Think about opening a door. You push near the handle, not near the hinges. Why? Because the same push is far more effective when applied farther from the hinge. That "turning effectiveness" of a force is exactly what torque measures.
If you push a door with 10 N of force right at the hinge, the door barely moves. Push with the same 10 N at the handle, and it swings open easily. The force is identical — what changed is the distance from the axis of rotation. Torque combines force and distance into one quantity that tells you how much a force will make something spin.
Note
Torque is to rotation what force is to translation. Force makes things move in a straight line; torque makes things rotate.
The Precise Definition
Torque τ about a point (or axis) is defined as the cross product of the position vector r (from the axis to the point where force acts) and the force vector F:
τ=r×F
The magnitude is:
τ=rFsinθ
where θ is the angle between r and F.
The direction of torque is given by the right-hand rule: curl your fingers from r toward F, and your thumb points along τ. This direction is perpendicular to both r and F.
Tip
For maximum torque, push perpendicular to the lever arm (θ=90∘, so sinθ=1). Pushing directly toward or away from the axis (θ=0∘ or 180∘) produces zero torque — that's why you can't open a door by pushing straight into it.
Why Torque Equals Rate of Change of Angular Momentum
This is the rotational analogue of Newton's second law: F=dp/dt.
For a particle of mass m at position r with velocity v, its linear momentum is p=mv. Angular momentum about the same point is:
L=r×p
Now differentiate with respect to time:
dtdL=dtdr×p+r×dtdp
Since dr/dt=v and p=mv, the first term is v×mv=0 (cross product of parallel vectors). The second term uses dp/dt=F, giving:
Torque is the turning effect of a force about a pivot or axis, and everyday examples usually involve applying a force at some distance from a hinge or pivot point to produce rotation. …
Torque (turning effect of a force) shows up whenever a force is applied at a distance from a pivot to produce rotation -- opening a door and turning a nut with a wrench are two everyday examples.
Torque is defined as tau = r x F, the cross product of the position vector (from the pivot/axis to the point of application) and the applied force; its magnitude is tau = r F sin(theta), and it produces angular acceleration (rotation) about the pivot.
Example 1 -- Opening/closing a door: The door rotates about the hinges. Pushing near the hinge requires a large force to open it, while pushing at the handle (far from the hinge, maximizing r) opens it easily with a small force, because torque = force x distance from the hinge.