Connected particles: pulley questions without tears

Every M1 paper has a pulley question. Every one of them can be solved in four steps. Here is the template.

The four-step template

Step 1 — Draw two SEPARATE free-body diagrams, one for each particle. Same tension $T$ on both (same string), pointing INTO the string in each case.

Step 2 — Assume a direction of motion. Draw an acceleration arrow $a$ in that direction on BOTH diagrams. If you assumed wrong, $a$ comes out negative — no drama.

Step 3 — Write Newton's 2nd law for each particle, along the direction of motion.

Step 4 — Add or subtract the two equations to eliminate $T$; solve for $a$; back-substitute for $T$.

The classic worked example

Two masses, 4 kg and 6 kg, connected by a light string over a smooth pulley. The 6 kg is falling. Find $a$ and $T$.

Step 1 — Two diagrams. On the 4 kg: $T$ up, $4g$ down. On the 6 kg: $T$ up, $6g$ down.

Step 2 — Motion: 6 kg falls, 4 kg rises. Both accelerate with the same magnitude $a$.

Step 3 — Newton's 2nd law:

• 6 kg (down positive): $6g - T = 6a$
• 4 kg (up positive): $T - 4g = 4a$

Step 4 — Add:

$2g = 10a \Rightarrow a = \dfrac{g}{5} = 1.96 \text{ m s}^{-2}$

Back-substitute:

$T = 4g + 4a = 4(9.8) + 4(1.96) = 47.04 \text{ N}$

Done.

The variations

One block on a horizontal table, one hanging over the edge. Table block has $T$ pulling it toward the pulley; hanging block has $T$ upward, $mg$ downward. Same template. Include friction on the table block if the surface is rough.

One block on an inclined plane, one hanging. Inclined block has $T$ up the slope, weight component $mg\sin\theta$ down the slope, friction $\mu R$ opposing motion. Everything else is identical.

Both blocks on inclined planes (Atwood variant). Two slopes meeting at a peak, string over the peak. Each block has $T$ up its slope, $mg\sin\theta$ down its slope. Same template, two slope angles.

The common traps

Tension isn't $mg$. Students routinely write $T = mg$ for the hanging block and then can't reconcile with the falling block. If a block accelerates, $T \ne mg$. That's the whole point.

Same tension throughout. A light, inextensible string transmits the SAME tension along its length. Both diagrams get the SAME $T$. Different tensions means the string is stretching or the pulley isn't smooth — a different question.

Sign of acceleration. Pick a direction, stick to it on both diagrams. If your $a$ comes out negative, the motion is opposite to what you assumed — but the magnitude is correct. Don't restart.

Your drill

Find three pulley questions from past M1 papers. Time yourself. Aim for under 6 minutes per question, including the two diagrams.

By the third question, you'll notice the four steps are the same every time. Only the free-body forces change. That's the pattern — commit to it and pulleys stop being a "topic" and become a template.

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