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To solve this problem, we need to determine the correct nodal forces for nodes 1 and 2 in a 2-node bar element subjected to a distributed load \( T = cx^2 \) over a length \( L = 1 \, \text{m} \).

### Steps to Solution

1. **Shape Functions**:
   The shape functions \( N_1 \) and \( N_2 \) for a linear 2-node element are given by:
   \[
   N_1 = \frac{x_2 - x}{x_2 - x_1}, \quad N_2 = \frac{x - x_1}{x_2 - x_1}
   \]
   Since \( x_1 = 0 \) and \( x_2 = L = 1 \), these become:
   \[
   N_1 = 1 - x, \quad N_2 = x
   \]

2. **Expression for the Nodal Force Vector**:
   The nodal force vector is defined by the integral:
   \[
   \{ f^{(e)} \} = \int_0^L \begin{bmatrix} N_1 \\ N_2 \end{bmatrix} T \, dx
   \]
   Substituting \( T = cx^2 \):
   \[
   \{ f^{(e)} \} = \int_0^1 \begin{bmatrix} 1 - x \\ x \end{bmatrix} cx^2 \, dx = c \int_0^1 \begin{bmatrix} (1 - x)x^2 \\ x^3 \end{bmatrix} dx
   \]

3. **Evaluating Each Component**:
   - For Node 1:
     \[
     f_1 = c \int_0^1 (1 - x)x^2 \, dx
     \]
     Expanding \( (1 - x)x^2 \):
     \[
     f_1 = c \int_0^1 (x^2 - x^3) \, dx = c \left[ \frac{x^3}{3} - \frac{x^4}{4} \right]_0^1 = c \left( \frac{1}{3} - \frac{1}{4} \right) = c \cdot \frac{1}{12} = \frac{c}{12}
     \]

   - For Node 2:
     \[
     f_2 = c \int_0^1 x^3 \, dx = c \left[ \frac{x^4}{4} \right]_0^1 = c \cdot \frac{1}{4} = \frac{c}{4}
     \]

4. **Result**:
   The nodal forces for nodes 1 and 2, respectively, are:
   \[
   f_1 = \frac{c}{12} \quad \text{and} \quad f_2 = \frac{c}{4}
   \]
   Both forces act towards the right.

### Answer
The correct answer is:
- **Option 1: \( \frac{c}{12}, \frac{c}{4} \) both acting towards right**.

Would you like further details on any part of the solution?

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#### Related Questions
1. How would the solution change if the distributed load were linear instead of quadratic?
2. What would happen to the nodal forces if the length \( L \) were increased?
3. How would you apply this method to a 3-node element?
4. Can you derive the expression for \( N_1 \) and \( N_2 \) for a general element length \( L \)?
5. What if \( c \) were negative? How would the direction of forces change?

#### Tip
For distributed loads, breaking down the load into components and applying shape functions systematically is key to finding nodal forces.

A 2-node bar element of length L = 1 m is subjected to a distributed load in the form T = cx^2 where c is a real constant. Determine the correct nodal forces for nodes 1 and 2, respectively.

Solve for nodal forces in a 2-node bar element subjected to a quadratic distributed load T = cx^2 over a length of L = 1 m. Learn about shape functions, finite element integration, and force calculation methods. Ideal for college engineering students studying finite element analysis.

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