Questions: Moment of a Force: Concepts and Calculation

Question 1

A horizontal force of 10 N acts at a point 3 m to the right of a pivot, at the same height as the pivot (so the position vector from pivot to application point is purely horizontal, parallel to the force). What is the moment about the pivot?

• A. 30 N·m, because the distance from the pivot to the application point is 3 m
• B. 0 N·m, because the force's line of action passes through the pivot — the moment arm is zero
• C. 30 N·m counterclockwise, by the right-hand rule
• D. 10 N·m, because only the perpendicular component of the distance matters

Answer: (B)

The moment arm is the perpendicular distance from the pivot to the *line of action* of the force — not the distance to the application point. Here, the force is horizontal and the application point is also directly horizontal from the pivot. The line of action (extended in both directions) passes straight through the pivot's height, so the perpendicular distance is zero and M = F × 0 = 0 N·m. Option A is the classic misconception: students use the distance to the application point (3 m) instead of the perpendicular distance to the line of action.

Question 2

A 20 N force is applied at 30° above horizontal at a point 2 m directly to the right of a pivot (at the same height as the pivot). Using Varignon's theorem, what is the moment about the pivot?

• A. 40 N·m, because M = F × d = 20 × 2
• B. 34.6 N·m, because M = F × d × cos30°
• C. 20 N·m, because only the vertical force component (20sin30° = 10 N) has a nonzero moment arm (2 m), giving 10 × 2 = 20 N·m
• D. 17.3 N·m, because only the horizontal component does work against the rotation

Answer: (C)

Varignon's theorem: decompose F into Fx = 20cos30° ≈ 17.3 N (horizontal) and Fy = 20sin30° = 10 N (vertical). The application point is 2 m to the right of and at the same height as the pivot. The moment from Fx: its line of action is horizontal at the pivot's height, so perpendicular distance = 0, moment = 0. The moment from Fy: it acts vertically at a horizontal distance of 2 m from the pivot, so moment = 10 × 2 = 20 N·m. Total = 20 N·m. Option A ignores the angle entirely; options B and D apply the angle incorrectly.

Question 3

The moment of a force about a point is zero whenever the line of action of that force passes through the point, regardless of how large the force is.

• T. True
• F. False

Answer: True

The moment is M = F × d, where d is the perpendicular distance from the reference point to the line of action. If the line of action passes through the reference point, d = 0, so M = 0 no matter how large F is. Physically: a force directed exactly at the pivot cannot cause rotation about that pivot — it can only push or pull the pivot itself. This is why you can push a door all day if you push along the hinge axis: zero moment, zero rotation.

Question 4

The moment produced by a force about a pivot depends on the distance from the pivot to the specific point where the force is applied to the body.

• T. True
• F. False

Answer: False

The moment depends on the perpendicular distance from the pivot to the *line of action* — not to the specific application point. Two forces with the same magnitude and direction but applied at different points along the same line of action produce identical moments about any reference point. This is the principle of transmissibility: a force can be 'slid' along its line of action without changing its moment about any external point. The location of the application point matters only when computing moments about a point not on the line of action.

Question 5

Explain Varignon's theorem and why it is useful. What does it allow you to do instead of finding the perpendicular distance to the line of action directly?

Model answer: Varignon's theorem states that the moment of a force about a point equals the sum of the moments of the force's rectangular components about the same point. Instead of constructing the perpendicular from the pivot to the oblique line of action (which requires trigonometry or geometric constructions), you decompose the force into horizontal and vertical components and compute each component's moment using simple right-angle geometry — usually one component acts through the pivot (moment = 0) and the other acts at a straightforward perpendicular distance.

In practice, most free-body diagram problems involve forces at angles, and the true perpendicular distance to an oblique line of action is cumbersome to find geometrically. Varignon's theorem turns every moment calculation into two simple multiplications: (x-coordinate of application point) × (vertical force component) minus (y-coordinate) × (horizontal force component), which is exactly the cross product M = x·Fy − y·Fx. This is why the cross product formulation and Varignon's theorem are equivalent approaches — they both decompose the moment into orthogonal contributions.