The bracket is subjected to the two forces shown. Express each force in Cartesian vector form and then determine the resultant force FR. Given that F2 = 410 N. Express your answer to three significant figures and include the appropriate units.
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a) Find the magnitude of the resultant force.
b) Find coordinate direction angle α of the resultant force.
c) Find coordinate direction angle β of the resultant force.
d) Find coordinate direction angle γ of the resultant force.
To solve this problem, we need to express each force in Cartesian vector form and then determine the resultant force. Let's follow these steps:
Step 1: Express each force in Cartesian vector form:
From the given picture, we can see that the forces F1 and F2 are acting at different angles.
Force F1 can be broken down into its Cartesian vector components as follows:
F1x = F1 * cosα
F1y = F1 * sinα
Similarly, Force F2 can be broken down into its Cartesian vector components as follows:
F2x = F2 * cosβ
F2y = F2 * sinβ
Step 2: Determine the resultant force (FR):
The resultant force can be found by adding the vector components of F1 and F2:
FRx = F1x + F2x
FRy = F1y + F2y
Step 3: Calculate the magnitude of the resultant force (FR):
The magnitude of the resultant force (FR) can be found using the Pythagorean theorem:
|FR| = sqrt(FRx^2 + FRy^2)
To find the values of α, β, and γ (the coordinate direction angles), we use the following formulas:
α = arctan(F1y / F1x)
β = arctan(F2y / F2x)
γ = arctan(FRy / FRx)
Now, let's apply these steps to solve the problem:
Given: F2 = 410 N
Step 1: Express each force in Cartesian vector form:
From the picture, it is not possible to accurately determine the values of α and β. Please provide the values of α and β.
Step 2: Determine the resultant force (FR):
FRx = F1x + F2x
FRy = F1y + F2y
Step 3: Calculate the magnitude of the resultant force (FR):
|FR| = sqrt(FRx^2 + FRy^2)
With the values of α and β, we can calculate the values of α, β, and γ using the formulas mentioned above.