Rotational Work & Power (AQA A Level Physics): Revision Note

Work Done & Torque

Work Done by a Rotating Object

• Work has to be done on a rigid body when a torque turns in through an angle about an axis

  • For example, rotating cranes and fairground rides

• In systems with linear acceleration, work W is the product of the force and the distance moved

• Therefore, the work done for a rotating object is defined by the equation

$W=\tau \theta$

• Where:

  • W = work done (J)

  • $\tau$ = torque (N m)

  • θ = angular displacement (the angle turned through by the rotating object) (rads)

• Work can also be calculated by finding the area under a torque-angular displacement graph

Torque-angular displacement graph

The work done is the area under the torque-angular displacement graph

• This is analogous to the work done being the area under a force-displacement graph

Power Output of a Rotating Object

• Power is the rate of doing work, and is defined by

$P=\frac{\mathrm{\Delta }W}{\mathrm{\Delta }t}=\frac{\mathrm{\Delta \tau \theta }}{\mathrm{\Delta }t}=\tau \frac{\mathrm{\Delta \theta }}{\mathrm{\Delta }t}$

$P=\tau \omega$

• Where:

  • P = power (W)

  • ω = angular velocity (rad s^–1)

• This equation is the angular version of the linear equation P = Fv

Frictional Torque

• In rotational mechanics, frictional forces produce a specific torque called frictional torque

  • This is the torque caused by the frictional force when two objects in contact move past each other

• Frictional torque can be defined as:

The difference between the applied torque and the resulting net, or observed, torque

• This means that the net torque is the difference between the applied and frictional torque

Net torque = applied torque – frictional torque

• In rotating machinery, power has to be expended to overcome frictional torque

  • This is due to resistive forces within the machinery

• In most cases, frictional torque is minimised to reduce the kinetic energy losses transferred to heat and sound

• The frictional force must always be subtracted from the torque resulting from an applied force to get the total or net torque in the system 

• Frictional torque is calculated using the same equations as torque

$\tau =Fr=I\alpha$

• The only difference is F is the frictional force instead of an externally applied force