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Torque Calculator

Torque equals force times distance

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Torque Equation

Torque is the rotational equivalent of force. It measures how effectively a force causes rotation about a pivot.

τ = F × d

How It Works

Torque (τ = F · d) is the rotational equivalent of force. It measures how effectively a force causes rotation about a pivot. A longer lever arm or a larger force produces more torque.

Example Problem

A mechanic applies 80 N of force at the end of a 0.3 m wrench. What torque is produced?

  1. Identify the known values: force F = 80 N, lever arm distance d = 0.3 m.
  2. Determine what we are solving for: the torque τ produced by the wrench.
  3. Write the torque equation: τ = F × d.
  4. Substitute the known values: τ = 80 N × 0.3 m.
  5. Compute the result: τ = 24 N·m (newton-meters).
  6. Interpret: if the bolt specification requires 24 N·m, this force is exactly sufficient to meet the torque requirement.

A simpler example: a 50 N force on a 0.2 m handle produces 50 × 0.2 = 10 N·m of torque.

When to Use Each Variable

  • Solve for Torque (τ) — when you know the applied force and lever arm distance, e.g., checking if a wrench produces enough torque to meet a bolt specification.
  • Solve for Force (F) — when you know the required torque and wrench length and need to find how much force to apply.
  • Solve for Distance (d) — when you know the required torque and available force and need to find the minimum wrench or lever arm length.

Key Concepts

Torque (τ = F × d) measures the tendency of a force to rotate an object about an axis. It depends on both the magnitude of the force and the perpendicular distance (moment arm) from the axis of rotation. Torque is a vector quantity — its direction follows the right-hand rule. In static equilibrium, the sum of all torques about any point must equal zero.

Applications

  • Automotive maintenance: calculating the force needed on a torque wrench to tighten fasteners to manufacturer specifications
  • Mechanical design: sizing shafts, gears, and bearings based on the torque they must transmit
  • Robotics: determining motor torque requirements for joint actuators based on arm length and payload weight
  • Structural engineering: analyzing moment (torque) at beam supports to verify structural adequacy

Common Mistakes

  • Using the total distance instead of the perpendicular distance — only the component of force perpendicular to the lever arm contributes to torque
  • Confusing torque with work — torque (N·m) and energy (J) have the same units but measure different things; torque is a static rotational tendency, work requires displacement
  • Ignoring the sign convention — clockwise and counterclockwise torques must be given opposite signs when summing moments for equilibrium

Frequently Asked Questions

What is the easiest way to find how much torque a wrench produces?

Use the formula τ = F × d. Multiply the applied force in newtons by the perpendicular distance (lever arm) in meters. For example, 100 N applied at 0.5 m produces 50 N·m of torque.

What equation relates torque, force, and lever arm?

The torque formula is τ = F × d, where τ is torque in newton-meters, F is force in newtons, and d is the perpendicular distance from the axis of rotation in meters. If the force is at an angle θ, use τ = F × d × sin(θ).

What is the difference between torque and moment?

In practice they are the same (force times distance). Engineers often use 'torque' for rotation about a shaft and 'moment' for bending in beams, but the math is identical.

Why does a longer wrench make bolts easier to turn?

A longer wrench increases the lever arm (d), so less force is needed to achieve the same torque. This is the mechanical advantage of a lever.

What units are used for torque?

SI: newton-meter (N·m). Imperial: foot-pound (ft·lbf). Note that N·m (torque) and J (energy) have the same dimensions but measure different things.

Is torque the same as work or energy?

No. Torque (N·m) and energy (J) share the same SI dimensions but measure different things. Torque is a rotational tendency — a static property. Work (energy) requires displacement. A bolt held at 50 N·m with no rotation does zero work.

How do you convert between N·m and ft·lbf?

Multiply newton-meters by 0.7376 to get foot-pounds. Multiply foot-pounds by 1.3558 to get newton-meters. For example, 100 N·m ≈ 73.76 ft·lbf. This calculator handles the conversion automatically.

Reference: Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.

Torque Formula

The torque equation relates rotational force to the applied force and lever arm distance:

τ = F × d

Where:

  • τ — torque, measured in newton-meters (N·m)
  • F — applied force, measured in newtons (N)
  • d — perpendicular distance from the axis of rotation (lever arm), measured in meters (m)

The formula assumes the force is applied perpendicular to the lever arm. If the force is at an angle θ, use τ = F × d × sin(θ). Only the perpendicular component contributes to rotation.

Worked Examples

Automotive Maintenance

What force is needed to torque a lug nut to 110 N·m with a 0.35 m wrench?

A mechanic must tighten lug nuts to 110 N·m using a 0.35 m (14-inch) torque wrench. How much force must be applied at the handle?

  • Rearrange: F = τ / d
  • F = 110 N·m / 0.35 m
  • F ≈ 314.3 N

That is about 70.7 lbf — easily achievable with one hand on a long wrench. Using a shorter wrench would require more force.

Mechanical Design

What tangential force acts at a motor shaft delivering 25 N·m at 0.04 m radius?

A motor must deliver 25 N·m of torque. If the drive shaft has an effective radius of 0.04 m, what tangential force acts at the shaft surface?

  • Rearrange: F = τ / d
  • F = 25 N·m / 0.04 m
  • F = 625 N

This tangential force determines the shear stress on the shaft and drives gear and bearing selection in the drivetrain.

Construction

What moment does a crane boom root resist when lifting 5,000 N at 12 m?

A crane lifts a 5,000 N load at the end of a 12 m boom. What moment does the boom root have to resist?

  • τ = F × d
  • τ = 5,000 N × 12 m
  • τ = 60,000 N·m

This is the static moment only. Dynamic loads, wind, and boom self-weight increase the actual design moment at the base.

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