Question

A beam is to be hoisted using two chains. Determine the magnitude of force(FA) and force(FB) acting on each chain in order to develop a resultant force of 600N. Force(FA) makes an angle of 30 degrees with the positive y axis and force(FB) makes an angle of 45 degrees with the positive y axis

Answer 1

The magnitude of force FA is 704.92 N, and the magnitude of force FB is 865.93 N.

Step-by-step explanation:

To determine the magnitude of forces FA and FB acting on each chain in order to develop a resultant force of 600 N, we can use vector addition. We'll use trigonometry to break down the forces into their vertical and horizontal components and then sum them up to find FA and FB.

Let's start with force FA, which makes an angle of 30 degrees with the positive y-axis. We can break down FA into vertical (FvA) and horizontal (FhA) components:

FvA = FA * sin(30°)

FhA = FA * cos(30°)

Similarly, for force FB, which makes an angle of 45 degrees with the positive y-axis, we can break down FB into vertical (FvB) and horizontal (FhB) components:

FvB = FB * sin(45°)

FhB = FB * cos(45°)

Now, we want the resultant force to be 600 N in the positive y-direction. So, the sum of the vertical components of FA and FB should equal 600 N:

FvA + FvB = 600 N

Substituting the expressions for FvA and FvB:

FA * sin(30°) + FB * sin(45°) = 600 N

Next, since there are no horizontal forces acting on the y-axis, the horizontal components should cancel each other out:

FhA + FhB = 0

Substituting the expressions for FhA and FhB:

FA * cos(30°) + FB * cos(45°) = 0

Now, we have a system of two equations:

FA * sin(30°) + FB * sin(45°) = 600 N

FA * cos(30°) + FB * cos(45°) = 0

We can use these equations to solve for FA and FB. Let's solve for FA in terms of FB from equation 2:

FA = -FB * cos(45°) / cos(30°)

Now, substitute this expression for FA into equation 1:

(-FB * cos(45°) / cos(30°)) * sin(30°) + FB * sin(45°) = 600 N

Simplify the equation:

-FB * cos(45°) * tan(30°) + FB * sin(45°) = 600 N

Now, solve for FB:

FB * (sin(45°) - cos(45°) * tan(30°)) = 600 N

FB = 600 N / (sin(45°) - cos(45°) * tan(30°))

Calculate the value of FB:

FB ≈ 865.93 N

Now that we have the value of FB, we can find FA using the expression we derived earlier:

FA = -FB * cos(45°) / cos(30°)

FA ≈ -(865.93 N) * (cos(45°) / cos(30°))

FA ≈ -865.93 N * (0.7071 / 0.866)

FA ≈ -704.92 N

So, the magnitude of force FA is 704.92 N, and the magnitude of force FB is 865.93 N.