A uniform beam 6.ometer long and weighing 4kg rests on supports at p and q placed left and right 1.0meter from each end of the beam. weight of mass 10kg and 8kg are placed near p and q respectively one at each end of the beam. calculate the reaction at p and q ?
To solve this problem, we can use the principle of moments. We know that the beam is in equilibrium, which means that the sum of moments about any point must be zero.
Let's choose point P as our reference point. The weight of the beam itself can be considered to act at its center of mass, which is at a distance of 3 meters from P.
The weight of the 10kg mass acts at a distance of 1 meter from P, and the weight of the 8kg mass acts at a distance of 5 meters from P. We can calculate the moments due to these weights as follows:
Moment due to 4kg beam = 4kg x 9.81m/s^2 x 3m = 117.72 Nm (clockwise)
Moment due to 10kg mass = 10kg x 9.81m/s^2 x 1m = 98.1 Nm (counter-clockwise)
Moment due to 8kg mass = 8kg x 9.81m/s^2 x 5m = 392.4 Nm (clockwise)
The reaction force at support Q can be calculated by considering the sum of vertical forces acting on the beam. We know that the beam is in equilibrium, which means that the sum of vertical forces must be zero. Therefore, we can write:
Reaction force at Q = Weight of beam + Weight of 10kg mass + Weight of 8kg mass - Reaction force at P
Plugging in the values, we get:
Reaction force at Q = 4kg x 9.81m/s^2 + 10kg x 9.81m/s^2 + 8kg x 9.81m/s^2 - Reaction force at P
Reaction force at Q = 294.84 N - Reaction force at P
Now, let's consider the sum of moments about point Q. Since the beam is in equilibrium, the sum of moments must be zero. We can write:
Moment due to 4kg beam = Moment due to 10kg mass + Moment due to 8kg mass + Moment due to Reaction force at P
Plugging in the values, we get:
117.72 Nm = 98.1 Nm + 392.4 Nm + Distance from Q to P x Reaction force at P
Solving for Reaction force at P, we get:
Reaction force at P = (117.72 Nm - 98.1 Nm - 392.4 Nm) / Distance from Q to P
Reaction force at P = -373.78 N / 4m
Reaction force at P = -93.445 N
Since the beam is in equilibrium, the reaction force at Q can be calculated using:
Reaction force at Q = Weight of beam + Weight of 10kg mass + Weight of 8kg mass - Reaction force at P
Reaction force at Q = 294.84 N - (-93.445 N)
Reaction force at Q = 388.285 N
Therefore, the reaction force at support P is -93.445 N and the reaction force at support Q is 388.285 N. Note that the negative sign for the reaction force at P indicates that it acts in the opposite direction to our assumed direction.
To calculate the reactions at supports P and Q, we first need to determine the total weight acting on the beam, which includes the weight of the beam itself and the masses placed near supports P and Q.
The total weight acting on the beam is the sum of the weight of the beam and the weights of the masses:
Total weight = weight of the beam + weight near P + weight near Q
Given:
Weight of the beam = 4 kg
Weight near P = 10 kg
Weight near Q = 8 kg
Total weight = 4 kg + 10 kg + 8 kg
Total weight = 22 kg
Now, we can calculate the reactions at supports P and Q using the principle of moments.
The moment about support P is equal to the sum of the anticlockwise moments and clockwise moments acting on the beam. The anticlockwise moment is the product of the weight near Q (8 kg) and its distance from support P (5.0 m = 6.0 m - 1.0 m), while the clockwise moment is the product of the total weight (22 kg) and its distance from support P (1.0 m).
Moment about P = (Weight near Q) * (Distance from P) - (Total weight) * (Distance from P)
Moment about P = (8 kg) * (5.0 m) - (22 kg) * (1.0 m)
Moment about P = 40 kg*m - 22 kg*m
Moment about P = 18 kg*m
Since the beam is in equilibrium, the total moment about support P must be zero.
Therefore, 18 kg*m = 0
The reaction at support P is 0.
To calculate the reaction at support Q, we can use the fact that the sum of the reactions at both supports must equal the total weight.
Reaction at Q = Total weight - Reaction at P
Reaction at Q = 22 kg - 0 kg
Reaction at Q = 22 kg
Therefore, the reaction at support P is 0 kg and the reaction at support Q is 22 kg.