Torque Calculator
Calculate torque, force, or lever arm length instantly using the fundamental torque equation τ = rF sin θ. Enter your values below to find the rotational force produced by any combination of applied force, distance, and angle.
Torque Calculator
Inputs
90° = max torque. 0° = no torque.
Enter Force, Radius, and Angle to calculate Torque.
Understanding Torque
Torque is the rotational equivalent of linear force. While a force causes an object to accelerate in a straight line, torque causes an object to acquire angular acceleration and rotate around an axis. The concept is fundamental in mechanics, engineering, and everyday life.
The Torque Equation
The magnitude of torque is determined by three quantities: the applied force, the distance from the pivot, and the angle of application. The mathematical formula is:
τ = r × F × sin(θ)- τ (tau) — Torque in Newton-metres (N·m)
- r — Lever arm or moment arm length in metres (m)
- F — Applied force in Newtons (N)
- θ (theta) — Angle between the force vector and the lever arm
The Moment Arm
The moment arm (or lever arm) is the perpendicular distance from the axis of rotation to the line along which the force acts. When the force is applied at an angle other than 90 degrees, only the perpendicular component of the force contributes to rotation. This is why the sin(θ) term appears in the equation. At 90 degrees the torque is maximized because sin(90°) = 1, and at 0 degrees the torque is zero because sin(0°) = 0.
Rotational Equilibrium
An object is in rotational equilibrium when the sum of all torques acting on it equals zero. This principle is used extensively in structural engineering, bridge design, and balancing problems. For example, a seesaw is balanced when the torque produced by the person on one side equals the torque produced by the person on the other side, even if their weights differ, by adjusting the distance from the pivot.
Torque in Everyday Life
Torque governs many actions we perform daily without thinking about the physics involved:
- Wrenches and bolts: A longer wrench provides a greater lever arm, generating more torque with the same effort. This is why mechanics use breaker bars for stubborn fasteners.
- Opening doors: Door handles are placed far from the hinges to maximize the moment arm. Pushing near the hinge requires significantly more force to produce the same torque.
- Steering wheels: A larger steering wheel diameter means a longer lever arm, making it easier to turn. Power steering reduces the force needed, effectively increasing the applied torque.
- Bicycle pedals: Longer crank arms increase torque at the expense of pedalling speed. Riders choose crank length based on their leg length and riding style.
Common Torque Values Reference Table
The following table lists typical torque values encountered in automotive, mechanical, and everyday applications. These values serve as a quick reference for engineers, mechanics, and students.
| Application | Torque (N·m) | Torque (lb·ft) |
|---|---|---|
| Car lug nut (passenger vehicle) | 80 – 110 | 59 – 81 |
| Bicycle pedal (right-side) | 35 – 40 | 26 – 30 |
| Spark plug (gasoline engine) | 15 – 30 | 11 – 22 |
| Small car engine (1.6 L) | 140 – 180 | 103 – 133 |
| V8 gasoline engine (5.0 L) | 400 – 550 | 295 – 406 |
| Heavy-duty diesel truck engine | 1,800 – 2,500 | 1,328 – 1,844 |
| Opening a door handle | 2 – 5 | 1.5 – 3.7 |
| Tightening a jar lid (hand) | 1 – 3 | 0.7 – 2.2 |
How to Calculate Torque
Follow these step-by-step worked examples to understand how the torque formula is applied in practical situations. Each example uses τ = r × F × sin(θ).
Example 1: Tightening a Lug Nut with a Wrench
A mechanic applies 200 N of force at the end of a 0.45 m wrench, perpendicular to the handle.
- Force (F) = 200 N
- Lever arm (r) = 0.45 m
- Angle (θ) = 90° → sin(90°) = 1
- τ = 0.45 × 200 × 1 = 90 N·m
This is within the typical 80–110 N·m range for passenger car lug nuts.
Example 2: Pushing a Door at an Angle
You push a door with 50 N of force at a point 0.75 m from the hinges, but at a 60° angle instead of straight on.
- Force (F) = 50 N
- Lever arm (r) = 0.75 m
- Angle (θ) = 60° → sin(60°) ≈ 0.866
- τ = 0.75 × 50 × 0.866 ≈ 32.5 N·m
Compared to pushing perpendicularly (37.5 N·m), the 60° angle reduces the effective torque by about 13%.
Example 3: Finding the Required Force for a Bicycle Pedal
A cyclist needs to produce 40 N·m of torque on a crank arm that is 0.17 m long. The force is applied perpendicularly. What force is needed?
- Torque (τ) = 40 N·m
- Lever arm (r) = 0.17 m
- Angle (θ) = 90° → sin(90°) = 1
- F = τ / (r × sin θ) = 40 / (0.17 × 1) ≈ 235.3 N
The cyclist must push with approximately 235 N (about 24 kg of force equivalent) on each pedal stroke.
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