Calculate the torque (rotational force) produced by a force applied at a distance from a pivot point, using τ = F × r × sin(θ).
📂 PhysicsFill in the fields on the left with your circuit's known values, then press the button to see your result instantly. No sign-up required, and no data is sent anywhere — everything is calculated right in your browser.
Torque (or moment of force) is a physical measure of a force's ability to rotate an object around a specific pivot point, calculated by the equation τ = F × r × sin(θ), where F is the magnitude of the applied force, r is the length of the lever arm (the distance from the pivot point to the point where the force is applied), and θ is the angle between the direction of the force and the direction of the lever arm. Torque's SI unit is the newton-meter (N·m). Torque reaches its maximum value when the force is exactly perpendicular to the lever arm (θ=90°, making sin(θ)=1), which explains why it's easier to loosen a nut with a wrench when pushing at a right angle to the handle rather than pushing along its length. The concept of torque is used in designing engines and hand tools (such as screwdrivers and wrenches), analyzing the balance of rotating objects, and calculating lever forces in mechanical engineering, where maximizing rotational effectiveness for a given applied force is a frequent design goal.
Torque's dependence on the angle between force and lever arm is one of those physics relationships that most people already understand intuitively from experience, even without knowing the formal equation behind it — anyone who has tried to loosen a stubborn bolt already knows that pushing straight along a wrench handle accomplishes almost nothing, while pushing perpendicular to it turns the bolt easily.
The sine term in the torque equation, τ = F × r × sin(θ), mathematically captures exactly this intuition: sin(90°) equals exactly 1, giving maximum torque when force is applied perpendicular to the lever arm, while sin(0°) equals exactly 0, meaning a force applied directly along the lever arm's length produces literally zero torque, no matter how large that force is.
This explains why wrench and tool design consistently positions the handle perpendicular to the fastener being turned — that geometry ensures the operator's applied force naturally lands close to the perpendicular angle that produces maximum torque for a given hand-applied force, rather than wasting effort at an inefficient angle that produces far less rotational effect for the same physical exertion.
The lever arm length r matters just as directly as the angle: doubling the distance from the pivot point to where force is applied exactly doubles the resulting torque for the same applied force, which is exactly why a longer wrench (or a length of pipe slipped over a wrench handle as a makeshift extension, sometimes called a 'cheater bar') makes it easier to loosen an extremely stubborn bolt — the same hand force applied farther from the pivot generates proportionally more torque.
This length-versus-force trade-off is a direct application of the broader lever principle in mechanical engineering: a longer lever arm lets a smaller applied force achieve the same torque as a larger force on a shorter arm, which is exactly the working principle behind everything from a simple crowbar prying up a heavy object to sophisticated torque wrenches used in precision mechanical assembly, where controlling the exact torque applied to a fastener (not just tightening it as much as possible) is critical to a joint's proper function and long-term reliability.
Only the component of force that's perpendicular to the lever arm contributes to rotation; a force applied parallel to the lever arm (0° or 180°) produces zero torque since it can't cause any turning effect.
Using a wrench to tighten a bolt: the longer the wrench handle (larger r) or the harder you push perpendicular to it (larger F), the greater the torque applied to the bolt.
Torque is measured in newton-metres (N·m), the same dimensional unit as the joule used for work and energy — but torque and work are conceptually different quantities (rotational force vs. energy transferred) and aren't interchangeable.