Torque is the rotational analog of force. Just as a net force causes linear acceleration, a net torque causes angular acceleration. But torque is not just about how large the force is — it depends on where the force is applied and at what angle.
The classic example: a door. Push near the hinge and the door barely moves. Push at the outer edge with the same force and it swings easily. Push at the outer edge but parallel to the door (toward the hinge) and again nothing happens. Three different outcomes from the same force magnitude — because r and theta both change.
The magnitude of torque is:
Where r is the distance from the axis of rotation to the point where the force is applied, F is the force magnitude, and theta is the angle between the position vector r and the force vector F.
The sin(theta) factor extracts the component of F that is perpendicular to r — the only component that produces rotation. The component of F parallel to r (F cos theta) points directly toward or away from the axis and produces no torque.
Drag the sliders to see how r, F, and theta affect torque. Watch the perpendicular component (F_perp) and lever arm update live. Drag theta to 90° for maximum torque; to 0° for zero.
At theta=90° torque is maximum — F is fully perpendicular to r. At theta=0° or 180° torque is zero — F points along r, no rotation results.
A 0.25 m wrench is used to tighten a bolt. A 80 N force is applied at the end of the wrench at 40° from the wrench handle. Calculate the torque.
The lever arm (also called the moment arm) is the perpendicular distance from the axis of rotation to the line of action of the force — the line through which the force acts, extended in both directions.
This is mathematically identical to tau = rF sin(theta). The lever arm d_perp = r sin(theta). Substituting gives tau = F * (r sin theta) = rF sin theta. The two forms are the same equation viewed geometrically in two different ways.
A 400 N child sits 1.5 m from the pivot of a seesaw. The weight force acts straight downward. What is the torque about the pivot?
Torques have direction — they tend to produce either CCW or CW rotation. By convention:
When multiple torques act on a rigid system, the net torqueis their algebraic sum. This is what drives angular acceleration in 5.4: tau_net = I * alpha.
A 2 m horizontal rod is pivoted at its left end. Force F1 = 30 N acts upward at the right end. Force F2 = 20 N acts downward at 0.8 m from the left end. Find the net torque about the pivot.
Torque force diagrams are like free-body diagrams, but with one critical additional requirement: forces must be drawn at their exact points of application, not at the center of mass. The location of the force on the object is what determines r — and r is what determines torque.