Intuition The ONE core ideaTwo weights hang from the two ends of a single rope draped over a wheel; because the rope cannot stretch, they are forced to move as one coupled system with the same speed . The heavier side wins, but it has to drag the lighter side along, so it falls slower than free-fall — and that slowing-down is the whole story.
Before you can read the derivation, you need to own every symbol it uses. This page builds each one from a picture, in the order they depend on each other. Nothing here assumes you have seen the parent derivation yet.
Definition "String" and "rope" mean the same thing hereThroughout this page (and the whole topic) the words string and rope refer to the same single idealised object : one thin cord that cannot stretch and has no weight. We use both words only for readability — every property (inextensible, massless) applies to it under both names.
Let's first fix the physical picture so every later symbol has a home. In the figure below, notice the labels: the wheel is the pulley , the cord is the string , and there are two blocks hanging from the two ends. The small arrows show the directions of motion when the left block is the heavier one. (We give the blocks their symbols m 1 m 2
Definition The setup in wordsA pulley (a wheel that spins freely) sits at the top. A string hangs over it. On the left end is one block; on the right end is another block. Gravity pulls both blocks down; the string pulls both blocks up toward the wheel. If the left block is heavier, it moves down and the right block moves up (the labelled arrows in the figure).
Everything below is just a careful naming of the things in that one picture.
m Plain words: how much stuff is in an object — how hard it is to get moving or to stop. Measured in kilograms (kg).
The picture: a block. A bigger block = bigger m Why the topic needs it: the two blocks are our whole cast, so we give each a mass symbol: the left block has mass m 1 m 2 1 2
Common mistake "Mass is the same as weight."Why it feels right: heavier things have more mass, so they blur together.
Why it's wrong: mass (m amount of stuff ; weight is a force the Earth exerts on that stuff. Same block on the Moon: same mass, less weight. Fix: keep m m g g
Why do the blocks move at all? Because the Earth pulls them down. We need a number for "how strongly."
Definition Gravitational field strengthg Plain words: how many metres-per-second of downward speed the Earth adds every second to anything it drops. Near Earth's surface g ≈ 9.8 m/s 2 10 The picture: a ball let go from rest — after 1 s it moves at g 2 g Why the topic needs it: g
Force needed to hold something up scales with its mass: twice the stuff → twice the pull. The proportionality constant is g = m × g
Before we can talk about "positive directions," we need the quantity that has a direction and that we ultimately solve for. It is built from one earlier word: velocity .
Definition Speed and velocityPlain words: speed is how fast something moves (e.g. 3 Velocity is speed plus its direction (e.g. 3 downward ). For our blocks, which move straight up or down, the two are the same number with a sign.
The picture: a speedometer needle (speed) together with a little arrow saying which way (direction).
Why the topic needs it: acceleration is defined as a change of velocity , so we must have "velocity" in hand first.
a Plain words: how quickly velocity changes — how fast the speed is ramping up. Units m/s 2 The picture: a speedometer whose needle is climbing . If it climbs by 2 a = 2 m/s 2 Why the topic needs it: a both blocks share the same a .
F Plain words: a push or a pull. It has a size (in newtons) and a direction .
The picture: an arrow. Length = strength, arrowhead = which way it pushes.
Why the topic needs it: every physics move here is "add up the arrows on one block."
Because forces have direction, we must decide which direction counts as positive before adding them. That single choice is where most beginners slip — so let's make it a picture too.
Definition Sign conventionPlain words: pick one direction and call it "+". Any force along it is positive; any force against it is negative. You may pick a different positive direction for each block — that is completely allowed and it is exactly what the derivation does.
The picture: a little "+" arrow drawn next to each block.
Why the topic needs it: we take down = + for the falling block and up = + for the rising block, so that both accelerations a same positive number .
The string pulls. That pull has its own name.
T Plain words: the pulling force carried inside a rope. A rope can only pull , never push — so tension always points away from the block, toward the pulley .
The picture: two arrows on the rope pointing outward to each end; on a block, the tension arrow points up along the string.
Why the topic needs it: T a
T same on both sides?
Because the string has (idealised) zero mass . Newton's law says F n e t = ma 0 no matter its acceleration , the net force on any piece of string must be 0 T Tension in Strings .
Two idealised words in the definition do enormous work. Let's earn them.
Plain words: the string cannot stretch — its total length never changes.
The picture: if the left block drops by a distance x somewhere , so the right block must rise by exactly the same x Why the topic needs it: equal displacement at every instant ⇒ equal speed ⇒ equal acceleration magnitude . This is the whole constraint that couples the blocks.
Definition Massless & frictionless pulleyPlain words: the wheel weighs nothing and spins with zero resistance.
The picture: the rope simply bends over a smooth peg.
Why the topic needs it: the pulley then only redirects the rope — it neither eats force nor adds any, so tension is unchanged going over it. See Pulley Systems & Mechanical Advantage .
Everything above pours into a single law.
F n e t Plain words: F n e t single leftover arrow you get after adding all the force arrows on one block, each with its sign. If gravity is − m g + T F n e t = T − m g The picture: many arrows on a block collapsed into one "winner" arrow.
Why the topic needs it: Newton's law below is stated in terms of this total, not of any single force.
Intuition Why THIS law and not another?We want to connect forces (gravity, tension) to motion (acceleration). Newton's Second Law is precisely the bridge between "what pushes" and "how it moves" — so it is the only tool that answers our question. See Newton's Second Law .
To apply it we draw a free body diagram : isolate one block, draw only the arrows touching it (gravity down, tension up), then add them into F n e t
Recall Free body diagram, in one lineQuestion: what two forces act on each hanging block? ::: gravity m g T Free Body Diagrams .
Intuition What if the two masses are equal?If m 1 = m 2 nothing accelerates : a = 0 a = 0
Intuition Which side is heavier does not change the physicsThe derivation usually names m 1 m 1 m 2 m 2 > m 1 m 2 down and m 1 up — just mirror the arrows. The very same formulas apply, because the physics only cares about the difference and total of the two masses, not which one you wrote first. Always let the heavier mass be the one that descends.
The tension answer will turn out to look like m 1 + m 2 2 m 1 m 2 g shape , and notice the g force (newtons), so it must carry a g
Definition Harmonic-mean-type combinationPlain words: the mass part m 1 + m 2 2 m 1 m 2 smaller of two masses; multiplying it by g The picture: for m 1 = 5 kg , m 2 = 3 kg 8 2 ⋅ 5 ⋅ 3 = 3.75 kg 3 5 3.75 g Why the topic needs it: the tension formula is symmetric in m 1 , m 2 Harmonic Mean .
Read this map top-to-bottom: the plain-language boxes at the top are the building blocks; the arrows show which idea feeds into the next; everything funnels into the final Atwood derivation at the bottom.
Weight equals mass times g
Newtons Second Law net force equals mass times acceleration
Sign convention plus and minus
Equal acceleration for both blocks
Massless frictionless pulley
Atwood derivation solve for a and T
Cover the right side and test yourself. If any answer surprises you, re-read that section before the derivation.
What does m Amount of stuff (inertia), in kilograms.
What is g How much downward speed (m/s) gravity adds each second; ≈ 9.8 m/s 2
Write the weight of a block of mass m m g
What is the difference between speed and velocity? Speed is how fast; velocity is how fast plus the direction.
What does the symbol a Acceleration — how quickly velocity changes, in m/s 2
What does F n e t The single leftover force after adding all force arrows on a body (with signs).
What can a string (rope) do — push, pull, or both? Only pull; tension points from the block toward the pulley.
Do "string" and "rope" mean different things here? No — they are the same idealised inextensible, massless cord.
Why is tension T The string is massless, so net force on any piece must be zero.
What does "inextensible" force about the two blocks? They have equal displacement, speed, and acceleration magnitude.
Why is a sign convention needed before adding forces? Forces have direction; you must fix which way is "+" to add them correctly.
State Newton's Second Law. F n e t = ma
What does a free body diagram show for one hanging block? Only gravity m g T
If m 1 = m 2 a = 0
If m 2 > m 1 m 2
Why is the tension formula symmetric in m 1 m 2 Swapping the two blocks cannot change the rope's pull.
