Levers, gears, springs and useful mechanical systems

Geometry connects the input to the output

A lever is a part that turns about a support or pivot. In the ideal static model, input force × perpendicular input arm balances load force × perpendicular load arm. A longer input arm can therefore balance a larger load force. The support also carries a force; leaving it out would give an incomplete force account even when the moments balance.

A compatible pair of external gears transmits motion through matching teeth. If 12 teeth pass the contact during one full turn of the small gear, a 36-tooth large gear turns one third of a revolution. For fixed centers and no loss of tooth engagement, rotation rates are inversely proportional to tooth counts. Directly meshing external gears turn in opposite directions. This describes the ideal kinematic relationship, not a guarantee of load capacity.

A spring can store energy supplied earlier

For a limited ideal linear spring, restoring-force magnitude = k × extension, where k is the spring stiffness and extension is measured from the unstretched position. The restoring force acts toward that position. A value k = 100 N/m means 1 N at 0.01 m extension and 3 N at 0.03 m. A real spring need not stay linear over all movements.

The force-extension graph explains the energy amount. Slowly extending this ideal spring from zero to 0.03 m requires force that rises from 0 to 3 N. Average force over that straight-line increase is 1.5 N, so supplied work is 1.5 × 0.03 = 0.045 J. Geometrically, it is the triangular area under the line. That work becomes stored elastic energy in the ideal model.

Each connection needs a consistent meaning

In a steady lossless gear pair, a threefold reduction in rotation rate can accompany a threefold increase in transmitted torque. The turning rate decreases as torque increases, keeping power consistent. Friction, deformation and acceleration change the actual energy account; the tooth ratio alone does not determine the complete operating performance.

A spring return also does not create energy. It can return some energy supplied during an earlier movement, while a real mechanism can dissipate part of it. To connect mechanisms, record which shaft, angle, force, travel and stored state belong to each side. If a stop prevents the movement assumed in the model, the earlier force and energy prediction may no longer describe the same operating condition.

Trace a motion ratio and an energy store

In the constructed gear pair, the 12-tooth input makes 90 revolutions per minute, abbreviated rpm. The 36-tooth output rate is 90 × 12 ÷ 36 = 30 rpm in the opposite direction. The pitch circles in the diagram have the same 1:3 size ratio; actual tooth profiles are not drawn.

For a separate ideal lever, 1 N at a 0.30 m arm balances 3 N at a 0.10 m arm: both moment magnitudes are 0.30 N·m. The pivot supplies 4 N upward because the two downward forces on the lever sum to 4 N. These are forces on the lever, not a mixture of forces acting on different bodies.

Finally, the ideal 100 N/m spring at 30 mm extension stores 0.045 J, as the graph showed. If the spring is released in a later operation, that stored amount can enter the next energy account. Do not count it again as newly supplied energy and also as an independent free output.

Try the reasoning

1. A 12-tooth gear drives a compatible 36-tooth external gear at 90 rpm. What is the output rate and direction?

   Show the answer and reasoning

   30 rpm, opposite to the input for this directly meshing external pair with fixed centers.

2. Why is the work to extend the ideal spring not 3 N × 0.03 m?

   Show the answer and reasoning

   The force is not 3 N throughout. It rises linearly from zero, so the average is 1.5 N and the work is 0.045 J.

3. Does a threefold ideal torque increase imply threefold power output?

   Show the answer and reasoning

   No. It accompanies a threefold lower rotation rate in the stated steady lossless model. The power account remains balanced.

What to carry forward

Mechanisms trade force with motion or store previously supplied energy. Identify the geometry, state and losses before combining them.