A uniform bar AB, 100cm long weighs 60N. An upward force of 50N is applied at a point 20cm from A, and downward force of 60N and 30N respectively. Compute the value and position of the equilibriant.
To solve this problem, we need to consider the principle of moments and the principle of equilibrium for the bar.
The principle of moments states that for an object in equilibrium, the sum of the anticlockwise moments about any point must be equal to the sum of the clockwise moments about the same point.
Let's assume that the equilibrium is at point X, which is at a distance of 'd' cm from A.
The anticlockwise moments about X are:
1. The force of 50N, which is applied 20cm from A, so its moment is 50N * 20cm = 1000 Ncm (clockwise).
The clockwise moments about X are:
1. The weight of the bar, which acts at its center (50cm from A), so its moment is 60N * 50cm = 3000 Ncm (anticlockwise).
2. The downward force of 60N, which acts at point B (100cm from A), so its moment is 60N * 100cm = 6000 Ncm (anticlockwise).
3. The downward force of 30N, which also acts at point B, so its moment is 30N * 100cm = 3000 Ncm (anticlockwise).
Now, according to the principle of moments, the sum of the anticlockwise moments should be equal to the sum of the clockwise moments.
Therefore, 1000 Ncm (clockwise) = (3000 Ncm + 6000 Ncm + 3000 Ncm) (anticlockwise)
1000 Ncm = 12000 Ncm (anticlockwise)
To find the value and position of the equilibrant force, we can calculate:
1. The value of the equilibrant force:
Since the moment of the equilibrant force is zero, we can add the moments of all the other forces and equate them to zero.
Total anticlockwise moments = Total clockwise moments
1000 Ncm = (3000 Ncm + 6000 Ncm + 3000 Ncm) + (equilibrant force * distance)
1000 Ncm = 12000 Ncm + (equilibrant force * distance)
Rearranging the equation, we get:
Equilibrant force * distance = -11000 Ncm
So, the value of the equilibrant force is:
equilibrant force = -11000 Ncm / distance
2. The position of the equilibrant force:
Since the bar is in equilibrium, the position of the equilibrant force must be between 0 cm and 100 cm, as it must act within the length of the bar.
Therefore, the position of the equilibrant force lies between A and B.
To find the exact position, we can use the moment calculations again.
The anticlockwise moments about A (point of application of the equilibrant force) are:
1. The force of 50N, which is applied 20cm from A, so its moment is 50N * 20cm = 1000 Ncm (clockwise).
The clockwise moments about A are:
1. The weight of the bar, which acts at its center (50cm from A), so its moment is 60N * 50cm = 3000 Ncm (anticlockwise).
2. The downward force of 60N, which acts at point B (100cm from A), so its moment is 60N * 100cm = 6000 Ncm (anticlockwise).
3. The downward force of 30N, which also acts at point B, so its moment is 30N * 100cm = 3000 Ncm (anticlockwise).
4. The equilibrant force * distance (let's call it F * X, where X is the position of the equilibrant force from A).
Now, according to the principle of moments, the sum of the anticlockwise moments should be equal to the sum of the clockwise moments.
Therefore, 1000 Ncm (clockwise) = (3000 Ncm + 6000 Ncm + 3000 Ncm) (anticlockwise) + (F * X)
1000 Ncm = 12000 Ncm (anticlockwise) + (F * X)
To find the position of the equilibrant force, we rearrange the equation:
(F * X) = -11000 Ncm
So, the position of the equilibrant force is:
X = -11000 Ncm / F
Now we can substitute the equation for the value of the equilibrant force we found earlier:
X = -11000 Ncm / (-11000 Ncm / distance)
Simplifying, we get:
X = distance
Therefore, the position of the equilibrant force is equal to the distance.