Torque, Levers, and Mechanical Advantage

From force to torque

Linear force pushes in a straight line, but robots mostly rotate things — arms swing, wheels spin, intakes roll. The rotational version of force is torque: the twisting effect a force makes about a pivot.

Torque depends on how hard you push and how far from the pivot you push:

τ = r × F × sinθ

• τ is torque, in newton-meters (N·m)
• r is the distance from the pivot to where the force is applied
• F is the force
• θ is the angle between the force and the arm

The quantity r × sinθ is the lever arm — the perpendicular distance from the pivot to the line of force. Torque is maximum when you push at 90° (sin 90° = 1) and zero when you push straight at the pivot. It's why a force out at the end of a long arm produces far more torque than the same force near the joint.

The lever

A lever is a rigid bar pivoting at a fulcrum. Because the torque on each side has to balance, a small force far from the fulcrum lifts a large load close to it:

F_input × d_input = F_output × d_output

Push with 50 N at 1.0 m and you can lift 100 N at 0.5 m. You traded distance for force.

Mechanical advantage

Mechanical advantage (MA) measures that trade:

MA = output force ÷ input force

• MA > 1 multiplies force — lifting heavy loads slowly.
• MA < 1 multiplies speed or distance instead — a light, fast-moving arm tip.

Physics enforces a catch. Work (force × distance) is conserved, so you never get something for nothing: multiply force by 3 and the input has to move 3 times as far (or 3 times slower). Real machines do worse than this ideal because friction bleeds off energy, so the actual mechanical advantage is always less than the ideal. Budget for that.

Why FRC builders live in torque

Torque is the through-line of mechanical design:

• Arms: a heavier game piece, or one held farther out, needs more torque at the pivot. So you either pull the load in closer or gear the motor down to multiply its torque.
• Gearboxes (next lesson): gears are just rotating levers. A reduction multiplies torque exactly like a longer lever arm, paying for it in speed.
• Motor specs: FRC motors are rated by stall torque (the most twist they make, at zero RPM) and free speed (max RPM at no load). They make very little torque on their own. Your job is to gear a motor so it delivers the torque the mechanism needs without stalling and cooking itself.

Real numbers make this concrete: a NEO makes only a couple of N·m at the shaft, but a 100:1 reduction turns that into plenty of torque to hold a heavy arm — at 1/100 the speed.

Once you internalize torque = force × lever arm and work is conserved, gear ratios stop being mysterious and start being obvious.