Given the FBD of member BC, what is the correct equilibrium equation from summing moments about point C?

Question

BlackTom AI · Accepted Answer

First, let’s restate the problem in our own words: we are summing moments about point C for the member BC, given the FBD with a force FAB acting at B at 60° to the beam and a 2000 N vertical load applied at the midspan (2 m from each end). The aim is to write the correct equilibrium moment equation about C.

Option A: - FABcos(60°) (4m) = 2000 (2m)
- This uses the cosine component of FAB rather than the sine component. If the perpendicular (moment arm) component of FAB about C is used, it should be the component perpendicular to the BC direction. Given the geometry, the contributing component to the moment about C is FAB sin(60°), not FAB cos(60°). Therefore this option misidentifies the correct perpendicular component and is incorrect.

Option B: - FABsin(60°) (4m) = 2000 (2m)
- This writes the moment due to FAB with a negative sign and equates it to 2000(2m) on the right. The left side captures the moment from FAB about C using the perpendicular component FAB sin(60°) with a lever arm of 4 m, which is plausible. However, the sign convention here places both terms on opposite sides of the equation without an explicit zero or balanced sum. Depending on the chosen sign convention (clockwise vs. counterclockwise), this form may be incomplete or not set to zero. By itself, as written, it isn’t the standard balanced moment equation about C (which is typically sum of moments equals zero). This makes it an incomplete representation under a standard equilibrium statement.

Option C: - FABsin(60°) (4m) + 2000 (2m) = 0
- This option expresses the sum of the two moments about C equaling zero, combining the FAB moment with the 2000 N moment with opposite senses. Using FAB sin(60°) with a 4 m moment arm is consistent with the perpendicular component and distance from C to B. The plus sign before 2000(2m) indicates opposite sense to the FAB moment, which aligns with a typical equilibrium statement (sum of moments about C equals zero). This is a plausible and correct format for the equation.

Option D: - FABcos(60°) (4m) + 2000 (2m) = 0
- This uses FAB cos(60°) instead of FAB sin(60°), which, as discussed earlier, does not represent the correct perpendicular component responsible for the moment about C given the geometry. Therefore, this option misrepresents the contributing moment and is incorrect.

Putting it together, the correct moment equation about C should balance the moment from the perpendicular component of FAB (which is FAB sin(60°)) with the moment from the 2000 N load (2000 × 2 m), arranged so that the sum equals zero. Among the choices, Option C provides the properly balanced sum with the correct perpendicular component and a zero-sum form. The other options either pick the wrong trigonometric component, misstate the sign convention, or fail to present a proper equilibrium equation.

Given the FBD of member BC, what is the correct equilibrium equation from summing moments about point C? 单项选择题

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