Muscles and levers

To hold 4 kg in the hand with the forearm level, the biceps pulls with about 470 N, more than seven times the weight it supports.

Bones as levers

A lever is a stiff bar that turns about a fixed point, the fulcrum. A force applied to it, the effort, moves or holds a resistance, the load. In the body, bones are the bars and joints are the fulcrums. The pull of a muscle is the effort, and the load is whatever resists the movement: a bag of shopping, or simply the weight of the limb itself.

Three classes of lever

Levers are sorted by what sits in the middle. In a first-class lever the fulcrum is between the effort and the load, like a seesaw. Nodding your head works this way: the skull rocks on the top of the spine, with neck muscles pulling behind and the weight of the face in front. In a second-class lever the load is in the middle, like a wheelbarrow. Rising onto your toes is an example: the ball of the foot is the fulcrum, the calf muscle pulls up on the heel, and the weight of the body sits between them. In a third-class lever the effort is in the middle. That is the forearm. The elbow is the fulcrum, the biceps pulls on the radius just in front of it, and the load is in the hand, much further out.

Balancing the turning effects

Whether a lever turns depends on torque, the turning effect of a force: the force multiplied by its perpendicular distance from the fulcrum. To hold the forearm still, the torque from the biceps must equal the torques from the weights pulling down. A worked physics example uses these values. The biceps inserts 4.00 cm from the elbow. The forearm has a mass of 2.50 kg, with its centre of mass 16.0 cm from the elbow. A 4.00 kg load is held 38.0 cm from the elbow, and the forearm is level. With g = 9.80 m/s², the balance reads like this.

F × 0.0400 m = (2.50 kg × 9.80 m/s² × 0.160 m) + (4.00 kg × 9.80 m/s² × 0.380 m) = 3.92 N·m + 14.90 N·m = 18.82 N·m, so F = 18.82 ÷ 0.0400 ≈ 470 N.

The forearm and the load together weigh 6.50 kg × 9.80 m/s² = 63.7 N, so the biceps is pulling about 7.4 times as hard as the weight it supports. The elbow joint takes a large force too: the humerus presses down on the forearm with the difference, about 407 N. The model uses a slightly different attachment geometry. With the forearm level and 4 kg in the hand, it shows about 468 N because its perpendicular biceps moment arm is about 4.02 cm rather than 4.00 cm.

Why attach so close to the joint?

Mechanical advantage is the load divided by the effort, and for a lever it equals the effort arm divided by the load arm. For the forearm that is 4 cm ÷ 38 cm, about 0.11. A mechanical advantage below 1 sounds like a bad deal, and for force it is. The trade is distance and speed. The distances moved are in the same ratio as the lever arms, so a 1 cm shortening of the biceps swings the hand about 9.5 cm. Small, quick contractions near the joint give large, fast movements of the hand. If the tendon were attached further down the forearm, the muscle could hold more, but the hand would move through a smaller range for the same shortening.

Try the Biceps attachment slider. In the textbook calculation, moving the insertion from 4 cm to 5 cm lowers the force needed with 4 kg from about 470 N to about 376 N, and the hand then moves only about 7.6 cm for each centimetre of shortening.

Angles change both arms

Change the elbow angle and both arms of the lever change. As the forearm tilts down from level, the weight in the hand comes closer to the vertical line through the elbow, so its load arm shrinks and the biceps needs less force. With the arm hanging straight down, the weights pull along the forearm and make almost no torque at all. Real elbows are more complicated than the model. The pivot point shifts slightly as the joint bends, and the force a muscle can produce also depends on its length, so the numbers here are a first estimate, not a measurement of anyone's arm.

This is educational material about how the body is built and works. It is not medical advice and is not for diagnosing or treating anyone.