page_title: Work (physics)
section_title: Torque and rotation
hierarchical_section_title: Work (physics) / Mathematical calculation / Torque and rotation
context_page_description: In physics, work is the process of energy transfer to the motion of an object via application of a force, often represented as the product of force and displacement. A force is said to do positive work if the force has a component in the direction of the displacement of the point of application. A force does negative work if the force has a component opposite to the direction of the displacement at the point of application of the force.
  For example, when a ball is held above the ground and then dropped, the work done by the gravitational force on the ball as it falls is equal to the weight of the ball multiplied by the distance to the ground. When the force F is constant and the angle between the force and the displacement s is θ, then the work done is given by W = Fs cosθ.
  Work transfers energy from one place to another, or one form to another.
  The SI unit of work is the joule.
context_section_description: A force couple results from equal and opposite forces, acting on two different points of a rigid body.  The sum (resultant) of these forces may cancel, but their effect on the body is the couple or torque T.  The work of the torque is calculated as
  where the T ⋅ ω is the power over the instant δt.  The sum of these small amounts of work over the trajectory of the rigid body yields the work,
  This integral is computed along the trajectory of the rigid body with an angular velocity ω that varies with time, and is therefore said to be path dependent.
  If the angular velocity vector maintains a constant direction, then it takes the form,
  where φ is the angle of rotation about the constant unit vector S.  In this case, the work of the torque becomes,
  where C is the trajectory from φ(t₁) to φ(t₂).  This integral depends on the rotational trajectory φ(t), and is therefore path-dependent.
  If the torque T is aligned with the angular velocity vector so that,
  and both the torque and angular velocity are constant, then the work takes the form,
  This result can be understood more simply by considering the torque as arising from a force of constant magnitude F, being applied perpendicularly to a lever arm at a distance r, as shown in the figure.  This force will act through the distance along the circular arc s = rφ, so the work done is
  Introduce the torque τ = Fr, to obtain
  as presented above.
  Notice that only the component of torque in the direction of the angular velocity vector contributes to the work.
caption: A force of constant magnitude and perpendicular to the lever arm