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Torque

General

Torque (Rotational Force)

The rotational equivalent of force — a measure of the twisting effect that causes an object to rotate, equal to the radius times force times the sine of the angle between them (τ = rF·sin θ).

Definition

Torque is the rotational equivalent of force — it describes the tendency of a force to cause an object to rotate around a pivot point or axis. Rather than measuring straight-line push or pull the way ordinary force does, torque captures the twisting effect that occurs when force is applied at some distance from a fixed point, such as turning a steering wheel, tightening a bolt, or spinning a bicycle pedal.

The Torque Calculator computes this rotational effect from three inputs — the applied force, the length of the lever arm (the distance from the pivot to where the force is applied), and the angle between the force and the lever arm — returning the result in newton-meters. This is a distinct calculation from the separate bolt-torque-calculator, which instead looks up or applies manufacturer-specified fastener tightening specifications rather than computing torque from first principles; the Torque Calculator here is for general rotational mechanics problems in physics and engineering.

Torque plays a central role in rotating machinery, where it combines with angular velocity to determine power output, following the same underlying relationship that connects Work and Power in linear motion. It's also a direct application of the rotational form of Newton's laws — just as Newton's Second Law relates linear force to linear acceleration (F = ma), torque relates to angular acceleration through an object's rotational inertia.

Formula

τ = r × F × sin(θ)

Where τ (tau) is torque (in newton-meters, N·m), r is the lever arm length — the distance from the pivot to the point of applied force (in meters, m), F is the applied force (in newtons, N), and θ is the angle between the force vector and the lever arm.

When force is applied perpendicular to the lever arm (θ = 90°), this simplifies to τ = r × F.

Worked Example

A mechanic applies 150 N of force perpendicular to a wrench handle that is 0.3 meters long. The torque generated is:

τ = 0.3 m × 150 N × sin(90°) = 0.3 × 150 × 1 = 45 N·m

If the mechanic instead pushed at a 45-degree angle to the wrench handle rather than perpendicular to it, the torque would drop to 0.3 × 150 × sin(45°) = 0.3 × 150 × 0.707 ≈ 31.8 N·m — noticeably less effective, which is why applying force perpendicular to a wrench produces the most turning power.

Key Things to Know

  • Maximized at a 90-degree angle: torque is greatest when force is applied perpendicular to the lever arm, since sin(90°) = 1 is the maximum value of the sine function.
  • Longer lever arms multiply torque: doubling the distance from the pivot doubles the torque for the same applied force, which is why longer wrenches and longer wheelbraces make turning stubborn bolts easier.
  • Distinct from the bolt-torque-calculator: that separate tool applies fastener-specific tightening specs (a manufacturer torque value for a given bolt size and material), while this general torque relationship (τ = rF·sinθ) applies to any rotational mechanics problem.
  • Connects to rotational Power: in engines and motors, power equals torque times angular velocity (P = τ × ω), linking rotational force directly to energy delivery rate.
  • Rotational analogue of Newton's Second Law: just as force causes linear acceleration via F = ma, torque causes angular acceleration via the rotational form of the same law, with rotational inertia replacing mass.

Related Calculators

Torque Calculator →Calculate rotational torque using τ = r × F × sin(θ). Enter lever arm length, force, and angle to get torque in newton-meters instantly, with a formula breakdown.

Related Terms

Newton's Second LawNewton's Second Law of Motion

Frequently Asked Questions

Torque is the rotational equivalent of force — it measures how much a force acting at a distance from a pivot point tends to twist or rotate an object. Turning a wrench to loosen a bolt, pushing a door open, and pedaling a bicycle all involve applying torque.
Torque equals the distance from the pivot (the lever arm) times the applied force times the sine of the angle between them, written as τ = r × F × sin(θ). When the force is applied perpendicular to the lever arm (θ = 90°), the formula simplifies to τ = r × F, since sine of 90 degrees is 1.
Torque is measured in newton-meters (N·m) in the SI system, representing 1 newton of force applied at a distance of 1 meter from the pivot point. In some regions, particularly the US, torque is also commonly expressed in pound-feet (lb-ft).
A longer lever arm produces more torque for the same applied force, which is why a longer wrench makes it easier to loosen a stubborn bolt. Doubling the distance from the pivot doubles the torque generated by an identical force.
The Torque Calculator computes general rotational torque from force, lever arm length, and angle for physics and mechanical engineering scenarios. The separate bolt-torque-calculator instead applies manufacturer-specified tightening torque values for fasteners, a distinct engineering spec rather than the general τ = rF·sin(θ) relationship covered here.
In engines and motors, power equals torque multiplied by angular velocity (P = τ × ω), which is why an engine's torque and RPM figures together determine its Power output. A high-torque engine at low RPM can produce the same power as a lower-torque engine spinning much faster.