Torque and Rotation

Torque is the rotational version of force, and it is the quantity that levers, gears and wheels are all really about. Naming it makes those three topics one.

The definition

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Torque = force x lever arm.

The units are pound-feet, pound-inches or newton-meters - a force unit times a distance unit, which is how you recognize a torque value.

The lever arm is measured from the pivot, perpendicular to the force's line of action.

A 20-pound force on a wrench 6 inches from the nut.

20 times 6 is 120 pound-inches.

The same force on a 12-inch wrench.

20 times 12 is 240 pound-inches - twice the turning effect for exactly the same push, which is the figure's middle panel and the whole reason long-handled tools exist.

The longer handle

Doubling the handle length doubles the torque. Nothing about your effort changed.

That is why a breaker bar loosens a bolt a short wrench will not, why a cheater pipe slipped over a handle works, and why a torque wrench has to be gripped at a particular point to read correctly.

It is also the wheel and axle and the lever, restated. All three are force times radius, and recognizing that is the point of teaching torque separately.

Angle matters

Only the perpendicular distance counts.

Pull a wrench straight out along its handle and it will not turn the nut at all, no matter how hard you pull, because the line of the force passes through the pivot and the lever arm is zero.

Pull at 90 degrees to the handle and every ounce of the force is doing useful work. Pull at an angle in between and only part of it is.

So the most effective pull is perpendicular to the handle, which is the figure's third panel and a question in its own right. Anything else wastes effort.

Rotational equilibrium

A balanced object has equal torques on both sides of its pivot.

Clockwise torque = counterclockwise torque.

That equation is what the levers lesson calls effort times arm equals load times arm, and it is the same thing.

A 10-foot beam pivots at its center. A 40-pound weight sits 3 feet to the left. Where must a 30-pound weight sit on the right to balance?

Left torque: 40 times 3 is 120.

Right: 120 divided by 30 is 4 feet from the pivot.

The lighter weight sits further out, which is the sanity check.

More than two forces

Add the torques on each side.

A beam pivots at its center. On the left, 20 pounds at 4 feet and 10 pounds at 2 feet. What single weight 5 feet to the right balances it?

Left: 20 times 4 is 80, plus 10 times 2 is 20, giving 100 total.

Right: 100 divided by 5 is 20 pounds.

Sum each side, then divide. That extends to any number of forces.

Rotational inertia

Mass further from the axis is harder to start and stop turning.

That is why a flywheel is heavy at its rim rather than at its center, and why a spinning figure pulls their arms in to speed up. The same mass, moved closer to the axis, spins more easily.

A question about why a wheel with its weight at the rim is harder to accelerate wants that idea, and the level asked is the comparison rather than a formula.

Rotating machinery

The torque topic also asks about the machinery that rotation drives. The engine's four strokes and parts are taught in the Auto and Shop engine lesson; these are the additional ideas asked here.

The crank turns straight-line motion into rotation. A piston moves back and forth - reciprocating motion - and the connecting rod and crankshaft turn that into rotation. A cam does the reverse, turning rotation into a controlled back-and-forth push on a follower; a steep ramp on the cam gives the follower rapid acceleration, and valve lift is how far the cam opens a valve.

Unbalanced parts vibrate. Uneven mass distribution in anything that spins makes it shake, which is why wheels, crankshafts and fans are balanced. A harmonic balancer on the front of a crankshaft damps its torsional (twisting) vibration. A drive shaft spun above its critical speed will whip and vibrate severely.

A governor regulates speed automatically. Its spinning flyweights are flung outward by the centrifugal force of rotation; as speed rises they move out and cut the fuel, and as it falls they move in and add fuel.

A Geneva mechanism turns continuous rotation into intermittent rotation - a step, a pause, a step - as in a film projector. A ratchet allows rotation one way only, which is how a socket wrench works.

Compression ratio has a ceiling in a gasoline engine: squeeze the mixture too hard and it ignites on its own, which is detonation, or knock.

What you can skip

Across the 89 questions on this topic in our bank:

• Rotational dynamics. The moment of inertia as a formula, radians and angular momentum never appear as calculations; angular momentum is named once.
• Metric torque units. Newton-meters never appear. Torque is in foot-pounds or inch-pounds.
• Gyroscopes never appear.

Where people lose points

Measuring the lever arm along the handle when the force is at an angle. The perpendicular distance is shorter.

Thinking pulling harder always helps. Direction matters as much as magnitude.

Adding torques on the same side as though they opposed each other.

Forgetting that a force through the pivot produces no torque.

Confusing torque with force. Torque has a distance in its units.

Putting the heavier weight further from the pivot in a balance question.

Work one in under a minute

A mechanic cannot loosen a bolt with a 10-inch wrench pulling 60 pounds. A 30-inch breaker bar is fitted. What pull is now needed for the same torque?

Original torque: 60 times 10 is 600 pound-inches.

Needed pull: 600 divided by 30 is 20 pounds.

Three times the handle, a third of the pull. Same torque, less effort - which is mechanical advantage in rotating form, and it is why the tool exists.

Where this leads

Torque is what gears multiply and what a wheel and axle delivers, so this lesson names the quantity those two were working with all along.