Questions: Find the magnitude of B in lb

Transcript text: Find the magnitude of B in lb

Solution

Step 1: Identify the forces and moments acting on the beam

Step 2: Set up the equilibrium equations

For a beam in static equilibrium, the sum of forces and the sum of moments must be zero.
Sum of vertical forces: \( \sum F_y = 0 \)
Sum of moments about point A: \( \sum M_A = 0 \)

Step 3: Calculate the reactions at the supports

Let \( R_A \) be the reaction at point A and \( R_B \) be the reaction at point B.
Sum of vertical forces: \( R_A + R_B - 20 \text{ lb} - 50 \text{ lb} = 0 \) \[ R_A + R_B = 70 \text{ lb} \]

Step 4: Calculate the moment about point A

Sum of moments about point A: \[ -20 \text{ lb} \times 3 \text{ ft} - 50 \text{ lb} \times 9 \text{ ft} + 140 \text{ ft-lb} + R_B \times 12 \text{ ft} = 0 \] \[ -60 \text{ ft-lb} - 450 \text{ ft-lb} + 140 \text{ ft-lb} + 12R_B = 0 \] \[ -370 \text{ ft-lb} + 12R_B = 0 \] \[ 12R_B = 370 \text{ ft-lb} \] \[ R_B = \frac{370}{12} \text{ lb} \] \[ R_B \approx 30.83 \text{ lb} \]

Step 5: Calculate the reaction at point A

Using the sum of vertical forces: \[ R_A + 30.83 \text{ lb} = 70 \text{ lb} \] \[ R_A = 70 \text{ lb} - 30.83 \text{ lb} \] \[ R_A \approx 39.17 \text{ lb} \]

The magnitude of the reaction at point B is approximately \( 30.83 \text{ lb} \).

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