Q.Give an example to show that the following statement is false: Any two forces acting on a body can be combined into a single force that would have the same effect.

Step 1. Restate precisely what the claim asserts, so it is clear what must be shown false.

The statement claims that for ANY two forces acting on a body, however they are applied, there exists some single equivalent force that could be applied instead and produce exactly the same overall mechanical effect (same net translational tendency AND same net rotational tendency). To disprove a universal claim like this, it is enough to exhibit ONE genuine counterexample where no such single force can possibly exist.

Step 2. Construct the counterexample — a couple.

Consider a thin rigid rod, and apply two forces of EQUAL magnitude $F$, in exactly OPPOSITE directions, but along two PARALLEL lines of action separated by some nonzero perpendicular distance $2r$ (for concreteness, one force pushing 'up' at one end of the rod, the other pushing 'down' at the other end) — exactly the configuration of a couple, as arises when turning a steering wheel with both hands, twisting a screwdriver, or opening a bottle cap.

Step 3. Compute the net force of this pair — it is exactly zero.

Since the two forces are equal in magnitude and exactly opposite in direction, their vector sum is:

The couple therefore produces NO net force whatsoever on the body — it does not tend to accelerate the body's center of mass in any direction at all.

Step 4. Compute the net torque of this pair — it is NOT zero.

Taking torques about the midpoint (or any point) on the rod: even though the two forces point in opposite directions, they act on OPPOSITE sides of the reference point, so both individually produce torque in the SAME rotational sense (both, say, anticlockwise) rather than cancelling. Adding the two individual torques (each of magnitude $Fr$) gives a net torque:

So the couple produces a genuine, nonzero turning effect — it tends to spin the body, purely rotationally, with no accompanying translation at all.

Step 5. Show that NO single force, of any magnitude or location, can reproduce both effects simultaneously.

Suppose, for contradiction, that some single equivalent force $\vec F_{eq}$ (applied at whatever point one likes) could reproduce the couple's combined effect. Two cases exhaust all possibilities:

• If $\vec F_{eq}=\vec 0$ (to match the couple's zero net force), then automatically its torque about any point is also zero ($\vec\tau=\vec r\times\vec 0=\vec 0$) — but the couple's actual torque is $2Fr\ne0$, so this fails to reproduce the torque. …

• Trying to find a counterexample where the two forces simply don't add to a 'nice' resultant, rather than specifically constructing a COUPLE — the couple is the sharpest counterexample because its net force is exactly zero while its torque is exactly nonzero, which is what makes single-force equivalence logically impossible, not just numerically awkward.
• Forgetting to check BOTH conditions (net force AND net torque) — a counterexample must show that no single force can match the force sum and the torque sum simultaneously, not just that the two forces 'look complicated'. …