A uniform beam 6m long weighing 4kg rests on supports p and q placed left and right 1metre from each end of the beam. Weight 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.
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To calculate the reactions at supports p and q, we need to consider the equilibrium of the beam. In other words, the sum of the forces and the sum of the moments acting on the beam should be zero.
First, let's calculate the weight of the beam itself. The weight of the beam is given as 4 kg, but to calculate the force, we need to multiply it by the acceleration due to gravity, which is approximately 9.8 m/s^2.
Weight of the beam = Mass of the beam * Acceleration due to gravity = 4 kg * 9.8 m/s^2 = 39.2 N
Now, let's calculate the reactions at support p and q.
1. Vertical forces (Sum of forces in the vertical direction):
At support p, we have:
Reaction at p + Weight of the beam + Weight of the 10 kg mass = 0
Reaction at p = -(Weight of the beam + Weight of the 10 kg mass)
At support q, we have:
Reaction at q + Weight of the beam + Weight of the 8 kg mass = 0
Reaction at q = -(Weight of the beam + Weight of the 8 kg mass)
2. Sum of moments (Sum of torques about any point, preferably one of the supports):
Taking moments about support p:
Clockwise moment = Weight of the 10 kg mass * Distance between the 10 kg mass and support p
Anti-clockwise moment = Reaction at q * Distance between support q and support p
Since the beam is in equilibrium, the sum of clockwise moments is equal to the sum of anti-clockwise moments.
Clockwise moment = Anti-clockwise moment
Weight of the 10 kg mass * 1 m = Reaction at q * 5 m
Now, we have two unknowns - Reaction at p and Reaction at q. We have two equations in terms of these unknowns:
1. Reaction at p = -(Weight of the beam + Weight of the 10 kg mass)
2. Weight of the 10 kg mass * 1 m = Reaction at q * 5 m
We can substitute the value of Reaction at p from equation 1 into equation 2 to find the value of Reaction at q.
Once we have the value of Reaction at q, we can substitute it back into equation 1 to find the value of Reaction at p.
Let's do the calculations:
Reaction at p = -(39.2 N + Weight of the 10 kg mass)
Reaction at q = (Weight of the 10 kg mass * 1 m) / 5 m
Finally, substitute the value of Reaction at q back into equation 1 to find the value of Reaction at p.
Note: Remember to consider the negative signs in the equations appropriately, as they depend on the direction of the forces.
calculate the moments about p to find the reaction at q
... and vice-versa