A simply supported beam \( AB \) of span \( L \) is shown in the figure. A moment \( M \) is applied at point \( C \). The magnitude of the reaction force at point A is:
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For a beam with a moment applied at a point, the reaction force at the support is calculated by balancing the moments about that support. The distance from the point of application of the moment to the support plays a crucial role in determining the magnitude of the reaction force.
We are given a simply supported beam \( AB \) with a span \( L \), and a moment \( M \) applied at point \( C \), which is located at a distance \( a \) from point \( A \). The goal is to determine the magnitude of the reaction force at point \( A \).
To solve this, we apply the basic principles of static equilibrium. For a beam subjected to external moments and forces, the sum of the moments about any point must be zero for equilibrium. We can use the following equations:
- The sum of forces in the vertical direction is zero, which means the reaction forces at \( A \) and \( B \) must balance any external forces.
- The sum of moments about any point (we choose point \( A \) for simplicity) must also be zero. Step-by-Step Solution:
1. Moment equilibrium about point A:
The moment applied at point \( C \) causes a reaction force at \( A \) that must balance it out. We can calculate the moment balance as:
\[
\text{Moment at A} = M
\]
2. Reaction at A:
The reaction force at \( A \), denoted \( R_A \), must create a moment that balances the applied moment \( M \). The distance from point \( A \) to the point of application of the moment is \( L \), so the reaction force at \( A \) creates a moment equal to:
\[
R_A \times L = M
\]
3. Solving for \( R_A \):
\[
R_A = \frac{M}{L}
\]
Thus, the magnitude of the reaction force at point \( A \) is \( \frac{M}{L} \), which corresponds to option (A).
