Levers & Gears

A lever is a rigid beam pivoting about a fixed point called the fulcrum. Any force applied at a distance from the fulcrum creates a torque: $\tau = F \cdot d$, where $d$ is the perpendicular distance from the fulcrum to the line of action of the force (the moment arm). For a lever in rotational equilibrium (not accelerating), the net torque must be zero: $\sum \tau = 0$. This gives the Law of the Lever: $F_{in} \cdot d_{in} = F_{out} \cdot d_{out}$. Rearranging, the mechanical advantage is $MA = F_{out}/F_{in} = d_{in}/d_{out}$. A long input arm and short output arm yields MA > 1, meaning a small input force produces a large output force.

Levers are classified by the relative positions of the fulcrum (F), effort (E), and load (L). A Class 1 lever has the fulcrum between effort and load (e.g., a seesaw or crowbar), and can have MA greater or less than 1 depending on arm lengths. A Class 2 lever has the load between the fulcrum and effort (e.g., a wheelbarrow), always giving MA > 1. A Class 3 lever has the effort between the fulcrum and load (e.g., tweezers or a fishing rod), always giving MA < 1 but gaining in speed and range of motion. In every case, the work-energy theorem ensures that ideal work in equals work out: $W = F_{in} \cdot d_{in} = F_{out} \cdot d_{out}$.

A gear is essentially a rotating lever. When two gears mesh, their teeth constrain their contact point to have the same linear (tangential) velocity: $v_t = \omega_1 r_1 = \omega_2 r_2$, where $\omega$ is angular velocity (rad/s) and $r$ is the pitch radius. This gives the gear ratio: $\frac{\omega_1}{\omega_2} = \frac{r_2}{r_1} = \frac{N_2}{N_1}$, where $N$ is the number of teeth. Because power is conserved in an ideal gear ($P = \tau \omega = \text{const}$), a gear that reduces speed increases torque by the same ratio: $\frac{\tau_2}{\tau_1} = \frac{N_2}{N_1}$. A large driven gear meshed with a small drive gear (gear ratio > 1) reduces speed but multiplies torque — crucial in vehicles, clocks, and industrial machinery.

Real levers and gears lose some energy to friction, deformation, and heat. Efficiency is defined as $\eta = W_{out}/W_{in} = P_{out}/P_{in}$, always less than 1 (or 100%). For gears, friction at meshing teeth dissipates power, so the actual output torque is $\tau_{out} = \eta \cdot (N_2/N_1) \cdot \tau_{in}$. Well-lubricated spur gears typically achieve efficiencies of 95–99%, while worm gears (used for very large ratios) may be only 50–90% efficient due to sliding contact. Lever efficiency depends on pivot friction and beam flexibility. Engineers must account for efficiency when designing machines that meet load requirements under real-world conditions.

• MA — Mechanical Advantage (dimensionless): Ratio of output force to input force; indicates how much a machine amplifies force.
• F_{out} — Output (load) force (N): The force exerted by the machine on the load.
• F_{in} — Input (effort) force (N): The force applied by the user to the machine.
• d_{in} — Input moment arm (m): Perpendicular distance from the fulcrum to the line of action of the input force.
• d_{out} — Output moment arm (m): Perpendicular distance from the fulcrum to the line of action of the output force.
• N_1 — Number of teeth on drive gear (dimensionless): Tooth count on the input (driving) gear.
• N_2 — Number of teeth on driven gear (dimensionless): Tooth count on the output (driven) gear.

Problem. A bicycle uses a front chainring with 48 teeth and a rear sprocket with 16 teeth. The cyclist applies a force of 120 N to each pedal at a crank arm length of 0.175 m. (a) What is the gear ratio? (b) What torque is delivered to the rear wheel axle (assume 97% efficiency)? (c) If the rear wheel has a radius of 0.34 m, what is the forward force on the ground?

Answer. Gear ratio = 3.0; Torque at axle ≈ 6.79 N·m; Forward driving force ≈ 20.0 N

• Torque and Rotational Equilibrium — Levers are analyzed by setting the net torque about the fulcrum to zero; understanding torque is essential to derive mechanical advantage.
• Angular Velocity and Angular Acceleration — Gear ratios relate the angular velocities of meshing gears through the constraint that contact points share equal tangential speeds.
• Rotational Power and Energy — The principle that P = τω is conserved in ideal gears explains the trade-off between torque and angular speed in any gear system.
• Work-Energy Theorem — Conservation of work (W = F·d = τ·θ) underlies why machines do not create energy but only redistribute force and displacement.
• Friction and Efficiency — Real levers and gears lose energy to friction; efficiency η = W_out/W_in quantifies how close a real machine is to the ideal case.