A uniform beam 6.0m long and weighing 40N at the center, rests on two supports, P and Q placed 0.1m from each end of the beam. Weights of 100N and 80N are hung from the ends of the beam near P and Q respectively. Calculate the reactions at the supports P and Q.
To calculate the reactions at the supports P and Q, we need to use the principle of moments and equilibrium. The key idea is that the sum of the clockwise moments about any point must be equal to the sum of the counterclockwise moments about the same point, and the sum of the upward forces must be equal to the sum of the downward forces.
Let's start by summing the moments about support P:
Taking moments about support P:
Clockwise moments = (Weight 100N) × (Distance from support P to weight 100N)
+ (Weight 40N) × (Distance from support P to weight 40N)
Counterclockwise moments = (Weight 80N) × (Distance from support Q to weight 80N)
Since the beam is in equilibrium, the sum of clockwise moments should be equal to the sum of counterclockwise moments.
Similarly, we can sum the moments about support Q:
Clockwise moments = (Weight 80N) × (Distance from support Q to weight 80N)
Counterclockwise moments = (Weight 100N) × (Distance from support P to weight 100N)
+ (Weight 40N) × (Distance from support P to weight 40N)
Again, the sum of clockwise moments should be equal to the sum of counterclockwise moments.
Solving these two equations will give us the reactions at supports P and Q.