dynamometer, and show it as a torque curve. Steam engines and electric motors tend to produce maximum torque close to zero rpm, with the torque diminishing as rotational speed rises (due to increasing friction and other constraints). Reciprocating steam-engines and electric motors can start heavy loads from zero rpm without a clutch. Relationship between torque, power, and energy If a force is allowed to act through a distance, it is doing mechanical work. Similarly, if torque is allowed to act through an angular displacement, it is doing work. Mathematically, for rotation about a fixed axis through the center of mass, the work W can be expressed as where τ is torque, and θ1 and θ2 represent (respectively) the initial and final angular positions of the body. Proof The work done by a variable force acting over a finite linear displacement is given by integrating the force with respect to an elemental linear displacement However, the infinitesimal linear displacement is related to a corresponding angular displacement and the radius vector as Substitution in the above expression for work gives The expression is a scalar triple product given by . An alternate expression for the same scalar triple product is But as per the definition of torque, Corresponding substitution in the expression of work gives, Since the parameter of integration has been changed from linear displacement to angular displacement, the limits of the integration also change correspondingly, giving If the torque and the angular displacement are in the same direction, then the scalar product reduces to a product of magnitudes; i.e., giving It follows from the work–energy principle that W also represents the change in the rotational kinetic energy Er of the body, given by where I is the moment of inertia of the body and ω is its angular speed. Power is the work per unit time, given by where P is power, τ is torque, ω is the angular velocity, and represents the scalar product. Algebraically, the equation may be rearranged to compute torque for a given angular speed and power output. Note that the power injected by the torque depends only on the instantaneous angular speed – not on whether the angular speed increases, decreases, or remains constant while the torque is being applied (this is equivalent to the linear case where the power injected by a force depends only on the instantaneous speed – not on the resulting