1. A square board ABCD is fixed in a vertical plane. Two pulleys P and Q on the board carry loads of 25 N and 50 N suspended by strings as shown. The coordinates of the centre of pulleys are P(1, 4 )m and Q(4, 1)m respectively. Determine the magnitude and direction of the resultant and the distance from A to the line of action of resultant.?

Question

Answer

Problem Statement:
A square board ABCD is fixed in a vertical plane. Two pulleys P and Q on the board carry loads of 25 N and 50 N suspended by strings as shown. The coordinates of the centre of pulleys are P(1, 4) m and Q(4, 1) m respectively. Determine the magnitude and direction of the resultant and the distance from A to the line of action of resultant.

Solution:
To determine the magnitude and direction of the resultant, we need to find the tension in each string and the angle between the strings.

Step 1: Find the tension in each string
Let T1 be the tension in the string attached to the 25 N load and T2 be the tension in the string attached to the 50 N load.
Using the principle of static equilibrium, we can write the following equations:

ΣFx = 0: T1 cosθ = T2 sinθ
ΣFy = 0: T1 sinθ + T2 sinθ = 25 + 50 = 75

Dividing the two equations, we get:
tanθ = T1/T2 = 1/2

Solving for T1 and T2, we get:
T1 = 25/√5 N
T2 = 50/√5 N

Step 2: Find the angle between the strings
Using the equation tanθ = 1/2, we get:
θ = 26.57°

Step 3: Find the magnitude and direction of the resultant
Let R be the magnitude of the resultant force. We can write the following equations:

ΣFx = 0: R sinθ = T2 cosθ
ΣFy = 0: R cosθ = T1 + T2 sinθ

Substituting the values of T1, T2 and θ, we get:
R = 75/√5 N ≈ 33.54 N
θR = 63.43°

Therefore, the magnitude of the resultant force is 33.54 N and the direction is 63.43° from the horizontal.

Step 4: Find the distance from A to the line of action of resultant
Let d be the distance from A to the line of action of resultant. We can use the equation:

d = (T1/T2) * h

where h is the distance between the centers of the pulleys, which is √18 m.
Substituting the values, we get:
d = (25/50) * √18 m ≈ 0.95 m

Therefore, the distance from A to the line of action of resultant is approximately 0.95 m.

Conclusion:
The magnitude and direction of the resultant force are 33.54 N and 63.43° from the horizontal respectively. The distance from A to the line of action of resultant is approximately 0.95 m.

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