Torque Formula
Reference for the torque formula τ = r × F × sin(θ) in N·m.
Covers right-hand rule, rotational equilibrium, and moment of inertia.
The Formula
Torque (also called moment of force) measures the tendency of a force to rotate an object around an axis or pivot point. It depends on both the magnitude of the force and the distance from the axis of rotation.
Variables
| Symbol | Meaning |
|---|---|
| T (or τ) | Torque (measured in newton-meters, N·m, or foot-pounds, ft·lb) |
| F | Force applied perpendicular to the lever arm (measured in newtons, N, or pounds, lb) |
| d | Distance from the pivot point to where the force is applied, also called the lever arm or moment arm (measured in meters or feet) |
Example 1
A mechanic applies 80 N of force to a wrench that is 0.3 m long. What is the torque on the bolt?
Identify the values: F = 80 N, d = 0.3 m
Apply the formula: T = F x d = 80 x 0.3
T = 24 N·m
Example 2
A door requires 15 N·m of torque to open. If the handle is 0.75 m from the hinge, how much force is needed?
Rearrange: F = T / d
F = 15 / 0.75
F = 20 N
Example 3 (Imperial Units)
A bolt specification requires 75 ft-lb of torque. If your wrench is 1.5 feet long, what force must you apply?
Rearrange: F = T / d = 75 / 1.5
F = 50 pounds of force
When to Use It
Use the torque formula whenever you need to calculate rotational force.
- Tightening bolts and nuts to specification (automotive, construction)
- Designing levers, gears, and mechanical systems
- Calculating the force needed to open doors, turn valves, or operate tools
- Comparing the effectiveness of different lever arm lengths
- Understanding why longer wrenches make bolts easier to turn
Important Notes
The formula T = F x d assumes the force is applied perpendicular (at 90 degrees) to the lever arm. If the force is at an angle, use the full formula: T = F x d x sin(theta), where theta is the angle between the force direction and the lever arm.
Unit conversions: 1 N·m = 0.7376 ft·lb. 1 ft·lb = 1.3558 N·m.