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To solve for the reactions \( R_1 \) and \( R_2 \) in the beam problem, we must perform static equilibrium analysis. Here's the approach:

### Given Data:
- **Triangular distributed load** with a maximum of \( 2 \, \text{kip/ft} \) over a length of \( 10' \).
- **Concentrated load** of \( 10 \, \text{kip} \) applied at \( 3'4'' \) (or \( 3.33' \)) from the right support.
- The beam has two supports: a **pin support** at \( R_1 \) and a **roller support** at \( R_2 \).

### Step 1: Calculate the Total Load of the Triangular Distributed Load
The area of the triangular load gives the total force:
\[
F_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10' \times 2 \, \text{kip/ft} = 10 \, \text{kip}
\]
This force acts at the centroid of the triangle, which is located \( \frac{1}{3} \) of the distance from the left (pin support).

Thus, the location of this force from \( R_1 \) is:
\[
\text{Location of centroid} = \frac{1}{3} \times 10' = 3.33' \text{ from the left}
\]

### Step 2: Write the Equilibrium Equations
We will use the three static equilibrium conditions:
1. **Sum of vertical forces**: 
\[
R_1 + R_2 = F_{\text{triangle}} + 10 \, \text{kip} = 10 + 10 = 20 \, \text{kip}
\]

2. **Sum of moments about \( R_1 \)** (to eliminate \( R_1 \) and solve for \( R_2 \)):
Taking counterclockwise moments as positive:
\[
\text{Moment at } R_1: 10 \, \text{kip} \times 3.33' + 10 \, \text{kip} \times (10' + 3.33') - R_2 \times (10' + 3.33') = 0
\]
Simplifying:
\[
33.3 + 133.3 - R_2 \times 13.33 = 0
\]
\[
R_2 = \frac{33.3 + 133.3}{13.33} = 12.5 \, \text{kip}
\]

3. **Sum of forces** (to find \( R_1 \)):
From the first equilibrium equation:
\[
R_1 + 12.5 = 20 \quad \Rightarrow \quad R_1 = 7.5 \, \text{kip}
\]

### Final Reactions:
- \( R_1 = 7.5 \, \text{kip} \)
- \( R_2 = 12.5 \, \text{kip} \)

Would you like any additional details or further clarification on the solution?

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Here are 5 related questions for deeper understanding:
1. How would the reactions change if the concentrated load were moved closer to the pin support?
2. What is the significance of the location of the centroid of the triangular load?
3. How do different types of loads (uniform vs triangular) affect the reaction forces?
4. How does the pin and roller support system differ from a fixed support system in terms of reaction forces?
5. What would be the effect of increasing the magnitude of the distributed load?

**Tip**: Always verify that the sum of moments and forces balances to ensure your calculations are correct.

Find the reactions R1 and R2 for the beam subjected to a triangular distributed load and a concentrated load as shown in the diagram.

Learn how to calculate the reactions R1 and R2 for a beam subjected to a triangular distributed load of 2 kip/ft over 10 feet and a concentrated load of 10 kip applied at 3'4" from the right support. The step-by-step solution includes moment calculations and equilibrium equations.

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