A uniform beam 6.0m long and weighing 4kg rests on supports P and Q placed left and right 1.0metre from each end of the beam.calculate the reactions at p and Q.
To calculate the reactions at supports P and Q, we can use the principle of moments. The principle of moments states that the sum of the clockwise moments is equal to the sum of the anticlockwise moments.
Let's assume that the reaction at P is Rp and the reaction at Q is Rq. We'll calculate the moments about point P, then set it equal to the moments about point Q.
The weight of the beam acts at its center of gravity, which is at a distance of 3.0m from either end of the beam. The weight of the beam can be calculated using the formula:
Weight = Mass × Gravitational Acceleration
Weight = 4kg × 9.8m/s^2 (taking acceleration due to gravity as 9.8m/s^2)
Weight = 39.2N
Now, let's calculate the moments about point P:
Clockwise moments:
Weight × Distance from center of gravity to point P = 39.2N × 3.0m = 117.6Nm
Anti-clockwise moments:
Rp × Distance from point P to point Q = Rp × 1.0m
According to the principle of moments, the clockwise moments are equal to the anticlockwise moments:
117.6Nm = Rp × 1.0m
Therefore, Rp = 117.6Nm / 1.0m = 117.6N
So the reaction at support P is 117.6N.
To calculate the reaction at support Q, we'll calculate the moments about point Q:
Clockwise moments:
Rq × Distance from point Q to point P = Rq × 1.0m
Anti-clockwise moments:
Weight × Distance from center of gravity to point Q = 39.2N × 3.0m = 117.6Nm
According to the principle of moments, the clockwise moments are equal to the anticlockwise moments:
Rq × 1.0m = 117.6Nm
Therefore, Rq = 117.6Nm / 1.0m = 117.6N
So the reaction at support Q is 117.6N.
In summary, the reactions at support P and support Q are both 117.6N.