A uniform beam 6.0m long and weighing 4kg rest on supports at P and Q placed left and right 1.0m from each end of the beam.weights of Mass 10kg and 8kg are placed near P and Q respectively one each end of the beam.calculate the reactions at P and Q?
Solve it
To calculate the reactions at the supports P and Q, we can use the principle of static equilibrium. For a beam in equilibrium, the sum of the clockwise moments about any point must be equal to the sum of the anticlockwise moments about the same point.
First, let's identify the forces acting on the beam:
1. The weight of the beam itself (4 kg) applied at its center (3.0 m from either end).
2. The weight of the 10 kg mass applied at point P (1.0 m from the left end).
3. The weight of the 8 kg mass applied at point Q (1.0 m from the right end).
Now, let's consider the equilibrium conditions at point P:
- The clockwise moment is the weight of the beam (4 kg) multiplied by the distance between P and the center of the beam (3.0 m).
- The anticlockwise moment is the weight of the 10 kg mass multiplied by the distance between P and the 10 kg mass (1.0 m). Since the 10 kg mass is placed left of P, the distance would be negative (-1.0 m).
Setting up the equation:
Clockwise moment = Anticlockwise moment
(4 kg * 3.0 m) = (10 kg * -1.0 m)
To calculate the reaction at support P, we need to find the sum of the forces acting on the beam vertically at P. The reaction at P balances the vertical forces:
Reaction at P = Weight of the beam + Weight of the 10 kg mass
Reaction at P = (4 kg + 10 kg) * 9.8 m/s²
Next, let's consider the equilibrium conditions at point Q:
- The clockwise moment is the weight of the beam (4 kg) multiplied by the distance between Q and the center of the beam (3.0 m).
- The anticlockwise moment is the weight of the 8 kg mass multiplied by the distance between Q and the 8 kg mass (1.0 m). Since the 8 kg mass is placed right of Q, the distance would be positive (+1.0 m).
Setting up the equation:
Clockwise moment = Anticlockwise moment
(4 kg * 3.0 m) = (8 kg * 1.0 m)
To calculate the reaction at support Q, we need to find the sum of the forces acting on the beam vertically at Q. The reaction at Q balances the vertical forces:
Reaction at Q = Weight of the beam + Weight of the 8 kg mass
Reaction at Q = (4 kg + 8 kg) * 9.8 m/s²
Calculating the reactions at supports P and Q:
Reaction at P = (14 kg) * 9.8 m/s²
Reaction at Q = (12 kg) * 9.8 m/s²