Torque Calculator

Calculate torque (rotational force) using force, lever arm distance, and angle. Essential for physics, engineering, and understanding rotational motion.

**Torque Formula:** τ = r × F × sin(θ) Where: • τ = Torque (N·m or ft-lb) • r = Distance from rotation axis (m or ft) • F = Applied force (N or lb) • θ = Angle between force and lever arm (degrees) **Vector Form:** τ = r × F (cross product) |τ| = |r| × |F| × sin(θ) **Special Cases:** • Perpendicular force (θ = 90°): τ = r × F (maximum torque) • Parallel force (θ = 0° or 180°): τ = 0 (no torque) **Effective Lever Arm:** r_eff = r × sin(θ) τ = r_eff × F **Rotational Dynamics:** τ = I × α Where I = moment of inertia, α = angular acceleration
**Example 1 (Wrench):** Apply 50 N perpendicular to a 0.3 m wrench. • Force = 50 N • Distance = 0.3 m • Angle = 90° • τ = 50 × 0.3 × sin(90°) = 15 N·m **Example 2 (Door):** Push a door 0.8 m from hinges with 20 N at 60° angle. • Force = 20 N • Distance = 0.8 m • Angle = 60° • τ = 20 × 0.8 × sin(60°) = 13.86 N·m **Example 3 (Bolt Tightening):** A torque wrench with 0.5 ft lever arm needs 75 ft-lb torque. What force? • Required torque = 75 ft-lb ≈ 101.7 N·m • Distance = 0.5 ft ≈ 0.152 m • Angle = 90° (perpendicular) • F = τ / (r × sin(θ)) = 101.7 / 0.152 = 669 N **Example 4 (Seesaw):** Two children: 300 N at 2 m, and 400 N at distance x. For balance: τ₁ = τ₂ 300 × 2 = 400 × x x = 1.5 m from pivot

What is torque?

Torque (τ) is a rotational force that causes objects to rotate around an axis. It depends on force magnitude, distance from the axis (lever arm), and the angle between them. Units are N·m (newton-meters) or ft-lb (foot-pounds).

What is the torque formula?

Torque τ = r × F × sin(θ), where r is the distance from the rotation axis, F is applied force, and θ is the angle between the force vector and lever arm. Maximum torque occurs at θ = 90°.

What is the difference between torque and work?

Both use N·m units, but torque is a rotational force while work is energy transfer. Torque causes rotation; work = torque × angular displacement. Never convert torque to Joules!

What is a lever arm?

The lever arm (or moment arm) is the perpendicular distance from the axis of rotation to the line of force action. It equals r × sin(θ), where r is the distance and θ is the angle.

How does angle affect torque?

Torque is maximum when force is perpendicular to the lever arm (θ = 90°, sin(90°) = 1). Zero torque occurs when force is parallel (θ = 0° or 180°). This is why wrenches work best when pulled perpendicularly.

What is rotational equilibrium?

Rotational equilibrium occurs when the sum of all torques equals zero (Στ = 0). The object doesn't rotate or rotates at constant angular velocity. Clockwise and counterclockwise torques balance.

How do I calculate net torque?

Sum all torques, assigning positive to counterclockwise and negative to clockwise (or vice versa, be consistent). Net torque τ_net = Στ determines angular acceleration via τ_net = I × α.

What is moment of inertia?

Moment of inertia (I) is rotational mass - resistance to angular acceleration. Just as F = ma for linear motion, τ = Iα for rotation. It depends on mass distribution relative to the rotation axis.

What are practical torque applications?

Car engines (engine torque), tightening bolts (torque wrenches), opening doors (doorknobs far from hinges), seesaws, steering wheels, bicycle pedals, and any machinery with rotating parts rely on torque principles.

Why are longer wrenches more effective?

Longer wrenches increase the lever arm distance (r), which increases torque for the same applied force. This is why a 1 ft wrench produces twice the torque of a 6-inch wrench with equal force.

Can torque be negative?

Yes! Sign convention: counterclockwise torque is positive, clockwise is negative (or vice versa). Negative torque indicates rotation in the opposite direction. The magnitude matters for calculations.

How do gears change torque?

Gears trade speed for torque. A large gear driven by a small gear rotates slower but with more torque (gear ratio). Car transmissions use this to match engine power to wheel requirements at different speeds.