Levers and Motion Mechanisms

How a lever changes force

A lever is a rigid bar that turns about a pivot called the fulcrum. The effort is the input force you apply; the load is the force you need to overcome or support. In a wheelbarrow, your hands supply the effort, the wheel’s axle acts as the fulcrum, and the weight of the contents provides the load.

The positions of these three parts determine what the lever does. Applying effort farther from the pivot gives it a greater turning effect. This is why a long handle can make lifting a load easier.

The three classes of lever

A lever’s class depends on which part lies between the other two.

A first-class lever can increase or reduce force, depending on the relative distances from the pivot. Moving the load closer to the pivot or applying effort farther away increases its force advantage.

In a second-class lever, the effort acts farther from the pivot than the load does. It therefore multiplies force in an ideal system. Long wheelbarrow handles and a load positioned close to the wheel reduce the effort needed.

In a third-class lever, the effort acts closer to the pivot than the load does. The output force is smaller than the input force, but the load moves farther for a given movement at the effort point. Tweezers use this arrangement to produce controlled movement and a gentle force at their tips.

Mechanical advantage: comparing forces

Mechanical advantage, or MA, compares the load force with the force:

$$MA = \frac{\text{load}}{\text{effort}}$$

Both forces must use the same unit, normally newtons, N. Their units cancel, so MA has no unit.

An MA greater than 1 means the mechanism multiplies force. An MA of 1 means the output and input forces are equal. An MA below 1 means the output force is smaller than the input force.

A lever does not create energy. Where it multiplies force, the effort must move through a greater distance than the load moves. Real mechanisms also lose some energy through friction.

Calculating forces using moments

A moment is the turning effect of a force about a pivot:

$$\text{moment} = \text{force} \times \text{perpendicular distance from pivot}$$

The distance is measured from the pivot to the force’s line of action. For a horizontal lever with vertical forces, this is the horizontal distance from the pivot to where each force acts. Force in N multiplied by distance in m gives a moment in N m.

The principle of moments states that a balanced lever has equal total clockwise and anticlockwise moments. For a simple lever with effort and load producing opposing turning effects:

$$E \times d_E = L \times d_L$$

Here, $E$ is effort, $L$ is load, and $d_E$ and $d_L$ are their perpendicular distances from the pivot. In an ideal calculation, friction and the lever’s own weight are ignored unless included in the question.

To find an unknown effort, calculate the load’s moment and divide it by the effort’s perpendicular distance from the pivot:

$$E = \frac{L d_L}{d_E}$$

The longer effort arm allows a smaller force to balance the load. To find an unknown load instead, rearrange to $L = \frac{E d_E}{d_L}$.

Rearranging the moments equation also gives a useful relationship for an ideal lever:

$$MA = \frac{L}{E} = \frac{d_E}{d_L}$$

This connects the lever’s geometry directly to its force advantage.

Rack and pinion: rotation into straight-line movement

A rack and pinion combines a circular gear, the pinion, with a straight toothed bar, the rack. Their teeth mesh. As the pinion rotates, its teeth push the rack along a straight path. Reversing the pinion’s rotation reverses the rack’s movement.

It can also work the other way: moving the rack can rotate the pinion. For example, a piston can push a rack, turning a pinion attached to an indexing table to move the table to its next position.

Unlike a mechanism that automatically moves back and forth, a rack continues moving in one direction while the pinion turns in one direction, until it reaches its travel limit.

Pawl and ratchet: preventing reverse movement

A pawl and ratchet controls the direction of rotation. The ratchet is a wheel with shaped teeth; the pawl is a pivoted, spring-loaded catch that engages them.

In the permitted direction, the sloping teeth pass beneath the pawl, lifting it. The spring then pushes the pawl into the next gap. If the wheel tries to turn backwards, the pawl bears against a tooth and blocks movement. This allows forward rotation while preventing unwanted reversal. The pawl must be disengaged if reverse rotation is required.

This mechanism can also be used to advance a rotary indexing table in steps, positioning it ready for successive operations.

Crank and slider: repeated push–pull movement

A crank and slider consists of a rotating crank, an off-centre crank pin, a connecting rod and a slider constrained to move along a straight guide.

As the crank rotates, its pin travels in a circle. The connecting rod transfers this movement to the slider, pushing it out and pulling it back. Continuous rotary input therefore produces reciprocating motion: repeated movement backwards and forwards along the same straight path.

The mechanism can work in reverse. A reciprocating piston can drive a crankshaft through a connecting rod, as in an engine. The connecting rod links the two parts throughout the movement rather than simply resting against a rotating surface.

Cams: controlling the pattern of movement

A cam rotates while a follower tracks its outer profile. Changes in the distance from the rotation axis to the contact surface make the follower rise and fall. Gravity or a spring can keep the follower in contact as the surface falls away.

The cam’s profile determines the pattern of output movement. A circular cam mounted off-centre produces a repeated rise and fall. A profile with a constant-radius section can hold the follower still for part of the rotation; this stationary period is called a dwell. A snail cam can raise a follower gradually and then allow it to drop suddenly.

This makes cams useful where the timing and shape of movement matter. Engine cams control when valves open and close; snail cams can lift stamping-machine weights before releasing them to fall. Both a crank and slider and a cam can turn rotation into reciprocation, but a cam provides greater freedom to shape the rise, fall and dwell of the output.

Lever classes

• First: fulcrum in the middle — seesaw, scissors.
• Second: load in the middle — wheelbarrow, nutcracker; ideal MA greater than 1.
• Third: effort in the middle — tweezers; ideal MA below 1, with greater movement at the load.

Force calculations

$$MA = \frac{L}{E}$$