Questions: Rigid Body Equilibrium: Planar Analysis

Question 1

A horizontal beam is supported by a pin at point A (left end) and a roller at point B (right end), with a downward load applied at the midpoint. You want to find the roller reaction at B using a single equation. What is the most efficient strategy?

• A. Sum all forces in the x-direction and set equal to zero
• B. Sum all forces in the y-direction and solve — B appears in the equation with other unknowns
• C. Take moments about point A, eliminating both pin reaction components and solving for B directly
• D. Take moments about the midpoint of the beam, where moments from symmetric loads cancel

Answer: (C)

Taking moments about point A is the efficient choice because the pin at A introduces two unknown force components (Ax and Ay), both of which act through point A and therefore contribute zero moment about A. They vanish from the moment equation entirely, leaving a single equation with B as the only unknown — solvable in one step. Summing forces in y (option B) includes both Ay and B as unknowns, requiring simultaneous equations. This 'strategic moment center' technique is the core skill of efficient planar equilibrium analysis.

Question 2

A beam is attached to a wall with a fixed (cantilever) support at one end. How many unknown reaction components does this support provide?

• A. One — a force perpendicular to the wall surface
• B. Two — horizontal and vertical force components
• C. Three — horizontal force, vertical force, and a moment couple
• D. Four — two force components and two independent moment components

Answer: (C)

A fixed support prevents all translation and all rotation at the attachment point. In 2D, preventing translation requires two force components (x and y); preventing rotation requires a moment couple. This gives three unknowns. A roller provides only one unknown (force perpendicular to the surface); a pin provides two (x and y force components, free to rotate). Correctly counting unknowns per support type is essential for determining whether a system is statically determinate before setting up equations.

Question 3

For a planar rigid body in equilibrium, the moment equation ΣM = 0 can primarily be validly applied about the body's centroid.

• T. True
• F. False

Answer: False

The moment equation ΣM = 0 is valid about any point in the plane — not just the centroid, not just a support point. The choice of moment center affects which unknowns appear in the equation (forces through the chosen point contribute zero moment), but the equilibrium condition itself holds about any point. This freedom to choose the moment center strategically — placing it where unknowns intersect to eliminate them from the equation — is the key to efficient analysis.

Question 4

A planar rigid body supported by two pins has more unknown reaction components than the three equilibrium equations can solve, making it statically indeterminate.

• T. True
• F. False

Answer: True

Each pin support in 2D provides two unknown force components (x and y). Two pins together give four unknowns, but planar equilibrium provides only three independent equations (ΣFx = 0, ΣFy = 0, ΣM = 0). With four unknowns and three equations, the system is statically indeterminate to the first degree — you cannot find the reactions from equilibrium alone. Solving it requires incorporating the body's deformation (compatibility equations from mechanics of materials). Recognizing this before writing equations saves significant wasted effort.

Question 5

What is the strategic advantage of choosing a moment center at the intersection of multiple unknown forces, and how does this simplify equilibrium analysis?

Model answer: A force contributes zero moment about any point on its line of action — because the perpendicular distance from that point to the force's line of action is zero. If the moment center lies at the intersection of two unknown forces, both of those unknowns drop out of the moment equation simultaneously, leaving a simpler equation (possibly with only one unknown) that can be solved directly without simultaneous equations. For a beam with a pin at A, taking moments about A eliminates both pin components at once, allowing the roller reaction to be found in one step. This reduces the algebra from a system of equations to a single equation.

This technique scales: if you choose a moment center at the intersection of all unknowns except one, you solve for that remaining unknown immediately. Strategic moment center selection is what separates an efficient statics solution from a clumsy one that solves three equations in three unknowns simultaneously when one equation would have sufficed.