An engineer needs to apply a pure 60 N·m torque to a bolt using a couple — two opposing 30 N forces separated by 2 m. At which location on the structural member should she apply this couple to ensure 60 N·m acts at the bolt?
ADirectly at the bolt — moments depend on position, so the couple must act there
BSymmetrically about the bolt centerline to cancel any net force
CIt does not matter — the couple's moment is a free vector and its effect is identical anywhere on the body
DAt least 2 m from the bolt to avoid geometric interference
A couple's moment is a free vector — it has magnitude and direction but no fixed point of application. Moving a couple anywhere on a rigid body produces the same rotational effect because all reference-point terms cancel when computing the total moment of two equal and opposite forces. The engineer can place it wherever is mechanically convenient; the 60 N·m torque is transmitted identically to the bolt regardless of placement.
Question 2 Multiple Choice
Two forces act on a rigid bar: 15 N upward at point A and 15 N downward at point B, separated by 4 m. What is the net mechanical effect on the bar?
AA net upward force of 30 N with no rotational effect
BNo net force and a pure moment of 60 N·m — a couple
CA net force of 15 N downward and a moment of 60 N·m about A
DNo effect — equal and opposite forces cancel completely
Equal-magnitude, opposite-direction, parallel forces form a couple. The net force is 15 − 15 = 0 (no translation). The moment is M = F × d = 15 × 4 = 60 N·m (pure rotation). Option D is wrong because the forces are offset — even with zero net force, the separation creates a moment. Option C is wrong because the forces are equal and opposite, giving zero net force.
Question 3 True / False
The moment produced by a couple about point A differs from the moment computed about a different point B elsewhere on the same rigid body.
TTrue
FFalse
Answer: False
This is the defining property of a couple. When computing the total moment of two forces F and −F at positions r₁ and r₂, the result is (r₁ − r₂) × F = d × F. The position vectors r₁ and r₂ both shift equally when the reference point changes, so their difference (r₁ − r₂) remains constant. The moment center cancels completely, giving M = F·d regardless of reference point — the same value about every point.
Question 4 True / False
A couple may be relocated to any point on a rigid body without changing the body's mechanical response.
TTrue
FFalse
Answer: True
Because a couple's moment is a free vector — independent of position — it can be placed anywhere on (or off) the rigid body without altering its rotational effect. This distinguishes couples from forces, which are sliding vectors that may only move along their line of action. The freedom of the couple vector is why, when simplifying complex force systems, any couple moment can be relocated to the most convenient reference point.
Question 5 Short Answer
Why does a single force's moment change when the reference point is moved, but a couple's moment remains the same regardless of reference point?
Think about your answer, then reveal below.
Model answer: A single force F at position r₁ produces moment r₁ × F about the origin. Move the reference to point P: the moment becomes (r₁ − P) × F, which depends on P. A couple has forces F and −F at positions r₁ and r₂. Total moment about P: (r₁ − P) × F + (r₂ − P) × (−F) = (r₁ − r₂) × F. The P terms cancel algebraically, leaving only the relative displacement between the two forces. The reference point has no effect on the result.
This algebraic cancellation is why the couple's moment is called a free vector — it genuinely has no preferred point of application. A single force's moment is bound to a specific line of action; the couple's moment floats freely in space while retaining its full mechanical meaning. This makes couples especially useful in equivalent force system analysis: the couple moment can slide to any convenient location during simplification.