Determine the location at the center of the column loadings of F1 and the magnitude of the resultant force.
Understand the Problem
The question asks to determine the location at the center of the column loadings of F1 and the magnitude of the resultant force.
Answer
The location is $1.64 \text{ m}$ and the resultant force is $700 \text{ N}$.
Answer for screen readers
The location at the center of the column loadings is $1.64 \text{ m}$ from point A, and the magnitude of the resultant force is $700 \text{ N}$.
Steps to Solve
- Calculate the resultant force
The resultant force $F_R$ is the sum of all the vertical forces. Here, the forces are 100 N, 200 N, 250 N, and 150 N. $$ F_R = 100 + 200 + 250 + 150 = 700 \text{ N} $$
- Calculate the moment about point A
The moment about point $A$ is the sum of the moments caused by each force. The distances are 0 m, 1 m, 2 m, and 3 m. $$ M_A = (100 \times 0) + (200 \times 1) + (250 \times 2) + (150 \times 3) $$ $$ M_A = 0 + 200 + 500 + 450 = 1150 \text{ Nm} $$
- Determine the location of the resultant force (x)
The location $x$ of the resultant force is given by the total moment about $A$ divided by the resultant force. $$ x = \frac{M_A}{F_R} = \frac{1150}{700} $$ $$ x = 1.64 \text{ m} $$
The location at the center of the column loadings is $1.64 \text{ m}$ from point A, and the magnitude of the resultant force is $700 \text{ N}$.
More Information
The center of the column loadings represents the point where the resultant force would act to produce the same overall effect as the individual forces. This is a key concept in statics for simplifying force systems.
Tips
A common mistake is to incorrectly calculate the moment by using the wrong distances or signs. Another mistake is to add the distances instead of multiplying them by the corresponding forces when computing the moment. Also, students sometimes forget units, which is critical.
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