Levers, gears and torque

Before you start

A door, a spanner, a seesaw and a bicycle’s gears all turn about a fixed point, and in each case where we apply a force matters as much as how hard we push. On the pulley page, a machine trades force for distance; a lever makes the same trade by turning. If you have not seen the pulley page, that is fine: the balance below starts again from the door. To compare turning effects, we need a quantity that combines the force with its distance from the pivot. That quantity, torque, returns on the rotation page, where it changes how fast things spin.

When opening a door, we naturally put a hand farther from the hinge. A balance turns that experience into a clearer comparison: keep the weights unchanged and move only the left pan. The force has not grown, so why can its turning effect change? Compare the distance from the pivot with the direction the balance tips. Torque records this connection. With two gears, our attention moves to the teeth in contact: how do they push each other and connect the two rotations? The relationship between distance and force will still be useful.

The setupTwo pans hang from one balance beam. Each weight is 1 kg. The right pan holds three weights and stays 2 m from the pivot. Drag the left pan along the beam to change its distance; left-click it to add a weight and right-click to remove one (on a touchscreen, tap to choose the number). The gear view replaces the beam with a small brass input gear driving a larger steel output gear.

What to do and noticeFirst keep three weights on the left too, and drag the left pan nearer to the pivot and then farther away. Watch which way the beam tips, and compare the two products of weight and distance shown above it. Next, change the number of weights on the left and look for the distance at which the beam balances again. When you switch to the gears, follow one tooth of the small gear as it pushes a tooth of the large gear, and compare how fast each gear turns with the turning effect it passes on.

01 / 03 · What torque measures

Predict firstWhy is the same force more effective farther from the pivot?

Preparing the model…

Guided lesson

A door is easier to turn farther from its hinge: position and direction matter as well as force.

Prerequisites: Both pans hang from the same balance beam on a wooden desk. Each weight represents 1 kg. The right pan has three weights and stays 2 m from the pivot. Drag the left pan to change its distance. Left-click adds a weight; right-click removes one; on a touchscreen, tap to enter a count. Then switch to the gears and compare the brass input with the steel output.

01 / 03

What torque measures

Predict firstWhy is the same force more effective farther from the pivot?

Keep the left weight count fixed and drag its pan outwards. Each weight represents 1 kg. The same mass has the same downward weight $F=mg$ , yet its turning effect grows. Each 1 kg weight is pulled down by about 9.8 newtons (N). Here $g$ is that pull per kilogram, about 9.8 N/kg, so three weights give about 29.4 N. Knowing the force alone is not enough; we also need its position.

The horizontal distance from the pivot to the vertical line of the weight is the perpendicular moment arm $d$ . The magnitude of the moment, or torque, is defined as $\tau=Fd$ . Let $r$ be the distance along the beam to the hanger and $\theta$ the beam’s angle above horizontal. Its horizontal projection is $r \cos \theta$ . For a level beam, $d=r$ , and the moment can be calculated directly. Three 1 kg weights at 2 m give a force of about 29.4 N and a moment of about $29.4\times2=58.8$ N·m. If the beam tilts, the arm is shorter than $r$ and the calculation needs $r\cos\theta$ . That step can be skipped here.

d=r\cos\theta,\qquad F=mg,\qquad \tau=mgd

A longer moment arm gives the same force more turning effect. Now hold the position fixed and left-click to add a weight or right-click to remove one; on a phone, tap and enter the count. Compare the change caused by $F$ instead. Torque is measured in N·m; it is not angular speed. Here are short answers to the three questions under How to observe. A longer arm needs a smaller force for the same turning job. But that force must move through a longer distance, so you pay for the smaller force with a longer push. In the gears, the 16-tooth input drives a 32-tooth output by default. The output turns at half the speed and passes on twice the torque. The panel’s mechanical advantage is the left arm divided by the right arm: $1.00\times$ means they are equal.

How to observe

Why is the same force more effective farther from the pivot?
Does a longer arm always mean less force?
Why does the larger output turn slower but transmit more torque?

What this model simplifies

This is an ideal teaching balance: the decorative beam, pans and chains are tared, so only the calibrated weights contribute static torque. Distances marked in metres are model units. Motion includes rotational inertia, damping and end stops. External involute gears neglect backlash, deformation and inertia; a matched load balances motor torque at the prescribed speed.

SourcesOpenStax · Mechanics