What if pulleys had friction?

A pulley is a wheel with a rope around it. Lots of pulleys together can make lifting easy.

But real pulleys rub as they turn. What does the rubbing do?

A pulley is a wheel with a rope running around it. Put several pulleys together, so that several pieces of rope hold up the load, and you can lift something heavy with a much smaller pull.

In a perfect pulley, nothing rubs. But real pulleys turn on axles, and the rope bends around the wheels, so some rubbing, or friction, is always there. So what if we add friction to an ideal pulley? In this lab you can turn friction up in a model and watch what changes.

Textbook pulley problems usually assume frictionless pulleys and massless, inextensible ropes. OpenStax notes that if the pulleys are friction-free, the force a pulley system delivers is about a whole-number multiple of the rope tension, set by how many segments pull on the load.

This lab drops the friction-free assumption. Using a simple-machines model, it compares the same four-segment block and tackle lifting 100 kg at 100 percent and at 60 percent efficiency, and asks what friction changes and what it does not.

Olly's crate weighs 100 kilograms. Four ropes hold it up.

Each rope holds a quarter. So the crate feels like only 25 kilograms!

In Olly's model, a 100-kilogram crate hangs from a moving pulley held up by four pieces of rope. With no friction, each piece carries a quarter of the load, so you only need to pull with about 245 newtons. That feels like lifting 25 kilograms instead of 100.

There is a catch: to lift the crate 1 meter, you must pull 4 meters of rope. You pull less hard, but for longer. The work you put in, force times distance, is the same 981 joules as lifting the crate straight up.

For an ideal system, the mechanical advantage equals the number of supporting segments, n = 4. The pull is m·g/n = 100 × 9.81 / 4 ≈ 245 N, and each meter of lift needs n = 4 m of rope. Work in = 245 N × 4 m = 981 J, exactly the work done on the load, m·g·h = 981 J.

This is the energy bookkeeping behind every simple machine. As OpenStax puts it, a machine cannot do more work than the energy put into it, but it can reduce the input force needed by spreading it over a longer distance.

Now Olly adds friction. The pulleys rub as they turn.

The crate is just as heavy, but Olly has to pull harder. It now feels like about 42 kilograms.

Now set the model's efficiency to 60 percent. That means only 60 percent of your work reaches the crate; friction takes the rest. With the same four ropes and the same crate, your pull rises from about 245 newtons to about 409. The crate now feels like lifting about 42 kilograms.

Some things stay the same. You still pull 4 meters of rope for each meter the crate rises, and the crate still gains the same 981 joules of energy. But the work you put in grows to about 1,640 joules. About 650 joules go into friction.

The model divides the ideal pull by the efficiency: F = m·g / (n·η) = 981 / (4 × 0.6) ≈ 409 N, which feels like 41.7 kg. Rope travel is unchanged at 4 m per meter of lift, because that follows from geometry, not friction. Work in rises to 409 N × 4 m ≈ 1,640 J against 981 J of work on the load, so about 654 J goes to friction.

A useful comparison: at 60 percent efficiency, even six segments need about 273 N, more than four ideal segments (245 N). At 50 percent, four segments need 491 N, the same as two ideal ones.

Rubbing makes things warm. Rub your hands together and feel!

Friction in a pulley turns some of your pulling into a little bit of heat.

Energy never disappears, but it can change into forms we can't use. When the axle and the rope rub as the pulley turns, friction turns some of your work into thermal energy: the parts warm up very slightly. That energy doesn't lift the crate, so you have to supply extra.

Scientists measure this with efficiency: the useful work out divided by the energy put in. A perfect pulley would be 100 percent efficient. Real machines are always less.

OpenStax defines efficiency as useful work output divided by total energy input. Friction is a nonconservative force: it converts mechanical energy into thermal energy that leaves the system and cannot be fully recovered as work. So η = W_out / W_in < 1, and with fixed W_out the input work and force must rise.

The model is simpler than real hardware. It uses one fixed efficiency, whatever the load or the number of pulleys. Real friction grows with load, and every extra sheave adds its own. The model also treats the ropes and pulleys as weightless and every rope segment as vertical, so it may understate the losses of a system with many pulleys.

Engineers try to make machines smooth and slippery.

Oil helps metal parts slide past each other much more easily.

Engineers work hard to cut friction in machines. Oil is one of the best tricks. OpenStax lists the friction between two sliding pieces of dry steel and two pieces of oiled steel: with oil, the sliding friction is about ten times smaller.

Some machines are very efficient. OpenStax gives a typical electric motor an efficiency of about 98 percent, while a gasoline engine is about 30 percent. Real pulleys always lose some work to friction too, so the real pull is always a bit more than the ideal one.

Lubrication changes friction dramatically. OpenStax's table of friction coefficients gives steel on dry steel a kinetic coefficient of 0.3 and oiled steel 0.03, ten times lower; it adds that atomic-scale research on friction could save energy now converted unnecessarily to heat.

Efficiencies vary widely between machines. OpenStax lists typical values of about 98 percent for an electric motor, 30 percent for a gasoline engine and 9 percent for the human body during weightlifting. For a pulley system, the efficiency depends on its bearings, the rope and how many sheaves it has; the model's single slider stands in for all of these.