Equilibrium and angular acceleration

Rotational equilibrium

A body is in rotational equilibrium if the sum of the anticlockwise torques is equal to the sum of the clockwise torques about a pivot:

\(\sum\Gamma_\textrm{anticlockwise}= \sum\Gamma_\textrm{clockwise}\)

A body is in translational equilibrium if no resultant force acts:

\(\sum F_\textrm{upward}=\sum F_\textrm{downward}\)

\(\sum F_\textrm{left}=\sum F_\textrm{right}\)

We can use these rules in combination to solve problems about objects on a pivot.

In this example, body is in rotational and translational equilibrium if the following apply:

  • \(Aa=Bb\) (using torques)
  • \(P=A+B\) (considering vertical forces)

Note that the beam has no mass. If the beam has mass, it the force \(mg\) acts vertically downwards in the centre.


Angular acceleration

Like torque for force and moment of inertia for mass, there is also a rotational equivalent for acceleration, angular acceleration.

Angular acceleration is defined as the rate of change of angular velocity:

\(a={\Delta \omega\over \Delta t}={{\omega _f-\omega_i}\over t}\)

  • \(\alpha\) is angular acceleration (rads-2)
  • \(\omega_f\) is final angular velocity (rads-1)
  • \(\omega_i\) is initial angular velocity (rads-1)
  • \(t\) is the time taken (s)

We can convert between tangential and angular acceleration:

\(a=r\alpha\)

  • \(a\) is tangential acceleration (ms-2)
  • \(r\) is the radius of the circle (m)
  • \(\alpha\) is angular acceleration (rads-2)

Recall from Oscillations and Circular motion, there are two methods for finding angular velocity:

  1. Angular velocity is the instantaneous rate of change of anglular displacement (rad): \(\omega={\Delta \theta \over \Delta t}\)
  2. For repeated circular motion, angular velocity is the total angle moved through for a complete circle (\(2\pi\) rad) divided by the time period (s): \(\omega = {2\pi \over T}=2\pi f\), where \(f\) is the frequency (Hz).

Note the three forms of acceleration:

  • Angular acceleration is the rate of change of angular velocity, or the rate of change in the angle subtended.
  • Angular acceleration is is proportional to the tangential acceleration, the rate of change of translational velocity. This requires a resultant force acting along a tangent to the circle.
  • These are not related to centripetal acceleration. Centripetal acceleration is towards the centre of the circle and thus requires a resultant force along a radius to the circle, perpendicular to the tangent.