Pulleys

Pulley questions look harder than they are. There is one counting rule, and once you apply it correctly every question in the topic is a division.

The three arrangements

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A fixed pulley gives no force advantage and is still useful, because pulling down is easier than lifting up - you can use your weight. That is the same point the mechanical advantage lesson makes about an advantage of 1.

A movable pulley gives an advantage of 2 because two rope segments share the load's weight.

The counting rule

Count the rope segments that support the moving load. That number is the mechanical advantage.

Three clarifications that decide most questions:

Count only the segments pulling up on the load or on the movable block. A rope running from an anchor down to a movable pulley and back up counts as two.

Do not count the segment you pull on unless it also runs to the load. In the movable-pulley panel the effort segment goes up and over a fixed point, so it does not support the load - two segments do, and the advantage is 2.

The number of pulleys is not the answer. A system can have four pulleys and an advantage of 4, 5 or something else, depending on how the rope is run. Count the rope, not the wheels.

The block and tackle in the figure has four segments marked between the two blocks, so its advantage is 4. A 400-pound load needs 100 pounds of effort.

The distance trade

The effort moves as many times further as the advantage.

An advantage of 4 means pulling four feet of rope to raise the load one foot. That is the trade from the mechanical advantage lesson, and it is frequently the actual question.

A block and tackle with an advantage of 5 lifts a load 3 feet. How much rope is pulled through?

5 times 3 is 15 feet.

Check with work: if the load is 500 pounds, the effort is 100 pounds. Work out is 500 times 3, which is 1,500 foot-pounds. Work in is 100 times 15, which is 1,500 foot-pounds. Equal.

Speed

The load moves as many times slower as the advantage.

Pulling rope at 4 feet per second in a system with an advantage of 4 raises the load at 1 foot per second. Force is multiplied, speed is divided, and a question asking about the rate of lift is asking about the same trade in different words.

Real systems

Friction reduces the actual advantage below the ideal count, as everywhere else. A block and tackle counted at 4 might deliver 3.5 in practice, and the question asking where the difference went wants friction in the sheaves and stiffness in the rope.

More pulleys means more friction, so there is a practical limit to how far the count can usefully be raised.

Weight of the movable block also counts against you, because the effort has to lift the hardware as well as the load.

What you can skip

Across the 45 questions on this topic in our bank:

• Differential pulleys and chain hoists never appear.
• Efficiency comes up once. The pulley questions count the supporting ropes and assume no friction.

Where people lose points

Counting pulleys instead of rope segments.

Counting the effort segment when it does not support the load.

Thinking a fixed pulley reduces the force. It does not; it redirects.

Forgetting the distance trade when asked how much rope must be pulled.

Expecting the ideal advantage in a real system. Friction takes a share.

Work one in under a minute

A system supports a 600-pound load on three rope segments. What effort is needed, and how much rope must be pulled to raise the load 2 feet?

Advantage is the segment count: 3.

Effort: 600 divided by 3 is 200 pounds.

Rope: 3 times 2 is 6 feet.

Check the work: 200 times 6 is 1,200; 600 times 2 is 1,200. Equal, which confirms both halves of the answer at once.

Where this leads

The segment count is the pulley's version of the arm ratio in a lever, and the distance trade is identical in both.