True or false on Newton's laws

Problem

True or false?

• (a) A zero force implies zero velocity.
• (b) If two forces have zero resultant, the body can move at constant velocity.
• (c) Action and reaction cancel each other out.
• (d) The weight of a body changes with latitude.
• (e) No forces act on a body at rest.

Solution

(a) FALSE. The second law relates the resultant force to the acceleration, not the velocity: $\vv{F} = m\,\vv{a}$. If the resultant is zero, then $\vv{a} = 0$, meaning the velocity remains constant, but not necessarily zero. A body can move indefinitely in uniform straight-line motion in the absence of forces (the principle of inertia). Confusing “constant velocity” with “zero velocity” is the Aristotelian error that Galileo and Newton corrected.

(b) TRUE. A zero resultant means $\vv{a} = 0$, so the velocity does not change: if the body was already moving, it continues at constant velocity (in magnitude, direction and sense). This is exactly the content of the first law. Rest and uniform straight-line motion are both compatible with a zero resultant.

(c) FALSE. Third-law forces (action and reaction) do indeed have equal magnitude and opposite direction, but they act on different bodies: the action on one body, the reaction on the other. For two forces to cancel each other out they would have to act on the same body. Precisely because they act on distinct bodies, the action-reaction pair never automatically produces equilibrium and allows motion (e.g. walking, rowing, recoil).

(d) TRUE. Weight is $P = mg$: the mass $m$ is invariant, but the acceleration due to gravity $g$ is not the same everywhere on Earth. Because of the planet’s not-quite-spherical shape (flattening at the poles) and the effect of Earth’s rotation, $g$ is slightly greater at the poles ($\approx 9{,}83\;\text{m/s}^2$) than at the equator ($\approx 9{,}78\;\text{m/s}^2$). So the same body weighs a little more at high latitudes.

(e) FALSE. “At rest” means zero resultant, not the absence of forces. A book on a table is at rest, yet the weight force (downward) and the table’s normal reaction (upward) both act on it: two non-zero forces that balance each other. Equilibrium is the result of forces that compensate one another, not of their absence.

$\ev{\text{(a) F} \quad \text{(b) T} \quad \text{(c) F} \quad \text{(d) T} \quad \text{(e) F}}$

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Topics: Dynamics Concepts: Newton’s first law · Newton’s second law · Newton’s third law · Weight force