You already know a force can transfer energy into or out of an object if it's exerted over a displacement — that's the definition of work. A torque works the same way, just one dimension over: if a torque is exerted on a rigid system while that system undergoes an angular displacement, energy is transferred into or out of the system.
🔑No angular displacement, no energy transfer — no matter how large the torque. A torque holding a system motionless (like your hand steadily gripping a stuck bolt) does zero work, exactly the way a force holding a box against a wall does zero work if the box never moves.
The direction matters too. A torque that acts in the same rotational sense as the system's rotation adds energy — positive work. A torque that acts opposite to the rotation removes energy — negative work. This is the exact rotational analog of a force doing positive or negative work depending on its angle relative to the displacement.
6.2.A.2Math
Computing Work Done by a Torque
The amount of work a torque does is related to its magnitude and the angular displacement through which the system rotates while that torque is exerted. In full generality:
W = ∫θ₁θ² τ dθ
💡That integral looks intimidating, but it collapses to something simple whenever the torque stays constant throughout the rotation — which is the case you'll use most often:
W = τΔθ
Use the tool below to build intuition for the constant-torque case, including how the sign of the torque relative to the rotation flips the sign of the work.
Set a constant torque and the angular displacement it acts through, then toggle whether the torque helps or opposes the rotation.
τ (N·m)8
Δθ (rad)3
W = (8)(3) = 24.0 J
ExampleGuided Example — Work Done Loosening a Bolt
A mechanic applies a constant torque of 15 N·m to a wrench, rotating a stuck bolt through 0.5 rad before it breaks free. How much work does the mechanic's torque do on the bolt?
Step 1 — Identify the given quantities
τ = 15 N·m (constant) · Δθ = 0.5 rad · the torque acts in the same direction the bolt turns.
6.2.A.3Concept⚠ Watch Out
Reading Work from a Torque-vs-Angle Graph
Just like the area under a force-vs-position graph gives work in the linear world, the area under a torque-vs-angular-position graph gives the work done on a rotating system — and this method works whether the torque is constant or changing throughout the motion.
⚠️This area is signed, not just a magnitude. If the graph dips below the θ-axis — meaning the torque points opposite to the direction of rotation over that interval — that portion of the area subtracts from the total work rather than adding to it. Always check which side of the axis you're shading before you report an answer.
ExampleGuided Example — Work from a Triangular Graph
A torque increases linearly from 0 N·m to 20 N·m as a system rotates from θ = 0 to θ = 3 rad. How much work is done on the system over this interval?
Step 1 — Recognize the shape
A torque that increases linearly from 0 traces a straight line on a τ-vs-θ graph — the region under it, from θ = 0 to θ = 3 rad, is a triangle.
← Back to Lesson 6.2Ready for 6.3? Angular Momentum and Angular Impulse looks at what happens when a torque acts over time instead of angle.