Lever Calculator

Force · Force Arm · Load Arm · Mechanical Advantage · Travel

Lever Calculator

Applied force at effort arm
Distance pivot → effort force F₁
Distance pivot → load

Formulas & Symbols

F₁ ↓                        ↓ F₂
━━━━━━━━━━━━━━╋━━━━━━━
     <── L₁ ──>△<─ L₂ ─>
Pivot point △ (Fulcrum)
Lever Law
Fundamental equation (moment equilibrium):
F₁ × L₁ = F₂ × L₂
Clockwise moment = counter-clockwise moment
Load force:
F₂ = F₁ × L₁ / L₂
Effort force:
F₁ = F₂ × L₂ / L₁
Mechanical advantage:
i = F₂ / F₁ = L₁ / L₂
i > 1 → force multiplication  |  i < 1 → distance multiplication
Travel ratio:
s₁ / s₂ = L₁ / L₂
s₂ = s₁ × L₂ / L₁
Energy conservation (ideal lever):
W = F₁ × s₁ = F₂ × s₂
No work is gained – only force is traded against distance
Symbol Reference
F₁Effort force / input force [N]
F₂Load force / output force [N]
L₁Effort arm – distance F₁ to pivot [m]
L₂Load arm – distance F₂ to pivot [m]
iMechanical advantage / ratio [–]
s₁Stroke / travel at effort arm [mm]
s₂Stroke / travel at load [mm]
WWork / energy [J]


Lever – Engineering Basics & Lever Law

What Is a Lever?

The lever is one of the six classical simple machines and the foundation of mechanics. It consists of a rigid body (lever arm) that rotates about a fixed point – the fulcrum or pivot. By choosing the arm lengths, you can either multiply force (long effort arm, short load arm) or multiply distance (short effort arm, long load arm).

The fundamental law is: F₁ × L₁ = F₂ × L₂. This is a direct result of moment equilibrium – the product of force and arm length (= torque) must be equal on both sides for the lever to be in balance.

Types of Levers

Class 1 Lever

Fulcrum lies between the two forces.

  • Seesaw, balance scale, crowbar
  • i = L₁/L₂ > 1 → force multiplication
  • Forces act on opposite sides
Class 2 Lever

Load lies between fulcrum and effort.

  • Wheelbarrow, nutcracker
  • Always i > 1 (force multiplication)
  • Fulcrum at one end
Class 3 Lever

Effort lies between fulcrum and load.

  • Tweezers, forearm (biceps)
  • Always i < 1 (distance multiplication)
  • Fast movements possible

Detailed Formula Derivation

1. Lever Law from Moment Equilibrium

At equilibrium, the sum of all torques about the pivot equals zero:

M₁ = M₂  →  F₁ × L₁ = F₂ × L₂
Solving for F₂:   F₂ = F₁ × L₁ / L₂
Example: F₁ = 100 N, L₁ = 2.0 m, L₂ = 0.5 m  →  F₂ = 100 × 2.0 / 0.5 = 400 N
2. Mechanical Advantage i
i = F₂ / F₁ = L₁ / L₂
Example: L₁ = 2.0 m, L₂ = 0.5 m  →  i = 2.0/0.5 = 4
With 100 N input you can lift 400 N – a fourfold force multiplication.
3. Energy Conservation – Force vs. Distance
W = F₁ × s₁ = F₂ × s₂  →  s₂ = s₁ × L₂ / L₁
Example: s₁ = 200 mm, L₁ = 2.0 m, L₂ = 0.5 m
s₂ = 200 × 0.5/2.0 = 50 mm
The load rises only 50 mm although F₁ moves 200 mm. Work is conserved!
4. Friction Losses – Real Lever
F₁_real = F₂ × L₂ / (L₁ × η)
η = bearing efficiency (typically 0.90–0.99 depending on bearing type)
Friction losses are minor for levers; significant for gears and lead screws.

Practical Example: Crowbar

Task:

A crowbar is 1.5 m long. The fulcrum (support block) is placed 0.1 m from one end. What load can be lifted with 120 N hand force?

Solution:
  • Effort arm L₁ = 1.5 − 0.1 = 1.4 m
  • Load arm L₂ = 0.1 m
  • i = 1.4 / 0.1 = 14
  • F₂ = 120 × 14 = 1,680 N ≈ 171 kg
  • Load lift when hand moves 30 cm: s₂ = 300 × 0.1/1.4 ≈ 21 mm

Frequently Asked Questions

No. A lever multiplies either force or distance, never both. The work W = F × s remains constant on both sides (conservation of energy). A long effort arm gives large force but small stroke. A short effort arm gives large stroke but small force. This rule applies to all simple machines.

Effort arm L₁: perpendicular distance from the pivot (fulcrum) to the point where the input force F₁ is applied.
Load arm L₂: perpendicular distance from the pivot to the point where the load F₂ acts. For inclined levers, always use the perpendicular (effective) arm length.

The lever law still holds, but you must use the effective arm lengths: the perpendicular distance from the line of action of each force to the pivot (= moment arm). The torque formula becomes M = F × L_eff, where L_eff = L × sin(α), with α being the angle between the lever arm and the direction of the force.

A torque calculator computes M = F × r for a single force about a pivot. A lever calculator finds the equilibrium of two opposing torques (F₁×L₁ = F₂×L₂) and determines the unknown output force or arm length. The lever is a special case of moment equilibrium applied to two forces.

Class 1 (pivot in middle): Seesaw, balance scale, crowbar, scissors, forklift tipping mechanism.
Class 2 (load in middle): Wheelbarrow, nutcracker, door handle, bottle opener.
Class 3 (effort in middle): Tweezers, forearm with biceps, fishing rod, drumstick. The human body predominantly uses class 3 levers.

Summary

Lever Law

F₁ × L₁ = F₂ × L₂
Moment equilibrium

Mech. Advantage

i = L₁ / L₂
Force or distance gain

Energy Conservation

W = F₁×s₁ = F₂×s₂
No gain, no loss

Typical Applications
  • Mechanical engineering: Toggle presses, clamping mechanisms, toggle clamps
  • Automotive: Brake pedal, clutch pedal, steering linkage
  • Medical technology: Joint prostheses, orthoses, surgical instruments
  • Construction: Crowbar, lifting tools, formwork clamps
  • Biology: Bone levers (forearm, kneecap, Achilles tendon)

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