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Let's solve the problem step-by-step.

The given beam has the following components:
- A **1300 N** point load at the left end.
- A **400 N/m** uniformly distributed load (UDL) over a **2 m** section.
- A **1600 N** point load at the far right end.
- A pin support at the **left end** (provides both vertical and horizontal reactions).
- A roller support at **2 m** from the pin support (provides only a vertical reaction).

We'll assume:
- The pin support provides the vertical reaction \(R_A\).
- The roller support provides the vertical reaction \(R_B\).

Now, let's calculate the reactions using **static equilibrium** equations:

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### Step 1: List all forces and moments
1. **1300 N** point load at 1 m from \(A\).
2. **400 N/m UDL** over 2 m (equivalent total load: \(400 \times 2 = 800 \, \text{N}\), acts at the center of the UDL span, i.e., 1 m from the left edge of the UDL).
3. **1600 N** point load at the far right end, 5.5 m from \(A\).

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### Step 2: Apply equilibrium equations

#### 1. Sum of vertical forces (\(\Sigma F_y = 0\)):

\[
R_A + R_B - 1300 - 800 - 1600 = 0
\]

\[
R_A + R_B = 3700 \, \text{N} \tag{1}
\]

#### 2. Sum of moments about \(A\) (\(\Sigma M_A = 0\)):

Taking counterclockwise moments as positive:
- Moment due to the 1300 N force: \(1300 \times 1 = 1300 \, \text{Nm}\).
- Moment due to UDL: \(800 \times 2 = 1600 \, \text{Nm}\) (acts at 2 m from \(A\)).
- Moment due to the 1600 N force: \(1600 \times 5.5 = 8800 \, \text{Nm}\).
- Moment due to \(R_B\) (acting at 2 m from \(A\)): \(R_B \times 2\).

\[
1300 + 800 \times 2 + 1600 \times 5.5 - R_B \times 2 = 0
\]

Simplifying:

\[
1300 + 1600 + 8800 = 2 R_B
\]

\[
11700 = 2 R_B \quad \Rightarrow \quad R_B = \frac{11700}{2} = 5850 \, \text{N}
\]

#### 3. Substitute \(R_B\) into Equation (1):

\[
R_A + 5850 = 3700
\]

\[
R_A = 3700 - 5850 = -2150 \, \text{N}
\]

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### Step 3: Interpret the results

Since \(R_A\) is negative, it means the direction we initially assumed for the reaction at \(A\) is incorrect. Thus, the actual reaction at \(A\) is **2150 N downward**.

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### Final Answers
- **Reaction at A**: \(R_A = -2150 \, \text{N}\) (downward)
- **Reaction at B**: \(R_B = 5850 \, \text{N}\) (upward)

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Would you like a deeper breakdown of any step?

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Here are 5 related questions you might find helpful:
1. How do you determine the centroid of a distributed load?
2. What happens if the UDL covers the entire beam?
3. How does the moment arm affect the reaction forces?
4. Can reactions be negative in statics problems?
5. How do pin and roller supports differ in function?

**Tip:** Always verify the direction of assumed reactions by checking the sign after calculation!

Determine the reaction for the beam shown.

Determine the reactions for a beam subjected to a 1300 N point load, a 400 N/m uniformly distributed load (UDL) over 2 m, and a 1600 N point load. This solution explains how to use static equilibrium equations to calculate reactions at pin and roller supports.

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