam has a 100 kgf load at its leftmost end.
- The beam is supported at points \( O \) (at the center) and \( A \) (1.2 meters from point \( O \)).

We are asked to:
1. **Calculate the force \( P \)** needed to balance the beam at point \( A \). 
2. **Find the reaction force at the support point \( O \)**.

### Step 1: Calculate the total weight of the beam.
The beam weighs \( 80 \, \text{kgf/m} \), and its length is \( 3.60 \, \text{m} \). Therefore, the total weight of the beam \( W_{\text{beam}} \) is:

\[
W_{\text{beam}} = 80 \, \text{kgf/m} \times 3.60 \, \text{m} = 288 \, \text{kgf}
\]

### Step 2: Set up the equations for static equilibrium.

For the beam to be in equilibrium, both the **sum of forces** and the **sum of moments** must equal zero.

1. **Sum of vertical forces:**
   The total upward force from the supports must balance the downward forces from the beam and the load.

2. **Sum of moments about point \( O \):**
   We can take moments about the support point \( O \) to solve for the unknown force \( P \). Let’s assume the distances and forces are positive when acting counterclockwise.

I'll calculate and provide the solution next.# Given values from the problem
length_of_beam = 3.60  # in meters
weight_per_meter = 80  # in kgf/m
left_load = 100  # in kgf
distance_O_to_A = 1.2  # in meters
distance_A_to_right_end = 2.4  # in meters

# Total weight of the beam
total_weight_of_beam = weight_per_meter * length_of_beam  # in kgf

# Position of the center of gravity of the beam (half the length)
center_of_gravity_position = length_of_beam / 2  # in meters

# Setting up the equation for moments about point O
# P * 1.2 (counterclockwise) - weight_of_beam * distance of center of gravity from O - left load * distance of load from O = 0

# Calculate the moment contributions
moment_due_to_beam_weight = total_weight_of_beam * center_of_gravity_position
moment_due_to_left_load = left_load * distance_O_to_A

# Sum of moments about point O to solve for force P
force_P = (moment_due_to_beam_weight + moment_due_to_left_load) / distance_A_to_right_end
force_P

Learn how to solve a static equilibrium problem involving a uniform beam with specific loads and support points. Calculate forces and moments to find equilibrium at various points along the beam. Includes step-by-step solution and key formulas.

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