A vertical load of 100N is supported by two chains, A and B, in the same vertical plane. The force in A is 50N and acts in a line at 30 degrees to the horizontal plane (i.e. at 120 degrees to the 100N load). Determine the force in chain B and the angle between A and B.
Quick one
To solve this problem, we can break down the forces acting on the load into their horizontal and vertical components. Let's start by finding the vertical component of the forces.
1. Determine the vertical component of the load:
The vertical component of the load can be found by multiplying the total load (100N) by the sin of the angle it makes with the horizontal plane:
Vertical component of the load = 100N * sin(90 degrees) = 100N
2. Determine the vertical component of force A:
The vertical component of force A can be found by multiplying the force in chain A (50N) by the sin of the angle it makes with the horizontal plane (30 degrees):
Vertical component of force A = 50N * sin(30 degrees) = 25N
3. Determine the vertical component of force B:
The vertical component of force B is equal to the total vertical load minus the vertical component of force A:
Vertical component of force B = Vertical component of the load - Vertical component of force A
Vertical component of force B = 100N - 25N = 75N
Now, let's move on to finding the horizontal component of the forces.
4. Determine the horizontal component of force A:
The horizontal component of force A can be found by multiplying the force in chain A (50N) by the cos of the angle it makes with the horizontal plane (30 degrees):
Horizontal component of force A = 50N * cos(30 degrees) = 43.3N
5. Determine the horizontal component of force B:
The horizontal component of force B can be found by multiplying the force in chain B (unknown) by the cos of the angle it makes with the horizontal plane (unknown):
Horizontal component of force B = Force in chain B * cos(angle between A and B)
Now, let's find the angle between force A and force B.
6. Determine the angle between A and B:
The angle between A and B can be found using the trigonometric identity:
tan(angle between A and B) = (Vertical component of force B) / (Horizontal component of force B)
Now, we can solve for the unknowns.
7. Solve for force in chain B:
Using the trigonometric identity mentioned earlier, we can rearrange the equation as follows:
Horizontal component of force B = (Vertical component of force B) / tan(angle between A and B)
Using the values we found earlier:
Force in chain B = (75N) / tan(angle between A and B)
8. Solve for the angle between A and B:
Using the values we found earlier:
tan(angle between A and B) = (Vertical component of force B) / (Horizontal component of force B)
By plugging in these values, we can determine the force in chain B and the angle between A and B.
To determine the force in chain B and the angle between chain A and chain B, we can use the principles of vector addition.
First, let's break down the given information:
- Vertical load: 100N
- Force in chain A: 50N
- Angle of chain A with the horizontal plane: 30 degrees
Here's a step-by-step explanation of how to solve the problem:
Step 1: Determine the vertical component of the force in chain A.
To do this, we need to calculate the sine of the angle of chain A with the horizontal plane. The formula is sin(angle) = opposite/hypotenuse. In this case, the opposite side is the vertical component and the hypotenuse is the total force in chain A. So, sin(30°) = vertical component/50. Rearranging the formula, we get vertical component = sin(30°) x 50.
vertical component = sin(30°) x 50 = 0.5 x 50 = 25N
Step 2: Determine the horizontal component of the force in chain A.
To do this, we need to calculate the cosine of the angle of chain A with the horizontal plane. The formula is cos(angle) = adjacent/hypotenuse. In this case, the adjacent side is the horizontal component and the hypotenuse is the total force in chain A. So, cos(30°) = horizontal component/50. Rearranging the formula, we get horizontal component = cos(30°) x 50.
horizontal component = cos(30°) x 50 = √3/2 x 50 = 25√3 N
Step 3: Determine the vertical forces acting on the load.
Since the load is vertically supported by two chains, the vertical forces must balance. The vertical component of the force in chain A is 25N, and the force in chain B contributes to the vertical support. Therefore, the force in chain B must be (100N - 25N) = 75N.
Force in chain B = 75N
Step 4: Determine the horizontal forces acting on the load.
Since the load is supported vertically, the horizontal forces must balance. The horizontal component of the force in chain A is 25√3 N, and the force in chain B contributes to the horizontal support. Therefore, the horizontal component of the force in chain B must be equal to 25√3 N.
Force in chain B (horizontal component) = 25√3 N
Step 5: Determine the angle between chain A and chain B.
To find this angle, we can use the inverse tangent function. The formula is tan(angle) = opposite/adjacent, where the opposite is the vertical component of chain B and the adjacent is the horizontal component of chain B. So, tan(angle) = (vertical component of chain B) / (horizontal component of chain B). Rearranging the formula, we get angle = arctan(vertical component of chain B / horizontal component of chain B).
angle = arctan(vertical component of chain B / horizontal component of chain B)
angle = arctan(75 / 25√3)
angle ≈ 62.9°
Therefore, the force in chain B is 75N, and the angle between chain A and chain B is approximately 62.9 degrees.