Angular Work and Power

Formula derivation for angular work

For linear motion, the work done is equal to the product of the force acting on a body and the linear displacement. For angular motion, we convert the linear displacement into angular displacement, using the arc length and radius relation. Hence, the angular work done is equal to the force ‘F’ multiplied by the angular displacement ‘\(\theta\cdot r\)’, with \(\theta\in [0,2\pi]\) as shown below: \(W_{linear} = Fx; \quad x = \theta \cdot r\)

\(W_{ang} = F(\theta \cdot r)\)

However, for circular motion, it is also known that torque ‘T’ is equal to a force acting on the body multiplied by its radius ‘F•r’. By replacing the torque relation with the angular work that was derived, we conclude that angular work and linear work equations have the same form. Furthermore, the torque is the reciprocal of force in linear work, and angular displacement is the reciprocal of linear displacement.

Angular work, just as linear work, is expressed in the unit of Joules.

\(T = F \cdot r\)

\(W_{ang} [J] = T \cdot \Delta \theta\)

Angular Work and Power: Torque direction

The direction of torque can be found by using the right-hand rule, where we point our fingers towards the direction of rotation. We then extend our thumb upwards. By pointing the fingers in the direction of rotation, the thumb points to a direction perpendicular to the direction of rotation. The direction of the torque is found by the direction of the thumb (see figure 1).

In vertical rotation, the following applies:

• If the direction of rotation is clockwise, the torque is downwards.
• If the direction of rotation is anticlockwise, the torque is upwards.

For non-vertical rotation, the direction of torque can be found using the right-hand rule. This states that four fingers follow the direction of rotation while the thumb is extended. The direction of the thumb shows the direction of the torque being either upwards or downwards. An example is shown in figure 2 below.

Angular Work and Power: Work-energy theorem

The work-energy theorem states that the total work done from external forces on a body is equal to the change of kinetic energy.

However, the change in kinetic energy must also include the translational kinetic energy ‘TKe’ and the rotational kinetic energy ‘RKe’. In the formula below, W is work, and ΔKe is the change in kinetic energy.

\(W = \Delta Ke = TKe - RKe\)

where \(RKe = \frac{1}{2} \cdot I \cdot \omega^2 [J]\)

Angular power

Linear power is the work done over time. Using the previous relations, we know that the work done for linear motion is equal to the product of the force and its linear displacement. Displacement over time equals velocity.

Formula derivation for angular power

For angular motion, we have proved earlier that the torque is the reciprocal of the force in linear motion, while angular velocity is the reciprocal of linear velocity.

Substituting the angular velocity into the equation, we get an equation in terms of torque in which torque, measured in Newtons per metre, is equal to the moment of inertia multiplied by the angular acceleration.

\[P[W] = \frac{w}{t}\]

We then modify the earlier equation for work, which is equal to the product of force and linear displacement. We substitute the force with torque over time, as by definition, the torque is equal to the product of the force and the radius, as seen below.

\[W = F \cdot x \space T = F \cdot r \Rightarrow F = \frac{T}{r} W_{ang} = \Big( \frac{T}{r} \Big) \cdot x\]

Here, x is the linear displacement, which is equal to linear speed multiplied by time. Since we are dealing with angular rotation, the linear speed v has to be replaced with its angular equivalent, which is utilised below where ω is the angular speed.

\[V = r \cdot \omega \cdot t\]

This will be substituted into the linear displacement term, which gives us the expression below.

\[W_{ang} = \Big( \frac{T}{r} \Big) \cdot x \qquad W_{ang} = \Big( \frac{T}{r} \Big) \cdot r \cdot \omega \cdot t\]

We can now substitute this derived work equation for angular rotation into the power equation.

\[P[W] = \frac{W_{ang}}{t} = \frac{\frac{T}{r} \cdot r \cdot \omega \cdot t}{t} = T \cdot \omega\]

At this point, the radius term and time are cancelled out, so we are left with a simpler expression where power is the product of torque T, and angular velocity ω is measured in rad/s².