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 ?

Answer 1

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.