Force Calculation

Part of Simple Machines

How to calculate forces, mechanical advantage, and work in simple machine systems using basic arithmetic.

Why This Matters

You cannot design a machine or structure safely without being able to calculate the forces involved. When you are building a block and tackle to lift a heavy stone, you need to know what force each rope segment carries — so you know whether your rope is strong enough. When you are designing a bridge beam, you need to know the bending force at the center — so you know whether your timber is thick enough. When you are building a screw press, you need to know the output force — so you know whether you are pressing hard enough to extract oil from seeds.

Force calculations do not require advanced mathematics. The concepts needed — multiplication, division, proportional reasoning — are taught in basic arithmetic. What requires learning is how to set up the calculation correctly: identifying which forces are acting, what their directions are, and how the geometry of the machine converts input forces to output forces.

This article teaches the calculation methods for every type of simple machine and for common combinations. Approach each new machine by drawing a diagram, labeling all the forces, and applying the relevant formula.

Fundamental Concepts

Force

Force is a push or pull. In everyday calculation, we measure it in kilograms-force (kgf) or Newtons (N). One kilogram-force is the force exerted by one kilogram of mass under Earth’s gravity. In this article, we use kgf because most practical measurements are in kilograms.

Key forces you will calculate:

• Weight (load): The gravitational force on an object. Equal to its mass in kg (on Earth).
• Effort: The force you apply to the machine.
• Reaction: The force the machine’s structure must carry (at anchors, pivots, supports).

Mechanical Advantage (MA)

Mechanical advantage is the ratio of output force (load) to input force (effort):

MA = Load / Effort

Rearranging:

Load = MA × Effort
Effort = Load / MA

If MA = 4 and effort = 30 kg: Load = 4 × 30 = 120 kg

Work

Work = Force × Distance moved in the direction of force.

The law of conservation of energy means that for a perfect machine (no friction): input work = output work.

Effort × Effort distance = Load × Load distance

This relationship tells you the distance tradeoff that goes with mechanical advantage:

Effort distance / Load distance = Load / Effort = MA

A 4:1 MA machine: for every 4 meters you pull the effort, the load moves 1 meter.

Lever Force Calculations

The Basic Lever Formula

Effort × Effort arm length = Load × Load arm length

This is simply the torque balance equation — the lever doesn’t accelerate when input torque equals output torque.

Example: A crowbar is used to lift a boulder. The boulder is 0.4 m from the fulcrum (load arm). The hand pushing is 2.2 m from the fulcrum (effort arm). What force does a 70 kg person need to apply?

Effort × 2.2 = Load × 0.4
70 × 2.2 = Load × 0.4
154 = Load × 0.4
Load = 154 / 0.4 = 385 kg

A 70 kg person can lift 385 kg with this lever geometry.

Finding effort for a known load: If you need to lift 500 kg and have a 3 m bar with the fulcrum 0.5 m from the load:

• Effort arm = 3.0 - 0.5 = 2.5 m
• Effort = Load × Load arm / Effort arm = 500 × 0.5 / 2.5 = 100 kg

Reaction Force at the Fulcrum

The fulcrum must carry the combined force of load and effort. For Class 1 lever:

Fulcrum reaction = Load + Effort

In the example above: Fulcrum reaction = 500 + 100 = 600 kg. The fulcrum, and whatever it rests on, must carry 600 kg.

Pulley Force Calculations

Single Pulley

Fixed pulley (MA = 1):

• Effort = Load (no force advantage, only direction change)
• Rope tension at both sides = Load

Movable pulley (MA = 2):

• Effort = Load / 2
• Each supporting rope segment = Load / 2

Block and Tackle

Count the rope segments supporting the movable block. That is the MA.

Effort = Load / Number of supporting rope segments

Example: 6-rope block and tackle lifting 300 kg:

Effort = 300 / 6 = 50 kg

With friction correction (real world): Each pulley loses 5-10% to friction. For 5% loss per pulley with 6 pulleys:

Efficiency = 0.95^6 = 0.735 = 73.5%
Actual effort needed = Load / (MA × Efficiency) = 300 / (6 × 0.735) = 68 kg

Forces on the Anchor

The anchor (the fixed block’s attachment point) carries the full load plus the hauling force:

Anchor force = Load + Effort

For the 6-rope example: Anchor force = 300 + 50 = 350 kg (or 300 + 68 = 368 kg with friction). Size the anchor point accordingly.

Inclined Plane Force Calculations

Effort = Load × (Height / Length of ramp)

This is equivalent to: Effort = Load × sin(angle from horizontal)

Example: Rolling a 200 kg barrel up a 4 m ramp to a height of 1 m:

Effort = 200 × (1/4) = 50 kg

Adding friction: Rolling friction on a wooden ramp is typically 5-10% of the load weight. Add this to the inclined plane effort:

Total effort = Inclined plane effort + Friction
= 50 + (200 × 0.07) = 50 + 14 = 64 kg

Screw Force Calculations

The screw converts rotary force (torque on the handle) to linear force (push or pull on the screw):

Output force = Input torque × 2π / thread pitch

Where:
Input torque = Force at handle × handle radius
Thread pitch = distance the screw advances per complete turn

Example: A screw press with a 60 cm handle (30 cm radius) and 8 mm pitch, pressed with 15 kg at the handle end:

Input torque = 15 kg × 0.30 m = 4.5 kg⋅m
Output force = 4.5 × 2π / 0.008 = 4.5 × 785 = 3,534 kg

In practice, screw friction reduces this to roughly 20-40% of theoretical: actual output ≈ 700-1,400 kg.

Gear Force Calculations

Gears trade speed for torque (or torque for speed):

Torque ratio = MA = Number of teeth on driven gear / Number of teeth on driver gear

Example: A 12-tooth driver gear meshes with a 36-tooth driven gear:

MA = 36 / 12 = 3

If you apply 5 kg⋅m of torque to the driver: output torque = 5 × 3 = 15 kg⋅m The output shaft turns at 1/3 the speed of the input shaft.

Calculating force at the gear tooth:

Tooth force = Torque / Pitch circle radius

For a 36-tooth gear with 20 mm pitch and gear radius 115 mm, carrying 15 kg⋅m torque:

Tooth force = 15 kg⋅m / 0.115 m = 130 kg

Each tooth in contact carries approximately 130 kg. This determines the required tooth strength and wood species.

Calculating Combined Systems

For machines in series, multiply the MAs:

Example: A 2:1 lever feeding a 4:1 block and tackle:

Combined MA = 2 × 4 = 8

Friction in series: Multiply efficiencies:

Combined efficiency = Lever efficiency × Block and tackle efficiency
= 0.90 × 0.72 = 0.648 = 65%

Actual output force:

Actual output = Input effort × Combined MA × Combined efficiency
= 20 kg × 8 × 0.65 = 104 kg

Structural Forces: Beam Bending

For a beam supported at both ends with a single load at the center:

Maximum bending moment = Load × Span / 4
Maximum shear force = Load / 2

Safe load calculation:

Safe load = (Allowable bending stress × Width × Height²) / (1.5 × Span)

Allowable bending stress for hardwood timber: approximately 700-1,000 kgf per square centimeter (7,000-10,000 kPa). For softwood: approximately 400-600 kgf/cm².

Example: Oak beam, 10 cm wide, 20 cm tall, spanning 3 m:

Safe load = (800 kgf/cm² × 10 cm × (20 cm)²) / (1.5 × 300 cm)
= (800 × 10 × 400) / 450
= 3,200,000 / 450
= 7,111 kgf

This shows an oak beam of those dimensions can safely carry 7,000 kg at its center — much more than typical loads, which is why timber-framed buildings rarely fail in normal use.

Always Include a Safety Factor

Calculate the maximum possible load on any component, then use material rated for at least twice that load (2:1 safety factor for static loads). For dynamic loads, sudden shock, or where failure would cause death, use 4:1 or higher. Engineering history is full of structures that failed because someone trusted the theoretical minimum without safety margin.