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Skill 3 of 9
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# Newton’s 3rd law vs 2nd law

• Newton’s 2nd law told us that unbalanced forces cause acceleration.
• But Newton’s 3rd law now tells that forces always come in pairs, equal but opposite to each other.

If forces always come in pairs, why are they then not always balanced? How can unbalanced forces ever exist if all forces always have an equal but opposite copy? Wouldn’t any force be cancelled out by its reaction force? How can anything ever accelerate according to Newton’s 2nd law if Newton’s 3rd law is true?

Because the two laws don’t apply on the same objects!

• Newton’s 2nd law applies on a single object (or system), whereas
• Newton’s 3rd law applies between two objects (or systems).

At a fair you throw a rice bag to knock down a tin can. It hits the top can with an impact force. Newton’s 3rd law tells that the bag itself feels the same impact force the opposite way. Newton’s 3rd law tells the relation between the two.

$\displaystyle \text{Newton’s 3rd law: }$
$\displaystyle \vec F_\text{A on B}=- \vec F_\text{B on A}\quad\Leftrightarrow\quad F_\text{bag on can}=F_\text{can on bag}$1

Newton’s 2nd law can then be used on one of them to figure out how they are influenced individually. The can feels only one impact force and thus accelerates; the bag also feels only one impact force the other way and thus accelerates.2

\text{Newton’s 2nd law:}\\ \begin{align} \text{Tin can: }\qquad \sum F_x&=m_\text{can}a_x\quad \Leftrightarrow \quad -F_\text{bag on can}=m _\text{can} a _\text{can} \quad \Leftrightarrow \quad -\overbrace{F_\text{can on bag}}^{ F_\text{bag on can} }=m _\text{can} a _\text{can} \\ ~&~ \\ \text{Rice bag: }\qquad \sum F_x&=m _\text{bag} a_x\quad \Leftrightarrow \quad F_\text{can on bag}=m _\text{bag} a _\text{bag} \end{align}

So, luckily, Newton’s 2nd and 3rd laws are not in conflict after all.