Foundations — Tension in inextensible strings

Before you can read the parent note Tension in Inextensible Strings, you must own every symbol it throws at you. Below, each symbol gets three things: plain words → a picture → why the topic needs it. They are ordered so each one leans on the one before.

1. Force $F$ — a push or a pull

The picture. Look at figure s01. The block gets a blue arrow pointing right — that arrow is the force. A longer arrow means a bigger push.

Why the topic needs it. Tension is a kind of force (a pulling one). If you don't picture a force as an arrow with size + direction, you cannot picture tension at all.

2. Mass $m$ and weight $mg$ — the "amount of stuff" and the pull of gravity

The picture. In figure s01, the hanging block has an orange arrow pointing straight down, labelled $mg$. That arrow never changes as long as the mass hangs on Earth.

Why the topic needs it. Every pulley problem starts with a mass being pulled down by gravity ($mg$) and held up by a string. You subtract these two arrows to find the net effect.

3. Acceleration $a$ — how fast the speed is changing

The picture. In figure s02, a green arrow shows the direction the object speeds up toward. It is separate from the velocity — an object can move up while accelerating down (slowing).

Why the topic needs it. The whole payoff of a pulley problem is a number: the acceleration $a$. And the "inextensible" trick is a statement about the two masses' accelerations being equal in size.

4. Newton's Second Law $F=ma$ — force causes acceleration

Reading it as a sentence. "Add every arrow on the object. Whatever's left over ($F_{\text{net}}$) makes it accelerate; a heavier mass accelerates less for the same push."

Why the topic needs it. Every single tension equation in the parent note — like $m_{1}g-T=m_{1}a$ — is just this law written for one body. See Newton's Second Law for the full treatment.

5. Free Body Diagram — drawing only the arrows on ONE object

The picture. Figure s03 shows a hanging block isolated: one orange arrow $mg$ down, one blue arrow $T$ up. That is the entire universe as far as that block is concerned.

Why the topic needs it. You cannot apply $F=ma$ until you know which forces to add. The FBD is the bookkeeping tool that lists them. See Free Body Diagrams.

6. Tension $T$ — the string's pulling force

The picture. In figure s03 the blue $T$ arrow points up, away from the block, along the string toward the pulley.

Why the topic needs it. Tension is the whole topic. Every other symbol here exists so you can write and solve equations involving $T$.

7. Normal force $N$ — the surface's push

Why the topic needs it. In the "block on a table" worked example, the vertical arrows ($N$ up, $mg$ down) cancel so only tension acts horizontally. You need $N$ to justify ignoring the vertical direction. See Normal Force.

8. The inextensible constraint — length never changes

Why the topic needs it. This is the glue. It lets you write one symbol $a$ for both masses instead of two unknowns. It is the seed of Constraint Relations.

9. The symbols $\dot{x}$ and $\ddot{x}$ — "rate of change" dots

Reading the parent's equation. "${\dot{x}}_1+{\dot{x}}_2=0$" simply says: the two ends' speeds are equal and opposite. Adding one more dot gives ${\ddot{x}}_1=-{\ddot{x}}_2$: the accelerations are equal in size, opposite in direction. That is the inextensibility fact in symbol form.

Prerequisite map

Read it top to bottom: forces + mass + acceleration build Newton's law; that plus the FBD lets you write a tension equation per body; the inextensible constraint glues the bodies with one shared $a$; together they are the topic.

Equipment checklist

Cover the right side and test yourself. If any answer is fuzzy, re-read that section before the parent note.

Connections

• Newton's Second Law — the law behind every tension equation.
• Free Body Diagrams — the tool that lists the arrows to add.
• Normal Force — the surface's perpendicular push, cousin of tension.
• Constraint Relations — the general form of "inextensible ⟹ same $a$".
• Atwood Machine — the first place all these symbols combine.
• Frictionless Pulleys vs Pulleys with Inertia — where uniform tension can break.
• Tension in Inextensible Strings — the parent topic this page equips you for.