11.1.8 Rotational Work & Power

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

• Where:
  • W = work done (J)
  •  = 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

• Where:
  • P = power (W)
  • ω = angular velocity (rad s^–1)
• This equation is the angular version of the linear equation P = Fv

• 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 sum of 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
• However, sometimes a frictional torque is applied, such as when using a screwdriver
  • When a screwdriver is gripped and turned, this increases its rotational kinetic energy
• The frictional force must always be added to the torque resulting from a force to get the total torque in the system 
• Frictional torque is calculated using the same equations as torque

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